Universal Prediction - Information Theory, IEEE ...
Embed Size (px)
Transcript of Universal Prediction - Information Theory, IEEE ...
2124 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
Universal PredictionNeri Merhav,Senior Member, IEEE, and Meir Feder,Senior Member, IEEE
Abstract—This paper consists of an overview on universalprediction from an information-theoretic perspective. Specialattention is given to the notion of probability assignment underthe self-information loss function, which is directly related tothe theory of universal data compression. Both the probabilisticsetting and the deterministic setting of the universal predictionproblem are described with emphasis on the analogy and thedifferences between results in the two settings.
Index Terms—Bayes envelope, entropy, finite-state machine,linear prediction, loss function, probability assignment,redundancy-capacity, stochastic complexity, universal coding,universal prediction.
CAN the future of a sequence be predicted based on itspast? If so, how good could this prediction be? These
questions are frequently encountered in many applications.Generally speaking, one may wonder why should the futurebe at all related to the past. Evidently, often there is sucha relation, and if it is known in advance, then it might beuseful for prediction. In reality, however, the knowledge ofthis relation or the underlying model is normally unavailable orinaccurate, and this calls for developing methods of universalprediction. Roughly speaking, a universal predictor is one thatdoes not depend on the unknown underlying model and yetperforms essentially as well as if the model were known inadvance.
This is a survey that describes some of the research work onuniversal prediction that has been carried out throughout theyears in several scientific disciplines such as information the-ory, statistics, machine learning, control theory, and operationsresearch. It should be emphasized, however, that there is no at-tempt to cover comprehensively the entire volume of work thathas been done in this problem area. Rather, the aim is to pointout a few of the highlights and the principal methodologiesfrom the authors’ personal information-theoretic perspective.Also, throughout the paper there are a few new results whosederivations are given in detail.
Manuscript received December 1, 1997; revised April 8, 1998. This workwas supported by the Israel Science Foundation administered by the IsraeliAcademy of Sciences and Humanities.
N. Merhav is with the Department of Electrical Engineering, Tech-nion–Israel Institute of Technology, Haifa 32000, Israel (e-mail: [email protected]).
M. Feder is with the Department of Electrical Engineering–Systems, TelAviv University, Tel Aviv 69978, Israel (e-mail: [email protected]).
Publisher Item Identifier S 0018-9448(98)05086-X.
Historically, the information-theoretic approach to predic-tion dates back to Shannon , who related prediction toentropy and proposed a predictive estimate of the entropyof the English language. Inspired by Haggelbarger, Shannon created later a “mind-reading” machine that predictshuman decisions. About that time, Kelly  showed theequivalence between gambling (which, in turn, is definitely aform of prediction) and information. Following Cover ,Rissanen , , and Rissanen and Langdon , it iswell recognized to date that universal prediction is intimatelyrelated to universal lossless source coding. In the last threedecades, starting from the pioneering work of Fittingoff and Davisson , and later Ziv , Lempel and Ziv, , , Rissanen and Langdon , Krichevskyand Trofimov , and others, the theory and practice ofuniversal coding have been greatly advanced. The state-of-the-art knowledge in this area is sufficiently mature to shedlight on the problem of universal prediction. Specifically,prediction schemes as well as fundamental performance lim-its (lower bounds), stemming from those of universal cod-ing, have been derived. It is the relation between universalcoding and universal prediction that is the main theme ofthis paper, from the point of view of both algorithms andperformance bounds.
Let us now describe the prediction problem in general.An observer sequentially receives a sequence of observations
over some alphabet . At each timeinstant , after having seen but notyet , the observer predicts the next outcome, or moregenerally, makes a decision based on the observed past
. Associated with this prediction or decision, andthe actual outcome , there is a loss function thatmeasures quality. Depending on the particular setting of theprediction problem, the objective would be to minimize thisinstantaneous loss, or its time-average, or the expected valueof either one of these quantities. Obviously, prediction in theordinary sense is a special case of this, where is anestimate of based on and is someestimation performance criterion, e.g., the Hamming distance(if is discrete) or the squared error(if is continuous).
Another special case, which is more general than the aboveexamples, is based on assigning weights or probabilities toall possible values of the next outcome. For example, theweatherman may assess 70% chance of rain tomorrow, in-stead of making a commitment whether it will rain or not.This is clearly more informative than the ordinary predictiondescribed above because it gives an assessment of the degree
0018–9448/98$10.00 1998 IEEE
MERHAV AND FEDER: UNIVERSAL PREDICTION 2125
of confidenceor reliability associated with the prediction. Interms of the above described prediction problem, hereisa conditional probability assignment of given , i.e., anonnegative function that integrates (or sums) tounity for every . Upon observing , the performanceof is assessed with respect to a suitable loss function, which should decrease monotonically with the probability
assigned to the actual outcome . A very importantloss function of this kind is theself-information lossfunction,which is also referred to as thelog–loss function in themachine-learning literature. For every probability assignment
over and every , this functionis defined as
where logarithms throughout this paper are taken to the baseunless otherwise specified. For reasons to be discussed in
Section II, the self-information loss function plays a centralrole in the literature on prediction and hence also throughoutthis survey.
Let us now return to the prediction problem in its generalform. Quite clearly, solutions to this problem are soughtaccording to the particular assumptions on the data-generatingmechanism and on the exact objectives. Classical statisticaldecision theory (see, e.g., ) assumes that a known prob-abilistic source generates the data, and so, a reasonableobjective is to minimize the expected loss. The optimumstrategy then minimizes the expected loss, given the past,i.e.,
where random variables are denoted by capital letters. More-over, under suitable assumptions on stationarity and ergod-icity, optimum prediction in the expected loss sense,is optimum also in the sense of minimizing the almost sureasymptotic time-average of (see, e.g., ). Given
, the quantity
is referred to as the conditionalBayes envelopegiven .For example, if is a binary source, , andis the Hamming distance, then
and the conditional Bayes envelope given is
If, in addition, the underlying source is known to beGaussian (or, if only the class of linear predictors is allowed),
then is well-known to be a linear function of given asa special case of the causal Wiener filter  (see also [86,Ch. 14-3]). In the self-information loss case,
minimizes , namely,the best probability assignment is the true one. The conditionalBayes envelope given , is the (differential) entropy ofgiven , i.e.,
While classical theory (e.g., Wiener prediction theory) as-sumes that the source is known, the more realistic andinteresting situation occurs when is either unknown, ornonexistent. In the second case, there is no probabilistic data-generating mechanism and the data are considered arbitraryand deterministic. Both cases fall into the category of theuniversal prediction problem, where the former is referred to astheprobabilistic settingand the latter is called thedeterministicsetting.Let us now elaborate on these two settings.
A. The Probabilistic Setting
In the probabilistic setting the objective is normally tominimize the expected cumulative loss asymptotically for large
simultaneously for any source in a certain class. A universalpredictor does not depend on, and at the sametime, keeps the difference between
vanishingly small for large . The cumulative Bayes envelopeof (4) represents the performance of the optimal predictortuned to . For a stationary and ergodic source, the sequence
has a limit , referred to as theasymptoticBayes envelope, that coincides (by the Cesaro theorem )with , which in turn exists by nonincreas-ing monotonicity. In the self-information loss case, isthe entropy rate of , which means that the goal of universalprediction is equivalent to that of universal coding.
There are essentially three levels of universality accordingto the degree of uncertainty regarding the source.
Universality with Respect to Indexed Classes of Sources:Suppose that the source is unknown except for being amember of a certain indexed class , whereis the index set. Most commonly, designates a parametervector of a smooth parametric family, e.g., the families offinite-alphabet memoryless sources,th-order Markov sources,
-state sources, AR Gaussian sources, but other index sets(e.g., finite sets) are possible as well. There are two interestingissues here. The first is to devise universal prediction schemesthat asymptotically attain in the above defined sense
2126 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
for every , and the second is performance bounds beyondthat apply to any universal predictor. Analogously
to the universal coding terminology, the extra loss beyondwill be referred to as theredundancy.Redundancy
bounds are useful to establish necessary conditions for theexistence of universal schemes as well as limitations on therate of convergence. Both are dictated by a certain measureof the richness of the class . Furthermore, even ifthe redundancy bound does not vanish as , andhence universal schemes in the above defined sense do notexist, the question of universality can be extended to thatof achieving this bound. For self-information loss prediction,we will explicitly characterize such bounds, and demonstrateachievability by certain universal schemes.
Universality with Respect to Very Large Classes of Sources:Suppose that all we know about the source is that it isMarkov of an unknown finite order, or that it is stationary andergodic, or mixing in a certain sense. For such large classes,quantitative characterizations of uniform redundancy rates donot exist , , . Here, one cannot hope for morethanweak universality, a term mentioned and defined in ,which means that universality is attained at a nonuniformconvergence rate. Sometimes even weak universality cannotbe obtained, and in  there are necessary and sufficientconditions for the existence of universal schemes.
Hierarchical Universality: In this level, the goal is to de-vise universal schemes with respect to a sequenceof index sets of sources, which may (though not necessarily)have some structure like nesting, i.e., for everypositive integer . Perhaps the most common example is wherefor every , is the class of all th-order Markov sources ofa given alphabet. Here the only prior knowledge that one mayhave on the source is that its indexbelongs to .The straightforward approach would be to consideras onebig class and to seek universal schemes with respect to. Thedrawback of this approach, however, is that it is pessimisticin the sense that the convergence rate towards , mightbe very slow, if at all existent, because could be a veryrich class. In the above Markov example, while eachfallswithin the category of the first level above, the unionfallsin the second level. Nonetheless, it turns out that in certainsituations it is possible to achieve redundancy rate that isessentially as small as if were knowna priori. This givesrise to an elegant compromise between the two former levelsof universality. It keeps the fast convergence rates of the firstlevel without sacrificing the generality of the class of sourcesof the second level.
B. The Deterministic Setting
In this setting, the observed sequence is not assumed tobe randomly drawn by some probability law, but is rather anindividual, deterministic sequence. There are two difficultiesin defining the universal prediction problem in this context.The first is associated with setting the desired goal. Formally,for a given sequence there is always the perfectprediction function defined as , and so, theprediction problem seemingly boils down to triviality. Thesecond difficulty is in the other way around. For a given
deterministic predictor , there is always the adver-sary sequence where at each time instant, is chosen tomaximize .
The first difficulty is fundamental because it means thatwithout any limitations on the class of allowed predictors,there is a severe overfitting effect, which tailors a predictor tothe sequence so strongly, that it becomes, in fact, anticipatingand hence completely misses the essence of prediction as acausal, sequential mechanism. Therefore, one must limit theclass of allowed predictors in some reasonableway. For example, could be the class of predictors thatare implementable by finite-state machines (FSM’s) withstates, or Markov-structured predictors of the form
, and so on. Such limitations make sense notonly by virtue of avoiding these trivialities, but also becausethey reflect real-life situations of limited resources, like mem-ory, computational power, and so on. Stated more formally, fora given class of predictors, we seek a sequential predictor
that is universal in the sense of being independent ofthe future, and at the same time, its average loss
is asymptotically the same as
for every . The universal predictor need not be necessarilyin but it must be causal, whereas the reference predictorin , that minimizes the average loss, may (by definition)depend on the entire sequence.
The second difficulty mentioned above is alleviated byallowing randomization. In other words, predictions are gener-ated at random according to a certain probability distributionthat depends on the past. Note that this is different from theabove discussed case wherewas a probability assignment,because now the assigned probability distribution is actuallyused for randomization.
Analogously to the probabilistic case, here we also dis-tinguish between three levels of universality, which are nowin accordance to the richness of the class. The first levelcorresponds to an indexed class of predictors which is dualto the above mentioned indexed class of sources. Examplesof this are parametric classes of predictors, like finite-statemachines with a given number of states, fixed-order Markovpredictors, predictors based on neural nets with a given numberof neurons, finite sets of predictors, etc. The second levelcorresponds to very large classes like the class of all finite-state predictors (without specifying the number of states),operating on infinitely long sequences, etc. Finally, the thirdlevel corresponds to hierarchical universality and parallels thatof the probabilistic setting. The nature of the reported resultsis somewhat similar to that of the probabilistic approach, butthere are several important differences in algorithmic aspectsas well as in existence theorems and performance bounds.
The outline of the paper is as follows. Section II is de-voted to the motivation and the justification for the use of
MERHAV AND FEDER: UNIVERSAL PREDICTION 2127
the self-information loss function as a performance criterionin prediction. In Section III, the probabilistic setting willbe discussed with a great emphasis on the self-informationloss case which is fairly well-understood. In Section IV, thedeterministic setting will be described with special attention tothe similarity and the difference from the probabilistic setting.Section V is devoted to the concept of hierarchical universalityin both settings. Finally, Section VI summarizes the paperalong with some open problems and directions for furtherresearch.
II. THE SELF-INFORMATION LOSS FUNCTION
We mentioned earlier the self-information loss functionand its central role in universal prediction. In this section,we discuss some motivations and justifications for using thisloss function as a measure of prediction performance. Asexplained in Section I, predictive probability assignment forthe next outcome is more general and more informative thanestimating the value of the next outcome, and a reasonable lossfunction should be monotonically decreasing with the assignedprobability of the actual outcome. The self-information lossfunction, defined in (1), clearly satisfies this requirement, butit also possesses many other desirable features of fundamentalimportance.
The first advantage of the self-information loss function istechnical. It is convenient to work with because the logarithmicfunction converts joint probability functions, or equivalently,products of conditional probabilities into cumulative sums ofloss terms. This suits the framework of the general predictionproblem described above.
But beyond this technical convenience, there is a deeper sig-nificance. As is well known, the self-information manifests thedegree of uncertainty, or the amount of information treasuredin the occurrence of an event. The conditional self-informationof the future given the past, therefore, reflects the ability todeduce information from the past into the future with minimumuncertainty.
Evidently, prediction under the self-information loss func-tion and lossless source coding are intimately related. Thisrelation stems from the fact that isthe ideal codelengthof with respect to a probability func-tion . This codelength can be implemented sequentiallywithin any desired precision using arithmetic coding .Conversely, any codelength function can be translated intoa probability assignment rule , , , . Anotherdirect application of self-information loss minimization to theproblem area of prediction, is that of gambling , , .In this case, represents the distribution of moneyinvested in each one of the possible values of the next outcome.The self-information loss function then dictates the exponentialgrowth rate of the amount of money with time.
The paradigm of predictive probability assignment is alsothe basis of Dawid’sprequential principle. However, themotivation of the prequential principle was not in predic-tion per se, but rather the use of probability assignment fortesting the validity of statistical models. A good probabilityassignment is one that behaves empirically as expected from
the true probabilistic model. For example, if are binary,then a good sequence of probabilities assignedto should satisfy
namely, the law of large numbers. As further discussed in–, other requirements are based on the central limittheorem, the law of iterated logarithm, behavior of confidenceintervals, and so on.
Interestingly, it turns out that predictive probability as-signment under the self-information loss criterion can beuseful also for the purpose of testing the validity of statis-tical models as described above. One reason is that whena certain source governs the data, then it is the trueconditional probability that mini-mizes . In simpler words, themaximum achievable assigned probability is also the trueone (a property shared by very specific loss functions, see). Moreover, by the Shannon–McMillan–Breiman theo-rem, under certain ergodicity assumptions, this is true notonly in the expected value sense, but also almost surely.Thus by combining the prequential principle with the Shan-non–McMillan–Breiman theorem, a good probabilistic modelfor the data must minimize
i.e., the average self-information loss.From another perspective, we observe that any sequential
probability assignment mechanism gives rise to a probabilityassignment for the entire observation vectorby
Conversely, any consistent probability assignmentfor(i.e., that satisfies for all and
), provides a valid sequential probability assignment by
Therefore, the choice of in self-information loss predic-tion is completely equivalent to the choice of that assignsmaximum probability to , that is, maximum-likelihoodestimation.
In our discussion thus far, we focused on motivating theself-information loss function itself. Yet another motivationfor studying universal prediction in the self-information losscase is that it sheds light on the universal prediction problemfor other loss functions as well. Perhaps the most direct way tolook at self-information loss prediction is as a mechanism thatgenerates a probability distribution when the underlying sourceis unknown or nonexistent. One plausible approach to the pre-diction problem with a general loss function is then to generate,at each time instant, a prediction that is a functional of theself-information-loss conditional probability assignment. For
2128 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
example, in the squared-error loss case, a reasonable predictorwould be the conditional mean associated with ,which hopefully tends to the true conditional probability asdiscussed above. As will be seen in the probabilistic setting,this technique is often successful, whereas in the deterministicsetting, some modification is required.
However, there is another way in which self-informationloss prediction serves as a yardstick to prediction under otherloss functions, and this is the notion ofexponential weighting.In certain situations, minimization of the cumulative loss
corresponds to maximization of the exponenti-ated loss , which in turn canbe treated altogether as an auxiliary probability assignment.In certain important special cases (though not always), thesolution to this probability assignment problem translates backas a solution to the original problem. We will also see theusefulness of the exponential weighting technique as a toolfor deriving lower bounds that are induced from correspondingstrong lower bounds of the self-information loss case.
III. T HE PROBABILISTIC SETTING
We begin with the problem of probability assignment forthe next outcome given the past, under the self-informationloss function. As explained above, this problem is completelyequivalent to that of finding a probability assignmentforthe entire data sequence.
As we mentioned earlier, if the sourcewere known, thenclearly, the optimal that minimizes the above expected self-information loss would be , i.e., the prediction inducedby the true underlying source
The average cumulative loss would then be the entropy. If is unknown and we wish
to assign a certain probability distribution that does notdepend upon the unknown, then the extra loss beyond theentropy is given by
where is the th-order information divergence (rel-ative entropy) between and . In the corresponding losslesscompression problem, is the coding redundancy,i.e., the normalized per-symbol difference between the averagecode length and the entropy. Of course, the minimizations of
for two or more sources at the same time mightbe contradictory. Thus the problem of universal probabilityassignment is that of finding a good compromisethat isuniformly as “close” as possible, in the information divergencesense, to every in a given class of sources. We shall elaboratelater on this notion of simultaneous divergence minimization.
As explained in Section I, the theory of universality splitsinto several levels according to the degree of uncertaintyregarding the source. We begin with the conceptually simplestcase where the source is known to belong to a given indexedclass of sources where is the index (e.g., aparameter vector) and is the index set. Since we look atprediction from the viewpoint of probability assignment and
we start from the self-information loss criterion, our survey inthis part is largely taken from the theory of universal coding.
A. Indexed Classes of Sources
1) The Self-Information Loss Function:We first describetwo common approaches to universal probability assignmentfor indexed classes of sources.
The Plug-in Approach versus the Mixture Approach:Onenatural approach to universal prediction with respect to anindexed class of sources is the so-calledplug-inapproach. According to this approach, at every time instant, the index (or the parameter) is estimated on-line from
(e.g., by using the maximum-likelihood estimator), andthe estimate is then used for prediction as if itwere the true parameter value, i.e., the conditional probabilityassigned to is given by .
The plug-in approach may work quite well under certainregularity conditions. Intuitively, if the estimator is statisti-cally consistent and is continuous in for every
and , then the estimated probability assignment mayconverge to the true conditional probability in the probabilisticsense. Nonetheless, this convergence property does not alwayshold (e.g., when is the center of a Cauchy density estimatedby the sample mean), and even if it does, the rate of conver-gence might be of crucial importance. Moreover, it is not true,in general, that better estimation of the conditional probabilitynecessarily yields better self-information loss performance.The plug-in approach is, in essence, a heuristic approachthat lacks a well-substantiated, deep theoretical justificationin general.
An alternative approach, henceforth referred to as themix-ture approach, is based on generating convex combinations(mixtures) of all sources in the class . Specifi-cally, given a certain nonnegative weight function thatintegrates to unity (and hence can be thought of as a prioron ), we define the mixture probability mass (or density)function over -tuples as
With an appropriate choice of the weight function, themixture , as we shall see later, turns out to possess certaindesirable properties which motivate its definition as auniversalprobability measure.This universal measure then induces aconceptually simple sequential probability assignment mech-anism defined by
It is interesting to note [72, Theorem 2] that the abovepredictive probability function induced by the mixture of
can also be represented as a mixture of the con-ditional probability functions , wherethe weighting function is given by theposterior probabilitydensity function of given , i.e.,
MERHAV AND FEDER: UNIVERSAL PREDICTION 2129
and where the last expression manifests the interpretation ofexponential weightingaccording to the probability assignmentperformance (given by ) on data seen thus far:points in that correspond to good performance in the past arerewarded exponentially higher weights in prediction of futureoutcomes. The exponential weighting is an important concept.We will further elaborate later on it in a broader context oflower bounds and algorithms for sequential prediction undermore general loss functions in the probabilistic as well as inthe deterministic setting.
For the class of binary memoryless (Bernoulli) sources with, the mixture approach, with being
uniform over , leads to the well-known Laplaceprediction , . Suppose that contains zeros and
which, in this case, can be thought of also as a plug-inalgorithm because can be interpreted as abiased version of the maximum-likelihood estimator of. Sucha bias is clearly desirable in a sequential regime because thenaive maximum-likelihood estimator wouldassign zero probability to the first occurrence of “” which,in turn, would result in infinite loss. Also, this bias gives riseto the plausible symmetry consideration that in the absenceof any data (i.e., ) one would assign equalprobabilities to “ ” and “ .” But this would be also the casewith any estimator of the form ,
. Indeed, other weight functions (from the Dirichletfamily) yield different bias terms and with slight differencesin performance (see also ). This discussion carries overto general finite-alphabet memoryless sources  (as will bediscussed later) and to Markov chains , . However, itshould be kept in mind that for a general family of sources
, the mixture approach does not necessarily boildown to a plug-in algorithm as above, and that the choice ofthe weight function might have a much more dramatic impacton performance . In this case, we would like to have sometheoretical guidance regarding the choice of.
This will be accomplished in the forthcoming subsection,where we establish the theoretical justification of the mixtureapproach in a fairly strong sense. Interestingly, in the nextsection, it will be motivated also in the deterministic setting,and for loss functions other than the self-information lossfunction.
Minimax and Maximin Universality:We have seen (6)that the excess loss associated with a given probability as-signment while the underlying source is is given by
. The first fundamental justification of the mixtureapproach (presented in ) is the following simple fact: givenan arbitrary probability assignment , there exists anotherprobability assignment in the convex hull of ,(that is, a mixture) such thatsimultaneously for every . This means that whenseeking a universal probability assignment, there is no lossof optimality in any reasonable sense, if we confine attentionmerely to the convex hull of the class . Nonetheless,this interesting fact does not tell us how to select the weightfunction of the mixture . To this end, we make a fewadditional observations.
As mentioned earlier, we wish to find a probability as-signment that is independent of the unknown, and yetguarantees a certain level of excess loss beyond the minimumachievable loss had been knowna priori (i.e., the th-orderentropy ). Referring again to (6), this suggests to solvethe following minimax problem:
The value of this quantity, after normalizing by, is called theminimax redundancyand is denoted by in the literature ofuniversal coding. At first glance, this approach might seemsomewhat pessimistic because it is a worst case approach.Fortunately enough, in many cases of interest, as
, which means that the minimax asymptoticallyachieves the entropy rate, uniformly rapidly in. Moreover,as we shall see shortly, the minimax approach, in the self-information loss case, is not at all pessimistic even ifdoes not tend to zero. Again, in view of the discussion in theprevious paragraph, the minimax-optimal is a mixture ofthe sources in the class.
An alternative to the minimax criterion is the maximincriterion, whose definition has a strong Bayesian flavor thatgives rise to the mixture approach from a seemingly differentpoint of view. Here is the idea: since is unknown,let us postulate some prior probability density functionover . The performance of a given probability assignmentwould be then judged with respect to the normalized weightedaverage redundancy , i.e.,
It is easy to see that for a given, the that minimizesis just the defined in (7), and that the resul-
tant average redundancy , is exactly the mutualinformation between random variables and
whose joint probability density function is given by. But is arbitrary and the question
that again arises is what would be an “appropriate” choice of? Let us adopt again a worst case approach and use the “least
favorable” prior that maximizes , that is, solvethe maximin problem
2130 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
whose value, when normalized by, is referred to as themaximin redundancyand denoted by . It is important tonote that , which is the supremum of overall allowable ’s, is given the interpretation of thecapacityofthe “channel” from to , defined by the class of sources.In this definition, each source is thought of as theconditional probability function of the “channel output” giventhe “channel input” . We will refer to this channel capacityas thecapacity of the classof sources and willdenote it by . Thus is identical to .
These notions of minimax and maximin universality werefirst defined by Davisson  in the context of universalcoding (see also , , , , , and others).Several years after Davisson’s paper  it was observed(first by Gallager , and then independently by Davissonand Leon-Garcia , Ryabko , and others) that theminimax and the maximin solutions are equivalent, i.e.,
. Furthermore, the mixture , where is thecapacity-achieving prior(i.e., ), is bothminimax and maximin optimal. This result is referred to asthe redundancy-capacity theoremof universal coding.
The capacity , therefore, measures the “richness” of theclass of sources. It should be pointed out, though, thatis not very sensitive to “distances” among the sources in theclass, but rather to the effective number of essentially distinctsources. For example, the source that generates’s onlywith probability one is at infinite divergence-distance from thesource that generates’s only. Yet their mixture(in the level of -tuples) is within normalized divergence of
from both, and so, the capacity of is verysmall. It is a remarkable fact that the theory of universalcoding is so intimately related to that of channel capacity.Moreover, the importance and significance of the redundancy-capacity theorem are fairly deep also in the broader context ofprobability assignment and prediction.
On the face of it, at this point the problem of universalprobability assignment, or equivalently, universal predictionunder the self-information loss function with respect to anindexed class of sources, is fairly well addressed. Nonetheless,there are still several important issues to be considered.
The first concern comes from a practical aspect. Explicitevaluation of the proposed minimax/maximin probability as-signment is not trivial. First of all, the capacity-achieving prior
is hard to evaluate in general. Furthermore, even when itcan be computed explicitly, the corresponding mixtureas well as the induced conditional probabilitiesmight still be hard to compute. This is in contrast to theplug-in approach, which is relatively easy to implement.Nevertheless, we shall return later to the earlier example ofthe mixtures of Bernoulli sources, or more generally, finite-alphabet memoryless sources, and see that fortunately enough,some satisfactory approximations are available.
The second technical point has to do with the evaluation ofcapacity, or at least, its asymptotic behavior, which is of crucialimportance. As mentioned earlier, the capacity measures the“complexity” or “richness” of the class of sources, and
if and only if uniform redundancy rates are achievable (i.e.,strong universality). This means that if the class of sources is
too rich so that does not vanish as grows without bound,one can no longer hope for uniformly small redundancy rates, . We shall see examples of this later.
Another problem that calls for attention is that the predictor,or the sequential probability assignment mechanism that weare proposing here, is not really sequential in the sense thatthe horizon must be prescribed in advance. The reasonis that the capacity-achieving prior depends on , ingeneral. A possible remedy (both to this and to the problemof computability) is to seek a fixed prior, independent of ,that achieves capacity at least asymptotically, i.e.,
Fortunately, this is possible in some important examples.Finally, we mentioned earlier that the minimax approach is
pessimistic in essence, a fact which seems to be of specialconcern when does not tend to zero as grows.The reason is that although for all ,minimaxity guarantees that the lower bound
is valid for one source in the class. The maximin pointof view tells us further that this holds true also in the senseof the weighted average of over with respect to
. Still, the optimality of is on seemingly somewhatweak grounds. Nonetheless, a closer inspection reveals thatthe right-hand side of (15) is essentially a lower bound in amuch stronger sense which will now be discussed.
A Strong Converse Theorem:It turns out that in the self-information loss case, there is a remarkable “concentration”phenomenon: It is shown in  that
for every and for -mostvalues of . Here, the term“ -most” means that the total probability mass of points withthis property, with respect to (or any asymptotically goodapproximation of ), tends to unity as . This meansthat if the right-hand side of (15) is slightly reduced, namely,multiplied by a factor , it becomes a lower bound for
-most values of . Referring again to the uniform upperbound, this means that -most sources in the class lie near thesurface of a “sphere” (in the divergence sense) of radius,centered at . Considering the fact that we have assumedvirtually nothing about the structure of the class of sources,this is quite a surprising phenomenon. The roots of this areexplained and discussed in detail in  and  in relationto the competitive optimality property of the self-informationfunction  (see also ).
There is a technical concern, however: for a class of finite-alphabet sources and any finite, the capacity-achieving priormust be discrete with support of at most points in[44, p. 96, Corollary 3]. Strictly speaking, the measurethen ignores all points outside its support, and the term “-most sources” is not very meaningful. Again, fortunatelyenough, in most of the important examples, one can finda smooth weight function , which is independent ofand asymptotically achieves capacity. This solves both this
MERHAV AND FEDER: UNIVERSAL PREDICTION 2131
difficulty and the horizon-dependency problem mentionedearlier. As an alternative remedy, there is another, moregeneral version of this strong converse result , whichallows for an arbitrary weight function . It tells that isoptimal for -most points in . But note that maydepend on for a general , and so the uniformity propertymight be lost.
The above result is, in fact, a stronger version of theredundancy-capacity theorem, as detailed in , and it gener-alizes the well-known strong converse to the universal codingtheorem due to Rissanen  for a smooth parametric family
whose capacity behaves like , whereis dimension of the parameter vector. Rissanen, in his award-
winning paper , was the first to show such a strongconverse theorem that applies to most sources at the sametime. The reader is referred to  (see also ) for detaileddiscussion on this theorem and its significance in general, aswell as in the perspective of Rissanen’s work in particular. Letus now examine a few examples in light of these findings.
Examples: Perhaps the simplest example is the onewhere , namely, there are sources
in the class, and the weight function isrepresented by a vector of nonnegativenumbers summing to one. In this case, the above described“concentration” phenomenon becomes even sharper [44,Theorem 4.5.1],  than in the general case because
for every for which . Inother words, -all sources lieexactly on the surface ofthe divergence sphere around . If the sources areeasily distinguishable in the sense that one can reliably identifywhich one of the sources generated a given vector, then theredundancy-capacity of the class is nearly , becausethe “channel input” can be “decoded” from the “channeloutput” with small error probability. In this case,tends to be uniform over and the best mixture
is essentially a uniform mixture. If the sources are noteasily distinguishable, then the redundancy-capacity is smaller.This can be thought of as a situation where the “channel” ismore “noisy,” or alternatively, that the effective number ofdistinct sources is smaller than. In the extreme case, where
, we have as expected, since wehave, in fact, only one source in the class.
Let us now revisit the Bernoulli example, or more generally,the class of memoryless sources with a given finite alphabetof size . This is obviously a parametric class whose naturalparameterization by is given by the letter probabilitieswith degrees of freedom. As mentioned earlier, isdiscrete in the finite-alphabet case, it depends on the horizon
, and it is difficult to compute. It turns out that for smoothparametric families with a bounded parameter set, like theone considered here, there is no much sensitivity to the exactshape of (used for ) as long as it is bounded awayfrom zero across . In fact, any such “nice” prior essentiallyachieves the leading term of the capacity, which is .Differences in performance for different choices of arereflected in higher order terms. Specifically, Clarke and Barron,  have derived a very accurate asymptotic formula for
the redundancy associated with a mixture
where is the determinant of the Fisher information ma-trix of (see also Takeuchi and Barron  for extensionsto more general exponential families). In the maximin setting,the weighted average of is then asymptoticallymaximized (neglecting the term) by a prior thatmaximizes the second term above, which is well known asJeffreys’ prior , , , 
In our case, is inversely proportional to the square rootof the product of all letter probabilities,
This, in turn, is a special case of the Dirichlet prior , whosegeneral form is proportional to the product of arbitrary fixedpowers of . Dirichlet mixtures and conditional proba-bilities derived from them have easy closed-form expressionsas well. Generalizing the earlier Bernoulli example to the size-
alphabet parametric family, and using Jeffreys’ prior, we getthe universal probability assignment
where is the number of occurrences of ,. The uniform prior that leads to the Laplace estimator
discussed earlier, is yet another special case of the Dirichletprior. It should be noted that Jeffreys’ prior asymptoticallyachieves capacity and so, it induces an asymptotically maximinprobability assignment. Interestingly, as observed in , it isnot asymptotically minimax, and it should be slightly modifiedto obtain minimax optimality. These results extend to moregeneral parametric families under certain regularity conditionsdetailed in the above cited papers.
But the main point to be remembered here is that forparametric classes, the choice ofis not crucial in terms ofperformance. This gives rise to the freedom of selecting a priorfrom implementational considerations, i.e., the availability ofclosed-form expressions for mixtures, namely, conjugate priors. We have just seen the example of the Dirichlet prior inclasses of memoryless sources. As another example, considerthe case where is a family of Gaussian memorylesssources with mean and variance . Clearly, with respectto a Gaussian prior is Gaussian itself in this case. The ideaof conjugate priors carries over in a natural manner to moregeneral exponential families.
It should be pointed out that there are other recent exten-sions , , , ,  of the redundancy-capacitytheory to more abstract classes of sources whose capacitiesare proportional to , where the number is attributed amore general notion of dimensionality that is induced by
2132 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
the Hellinger distance, the Kullback–Leibler distance, theVapnik–Chervonenkis (VC) dimension, etc. Other extensionsto wider classes of sources exhibit different behavior of theredundancy-capacity , . Still, the general underlyinginformation-theoretic principle remains the same; the richnessof the class is measured by its Shannon capacity. Other exam-ples of classes of sources that are not necessarily parametric,are given in  and .
2) General Loss Functions:It turns out that satisfactorysolutions to the universal prediction problem under the self-information loss function, may prove useful for more generalloss functions. Intuitively, under suitable continuity conditions,an optimal predictor with respect to, based on a goodestimator of , should be close to optimum underthe true conditional probability. Generally speaking, sinceminimum self-information loss probability assignments are es-sentially maximum-likelihood estimates (cf. Section II), whichare statistically consistent in most situations, this requirementis satisfied.
Specifically, in the discrete alphabet case, let denotethe underlying source and consider the universal probabilityassignment for which forall . Using Pinsker’s inequality (see, e.g., [24, Ch. 3,Problem 17]) and the concavity of the square root function,we have (20) shown at the bottom of this page. Now, for ageneral loss function, let
where denotes expectation with respect to, and
where denotes expectation with respect to. Assume thatis nonnegative and bounded by some constant . Then,
by the inequality above, we get
In words, the optimum predictor with respect to the universalprobability assignment is within close tooptimum simultaneously for every . The importantconclusion from this result is the following:The existence ofuniversal predictors with uniformly rapidly decaying redun-dancy rates under the self-information criterion, is a sufficientcondition for the existence of such predictors for general lossfunctions.
At this point, two comments are in order: first, the aboveassumption on boundedness ofcan be weakened. For ex-ample, the leftmost side of (23), which can be thought ofas a generalized divergence between and , canoften be upper-bounded in terms of the variational distancebetween and . We have adopted, however, the bound-edness assumption to simplify the exposition. The secondcomment is that the upper bound of (23) might not be tightsince the true redundancy rate could be faster in certainsituations. For example, minimum mean-square error, fixed-order, universal linear predictors ,  have redundancyrates as small as , whereas the above upper boundgives . The question that arises now is whetherwe can provide a more precise characterization of achievableredundancy rates (tight upper and lower bounds) with respectto general loss functions.
A natural way to handle this question is to take the mini-max–maximin approach similarly to the self-information loss
MERHAV AND FEDER: UNIVERSAL PREDICTION 2133
case. The minimax predictor is the one that minimizes
Unfortunately, there is no known closed-form expression forthe minimax predictor for a general loss function. Nonetheless,game-theoretic arguments tell us that sometimes the minimaxproblem is equivalent to the maximin problem. Analogously tothe self-information loss case, the maximin problem is definedas the supremum of
over all nonnegative weight functions that integrate tounity. In general, the minimax and maximin problems are wellknown to be equivalent for convex–concave cost functions. In our case, since (25) is always affine and hence concavein , the remaining condition is that the set of allowablepredictors is convex, and that is convexin for every . The latter condition holds, for example,if , .
The maximin-optimal predictor is clearly the one that mini-mizes for the worst case choiceof , i.e., the one that maximizes
In general, the maximizing may not agree with thecapacity-achieving prior that has been defined for the self-information loss case. Nonetheless, similarly as in (22), theseminimax–maximin considerations again justify the approachof Bayes-optimal prediction with respect to a mixture of
. It should be pointed out that in certain cases (e.g., theparametric case), prediction performance is not sensitive tothe exact choice of .
By definition, vanishingly small minimax redundancy ratesguarantee uniform convergence to the Bayes envelope. How-ever, unlike the self-information loss case, for a general lossfunction, there is not necessarily a “concentration phenom-enon” where -most points of lie at nearly the sameredundancy level. For example, in the Bernoulli case with
being the Hamming distance betweenand there are only two optimal predictors: one predicts always“ ” and the other predicts always “,” according to whether
is smaller or larger than . Thus it is easy tofind a zero-redundancy predictor for one half of the sources inthe class, and hence there cannot be a nontrivial lower boundon the redundancy that applies to most sources. Nevertheless,by using the concept of exponential weighting, in some cases itis possible to derive strong lower bounds that hold for-mostpoints in at the same time.
Specifically, let us assume thatis an estimate of , thesubtraction operation is well-defined, and that the lossfunction is of the form , where the function
is monotonically increasing for , monotonicallydecreasing for , and . We next derive a lowerbound on
which holds for -most points in , and for any predictorthat does not depend on. This will extend the lower
bound on universal minimum mean-square error prediction ofGaussian autoregressive moving average (ARMA) processesgiven by Rissanen .
We assume that is sufficiently “steep” in the sense thatfor every , and define the log-moment
The function can be interpreted as the (differential)entropy associated with the probability function
where is tuned so that , being the expectationoperation with respect to . For a given predictor ,consider the following probability assignment:
where is a locally bounded away from zero “prior” on. According to , can be approximated as
where is a small remainder term. Iffor all , then following the strong converse of the self-information loss case (16), we have that for-most pointsof
2134 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
for every and all sufficiently large . Since isconcave , interchanging the order between the expectationoperator and the function would not decrease the expressionon the right-hand side of the first line of (31), and so
for every , sufficiently large, and -most . Sinceis monotonically nondecreasing, this gives a lower bound on
The above lower bound is not always tight. Evidently,tightness depends on whether the above definedalso satis-fies the reverse inequality in (31) for some predictor. This,in turn, is the case whenever the self-information lowerbound is achievable by universalpredictive coding, whichmodels the prediction error as a memo-ryless process with being the marginal for some .Referring to the case where , the above boundis nontrivial if , the entropy rate of
. When this is the case, our lower bound suggests aconverse to the previous statement on conditions for uni-form redundancy rates:The existence of universal predictorswith uniformly rapidly decaying redundancy rates under theself-information criterion (i.e., ), is a necessarycondition for the existence of such predictors for generalloss functions.In summary, under suitable regularity con-ditions, there is a uniform redundancy rate for a general, if and only if there is one for the self-information loss
function. Furthermore, even if , there isanother requirement for the bound to be nontrivial, whichis . Indeed, in the Bernoulli case, where it ispossible to achieve zero redundancy for half of the sources(as mentioned earlier), and the bound becomesmeaningless.
Let us consider an important example where the abovebound is useful. For , is the zero-mean Gaussiandensity function with variance . Therefore, the log-moment generating function is given by , andthe differential entropy is . Thus we have
If is the class of Gaussian ARMA(, ) sources withdriving noise of variance , then
and , and we further obtain
This bound has been obtained by Rissanen , and it isknown to be tight at least in the autoregressive case .Another example of a class of Gaussian sources is the onewhere , being zero-mean independent andidentically distributed (i.i.d.) Gaussian noise with power,and is a deterministic signal with power,
limited to and relative bandwidth (normalized by) limitedto . Here again,
for every , but now
the capacity of the band-limited Gaussian channel, which gives
As for achievability of the above bound, recall that thecorresponding universal probability assignment problem issolved by the mixture with respect to the capacity-achieving input which is Gaussian, and therefore itselfis Gaussian. When is in turn factored to a productof , each one of these conditional densities isagain a Gaussian density, whose exponent depends only on
, where is a linear predictor, and theasymptotic variance is given by
being the power spectral density of the capacity-achieving input process. It can be shown (using techniquessimilarly as in ) that this Bayesian linear predictorasymptotically attains the above bound.
MERHAV AND FEDER: UNIVERSAL PREDICTION 2135
Another approach to derivation of lower bounds on perfor-mance of universal schemes has been proposed in the broadercontext of the multi-armed bandit problem , , ,. In this line of work, tight upper and lower bounds onredundancy rates have been given for a class ofuniformly goodschemes in the sense of adapting to the underlying source.However, these results are confined to the case whereis afinite set.
B. Very Large Classes of Sources
So far we have discussed classes of sources where thereexists a uniform redundancy rate, which is given in terms ofthe capacity , at least in the self-information loss case. Thecapacity may or may not tend to zero as , but even ifit does not, the predictive self-information performance, orthe compression ratio of the corresponding universal code,
, might still be less than (whereis the alphabet size) for all , provided that issufficiently large. This means thatsomedegree of compression(or nonuniform probability assignment) is still achievable forall sources at the same time, although there is no longer hopeto approach the entropy for every.
In this section, we focus on much wider classes of sourceswhere even this property does no longer exist. These classesare so rich that, in the self-information loss case, for everyfinite and every predictive probability assignment, thereexists a source in the class such that
In other words, there is a total “breakdown” in terms of self-information loss performance, and similar behavior with otherloss functions. This happens, for instance, with the class of allstationary and ergodic sources , ,  the class of allfinite-order Markov sources (without limiting the order), andmany other classes that can be represented as infinite unionsof nested index sets . Nonetheless, universalschemes that approach the entropy rate, or more generally,the asymptotic Bayes envelope, may still exist if we do notinsist on uniform redundancy rates. In other words,weaklyuniversalschemes  are sometimes available. For example,the Lempel–Ziv algorithm (and hence also the predictiveprobability assignment that it induces ) is weakly universalover the class of all stationary and ergodic sources with a givenfinite alphabet . Necessary and sufficient conditions forthe existence of weak universality can be found in .
One straightforward observation that we can now make froman analysis similar to that of (23), is that a sufficient conditionfor the existence of a weakly universal predictor for a general(bounded) loss function is the existence of such predictorfor probability assignment in the self-information case. Thusthe predictive probability assignment with respect to the self-information loss function is again of crucial importance. Inview of this fact, the fundamental problem, in this context, isthat of estimating conditional probabilities.
Cover  has raised the question whether it is possible toproduce consistent estimates of conditional probabilities with
almost surely as
. Bailey  gave a negative answer to this question (seealso Ryabko [98, Proposition 3]), but pointed out a positiveresult (Orenstein ) to a similar question. It states that for atwo-sided stationary binary process, it is possible to estimatethe value of strongly consistentlyas . The proposed estimates are based on finite-orderMarkov approximations where the order depends on the dataitself. A similar estimator for turns out toconverge to the true value in the sense, which is weakerthan the almost sure sense. This estimator has been shown byBailey  to give
almost surely as . Algoet  gave an extension ofOrenstein’s results to more general alphabets, which was latersimplified by Morvai et al. . In a more recent paper,Morvai et al.  have simplified the estimator (which isbased on empirical averages) for the finite-alphabet case, atthe expense of losing the strong consistency property. Theirestimator is consistent in the self-information sense, i.e., forevery stationary
which implies consistency in the sense.Another line of research work concentrates on the square-
error loss function. Since the minimum mean-square-errorpredictor for a known source is the conditional mean
most of the work in this direction focuses on consistent esti-mation of the conditional mean. For Gaussian processes withunknown covariance function, Davisson  has shown thata th-order linear predictor, based on empirical covariancesgives asymptotic cumulative mean-square error that behaveslike , where is the residual error ofoptimal th-order linear prediction with known covariances.Thus by letting grow sufficiently slowly with time, theconditional mean, given the infinite past, can be eventuallyattained. For general stationary processes, Scarpellini used sample averages with certain spacing between timeinstants in order to estimate where
is a fixed time instant. Modha and Masry considered mixing processes and proposed an estimator basedon slow increase of the prediction memory, using complexityregularization methods. The limitation of their method is that itdepends on knowledge of the mixing rate. Meir  proposeda complexity regularization method in the same spirit, wherefor a given complexity, the class of allowable predictors islimited by a finite Vapnik–Chervonenkis (VC) dimension.
Finally, for a general loss function, Algoet  (see also for the special case of log-optimum investment) has provedstrong ergodic theorems on the cumulative loss. First, for aknown stationary and ergodic source, it is shown that thestrategy that minimizes the conditional mean of giventhe past, is also optimal in the almost sure (and) limit of the
2136 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
time-average loss. When is unknown, empirical estimatesof the conditional probability are provided. By plugging inthese estimates instead of the true, universal schemes areobtained with the same ergodic property as above.
IV. THE DETERMINISTIC SETTING
In the traditional, probabilistic setting of prediction, thatwas described in the previous section, one assumes that thedata are generated by a mechanism that can be characterizedin statistical terms, such as a memoryless source, Markovsource, or more generally, an arbitrary stationary and ergodicsource. As we have seen, the observer estimates on-line eitherexplicitly (plug-in approach) or implicitly (mixture approach)the conditional probability of the next outcome given the past,and then uses this estimate for prediction of future outcomes.
But when it comes to the deterministic setting of indi-vidual data sequences, the underlying philosophy must besubstantially different. There is no longer an assumption ofan ensemble of sequences generated by an underlying proba-bilistic mechanism, but rather only one arbitrary, deterministic,individual sequence. What is the best prediction strategy thatone can possibly use for this fixed sequence?
We realize that, as stated, this question is completely trivialand meaningless. As explained in Section I, formally, for anysequence, there is a perfect predictor that suffers zero lossalong this particular sequence. But at the same time, thisparticular predictor might be extremely bad for many othersequences. Evidently, we are over-tailoring a predictor to oneparticular sequence, and there is no hope to track the strategyof this predictor in the sequential regime that is inherent tothe task of prediction. The root of this “overfitting” effect liesin the fact that we allowed, in the above discussion, too muchfreedom in the choice of the predictor. Loosely speaking, somuch freedom that the amount of information treasured in thechoiceof this predictor is as large as the amount of informationconveyed by the sequence itself! Roughly speaking, in thesesituations the algorithm “learns the data by heart” instead ofperforming the task we expect. The unavoidable conclusion isthat we must limit the freedom of the choice of predictors toa certain class. This limited class of allowable predictors willbe henceforth referred to as thecomparison class(or targetclass) and will be denoted by .
We would like to have asingle universal predictor thatcompetes with the best predictor in, simultaneously forevery , in the sense that
is asymptotically the same as
The universal predictor need not be necessarily inbut itmust be the same predictor for every, whereas the choice ofthe reference predictor in , that minimizes the average loss,may depend (by definition) on the entire sequence. Thedifference between the performance of the sequential universalpredictor and the best predictor in for actually manifests
our regret, because the choice of this optimal predictor is thebest we could have done in retrospect withinhad we knownthe entire sequence in advance.
Loosely speaking, there is a fairly strong duality betweenthe probabilistic and the deterministic setting. While in theformer, we make certain assumptions and limitations on thedata sequences that we are likely to encounter, but no priorlimitations on the class of prediction algorithms, in the latter,it is the other way around. Yet, the deterministic setting isfrequently considered stronger and more appealing, becausethe underlying model seems to be better connected to practicalsituations: There is no (known) probabilistic mechanism thatgenerates the data, but on the other hand, our algorithmicresources are, after all, limited.
Perhaps one of the facts that shed even more light on thisduality between the probabilistic and the deterministic setting,is that quite frequently, the comparison classis defined as acollection of predictors that are obtained as optimal solutionsfor a certain class of sources in the parallel probabilisticsetting. For example, fixed predictors, where is aconstant independently of , are optimal for memorylessstationary sources, linear predictors are sufficient for theGaussian case, Markov predictors are adequate for Markovprocesses, and so on. In these cases, there is a remarkabledegree of duality and analogy between results obtained inthe deterministic setting and those of the corresponding prob-abilistic setting, notwithstanding the considerable differencebetween the two concepts. Specifically, many of the resultsof the individual-sequence setting are completely analogousto their probabilistic counterparts, where the probabilisticsource is replaced by the empirical measure extracted from theindividual sequence with respect to certain sufficient statisticsthat are induced by . Indeed, the structure of this section issimilar to that of the previous section, so as to manifest thisanalogy. Nonetheless, there are still certain aspects in whichthe two scenarios diverge from each other, as we shall see later.
Similarly as in the previous section, our emphasis here is onthe information-theoretic point of view, and as such, it againlargely focuses on the self-information loss function.
A. Indexed Comparison Classes
In analogy to the indexed class of sources, that was exten-sively discussed in the previous section on the probabilisticsetting, there has been considerable attention in the literatureto the dual comparison classes in the deterministic setting. Anindexed comparison class of predictors is a classthat canbe represented as , where designates the indexand is the index set. Similarly as in Section III-A, the indexset could be a finite set ( —positive integer),where may or may not grow with , a countably infiniteset, a continuum, e.g., a compact subset of the real-line or ahigher dimensional Euclidean space (whenis a parameter ofa smooth parametric class), or some combination of these. Aswas already noted above, in many cases,could be defined asthe optimum predictor for a certain member of an indexedclass of sources (cf. Section III-A).
1) Self-Information Loss:In analogy to Section III, let usconsider first the self-information loss function, or equiv-
MERHAV AND FEDER: UNIVERSAL PREDICTION 2137
alently, the probability assignment problem for individualsequences. In other words, our goal is to sequentially assigna universal probability mass function
to the observed sequence, so that would beessentially as small as
for every sequence , uniformly if possible.Shtarkov  has demonstrated that this is indeed possible
by minimizing over the quantity
Specifically, the minimax-optimal probability assignment isattained by the normalized maximum-likelihood function
where is a normalization factor, i.e.,
Indeed, it is readily seen that, by definition of
and so, the universal probability function essentiallyassigns uniformly as high probabilities as those assigned bythe best member in the comparison class, provided thatdoes not grow exponentially rapidly with.
If, for example, is the class of finite-alphabet memory-less probability assignments (i.e., ) with
designating the vector of free letter probabilities,then it is easy to show (e.g., by using the method of types) that grows asymptotically in proportion to andthus (38) behaves like . This in turn is the samebehavior that was obtained for smooth parametric families inthe probabilistic setting.
The number is therefore given theinterpretation of the deterministic analog to the minimaxredundancy-capacity , where the maximization of redun-dancy over in the probabilistic setting is now replaced bymaximization over all possible sequences. Intuitively,is another measure for the richness of the comparison class ofpredictors, in addition to the capacity of the probabilisticsetting. Moreover, it turns out that there are relations betweenthese two quantities. To demonstrate this relation betweenand the operational notion of capacity as the maximum reliabletransmission rate, we note that when , thequantity can be interpreted as where is the
probability of correct decision of an -hypotheses testingproblem involving the sources
that are induced by the predictors, with a uniform prior on.This is true because
This means that if the sources are “far apart” and distin-guishable with high probability, then the minimax redundancyis essentially (compare with the first example in SectionIII). If is countably infinite or a continuum, then any finitesubset of gives a lower bound onin the above manner. As grows, normally decreases, butthe product can be kept large at least as long asissmaller than so as to “transmit” at a rate below capacity,which allows for keeping close to unity. But the maximumachievable product might be achieved at rates beyondcapacity.
It is easy to show directly that is never smaller thanfor the same class of sources or probability assignments
indexed by . This implies that a necessary condition for theexistence of minimax universality in the deterministic settingis the existence of the parallel property in the dual probabilisticsetting. In the smooth parametric case bothand behavelike . More precisely (see, e.g., Rissanen )
It turns out, however, that richer indexed classes may exhibita considerably larger gap between these two quantities (see,e.g., the example of arbitrarily varying sources in ).
The main drawback of the maximum-likelihood (ML) prob-ability assignment is obviously on the practical side: notonly is hard to compute in general, but more importantly, itis again horizon-dependent, i.e., the sequence lengthmust beprescribed. To alleviate this difficulty, the maximum-likelihood
can be exponentially approximated bya mixture using Laplace integration . Specifically, forthe case of stationary memoryless probability assignments,Shtarkov  proposed, following Krichevsky and Trofimov, the Dirichlet- (Jeffreys’ prior) mixture, whichleads to the purely sequential probability assignment
where is the number of occurrences of the letterin. We have mentioned earlier, in Section III, the family of
sequential probability assignments that arise from Dirichletweighting in general. But the interesting property of theDirichlet- (in addition to being Jeffreys’ prior for
2138 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
this family), is that it is asymptotically as good as the MLprobability assignment. Specifically, with definedas above
where only the constant here is larger than the one obtainedby .
Further refinements and extensions of this result have beenrecently carried out, e.g., in  and . Specifically,Xie and Barron  introduce also the dual notion of themaximin redundancy (or regret) whose value coincides with
as well, and show that Jeffreys’ mixture is asymptoticallymaximin with asymptotically constant regret for sequenceswhose empirical pmf’s are internal to the simplex. Similarlyas in the probabilistic setting, it is not asymptotically minimaxthough because of problematic sequences on the boundary ofthe simplex. Nevertheless, a slight modification of Jeffreys’mixture (which again, depends on and hence makes itagain horizon-dependent), is both asymptotically minimax andmaximin.
Finally, Weinberger, Merhav, and Feder  have studiedthe problem of universal probability assignment for individualsequences under the self-information loss function with respectto the comparison class of all probability assignments that areimplementable by finite-state machines with a fixed number ofstates. There are no such accurate formulas therein regardingthe higher order redundancy terms. However, it is shown thatthe behavior is not only minimax over all sequences,but moreover, it is a tight lower bound formostsequences ofmosttypesdefined with respect to those finite-state probabilityassignments. This result parallels the-almost everywhereoptimality of universal probability assignments in the proba-bilistic setting (cf. Section III). In this context, it is interestingto note, as shown in , that in contrast to the probabilisticsetting, the plug-in approach fails, in general, when it comes toindividual sequences. We will elaborate on these results furtherin Section V in the context of hierarchical comparison classes.
2) General Loss Functions:The problem of universalsequential prediction or decision-making for individualsequences under general loss functions, is definitely a muchwider problem area than that of the special case of probabilityassignment under the self-information loss function that wediscussed thus far in this section. In fact, most of the classicalwork in this problem area, in various scientific disciplines, hasconcentrated primarily on the case ofconstantpredictors, i.e.,predictors for which each yields a certain fixed prediction,regardless of the observed past. For example,, for a certainvalue of , may suggest to predictalways “ ” as the nextoutcome of a binary sequence, or, it may always assign aprobability of for the next outcome being “.” This isseemingly not a very interesting comparison class becausepast information is entirely ignored.
Nonetheless, the motivation for carefully studying this sim-ple comparison class is that it is fundamental for exam-ining comparison classes of more sophisticated predictors.For example, a first-order Markov predictor, characterized by
, can be thought of (in the binarycase) as a combination of two fixed predictors operating,respectively, on two subsequences of: the one correspond-ing to all time instants that follow , and theother—where . Having made this observation, theproblem then boils down back to that of constant predictors.
One example, which is still closely related to the self-information, is that of portfolio selection for optimal invest-ment in the stock market –, . In this model, the goalis to maximize the asymptotic exponential growth rate of thecapital, where the current investment strategy depends on thepast. The corresponding loss function, in our framework, isthen , with both and being -dimensional vectors of nonnegative components, where inthe former these components sum to unity. The vectorrepresents the return per monetary unit in several investmentopportunities (stocks), whereas the vectorcharacterizes thefraction of the current capital allocated to each stock. Cover and Cover and Ordentlich  have used techniquessimilar to those of the self-information loss described above,to develop a sequential investment algorithm and related itagain to universal coding with results of a similar flavor.Again, their universal sequential strategy competes with thebest constant investment strategy. These results can be viewedas an extension of the self-information loss because the latteris actually a special case where the vectoris always all-zero except for one component (corresponding to the currentalphabet letter), which is .
3) The Sequential-Compound Decision Problem:Other ex-amples of loss functions are not so closely related to that ofthe self-information loss, and consequently, the techniques andthe results are considerably different. The comparison class ofconstant strategies for more general loss functions has beenstudied in a somewhat more general setting, referred to as thesequential-compound decision problem, which was first pre-sented by Robbins  and has been thoroughly investigatedlater by many researchers from disciplines of mathematicalstatistics, game theory, and control theory (see, e.g., ,, , , and ). Perhaps the most fundamentalfindings of the compound sequential decision problem aresummarized in the theory of Bayes decision rules, that includesthe notion of Bayes envelope(that is, the best achievabletarget performance as a functional of the empirical pmf of thesequence) and an analysis of its basic properties. This in turnhas been combined with approachability–excludability theory,that provides simple necessary and sufficient conditions underwhich one player (in our case, the predictor) of a repeated zero-sum game can reach a certain performance level (in our case,the Bayes envelope) for every strategy of the opponent player(in our case, Nature that chooses an adversary sequence).
The sequential compound decision problem is more generalthan our setting in the sense that the observer is assumed toaccess only noisy versions of the sequence, yet the lossfunction to be minimized is still associated with the clean se-
MERHAV AND FEDER: UNIVERSAL PREDICTION 2139
quence (e.g., the expected cumulative loss, or its probabilisticlimit with respect to the ensemble of noise processes). Hannan has taken a game-theoretic approach to develop upperbounds on the decay rate on the regret, showing a convergencerate of in the finite-alphabet, finite-strategy spacecase, and a rate of in the continuous case,provided that the loss-minimizing strategyas a functional ofthe underlying empirical pmf of , that is, theBayes response,satisfies a Lipschitz condition of order . Thus for ,which is normally the case, this means a convergence rate of
, similarly to the self-information loss case that wehave seen above.
One of the essential ideas underlying the analysis tech-niques, is the following simple “sandwich” argument (see, e.g.,): It is easy to show that
i.e., the Bayesian envelope, is upper- and lower-bounded bythe average loss associated with two strategies. The currentstrategy for the upper bound is optimal within for the dataseen thus far , and for the lower bound, it is an (imagined)strategy that is allowed to accessfor this optimization within
. Thus the strategy of the lower bound sees merely one moreoutcome than that of the upper bound. When the comparisonclass is that of constant strategies, the Bayes envelope dependson the sequence only through its empirical pmf, and thisadditional observation perturbs the current empirical pmf bya term proportional to . Therefore, under the appropriatesmoothness conditions ( above), the instantaneous lossesof the upper and lower bound differ also by a quantity thatscales proportionally to , which when averaged over theintegers , gives . A fortiori, the differencebetween the upper bound and the Bayes envelope, i.e., theregret, cannot exceed .
In some important special cases, however, the loss functionand the Bayes response are discontinuous. This happens, forexample, in prediction of binary sequences under the criterionof relative frequency of mispredicted outcomes, where theBayes response with respect to the class of constant predictorsis binary itself and it depends on whether the relative frequencyof zeros is below or above . In this case, randomizationof the sequential prediction strategy around the discontinuitypoint (see, e.g., , , and ) is necessary in orderto achieve the target performance for problematic sequenceswhose empirical pmf’s visit infinitely often (as ) thesediscontinuity points. The cost of this randomization, however,is a considerable slowdown in the rate of convergence towardsthe Bayes envelope. In the above binary case, for example,the rate of convergence is , whereas in the parallelprobabilistic setting, where such a randomization is not needed(cf. Section III), it is as fast as .
Van Ryzin  has shown that even in the former caseof smooth loss functions, the convergence rate can be moretightly upper-bounded by under certain regularityconditions on the channel through which the observer receivesthe noisy measurements. Gilliland  further investigated
convergence rates for the special case of the square lossfunction under various sets of assumptions.Several later papers ,  deal with the more generalcase where the comparison class consists of Markov strategies,whose importance will be emphasized later.
On-Line Prediction Using Expert Advice:A completelydifferent point of view has been taken more recently, primarilyby learning theorists in their studies of a paradigm referred toason-line prediction using expert advice(see, e.g., , ,, , , , and ). In the previously definedterminology, the basic assumption is that the comparisonclass consists of finitely many predictors , referredto as experts.There are absolutely no assumptions on anystructure or relationships among these experts. The goal isto devise a sequential universal prediction algorithm thatperforms essentially as well as the best of these experts alongevery individual sequence.
We have actually examined earlier this scenario in thecontext of the self-information loss function and a finiteindex set , where our conclusion was thatthe necessary minimax price of universality need not exceed
in the worst case, namely, when the probabilityassignments correspond to distinguishable sources. Inter-estingly, this behavior essentially continues to take place forgeneral (but sufficiently regular) loss functions. Vovk and Littlestone and Warmuth  proposed independentlya sequential prediction algorithm, whose regret with respectto the best expert never exceeds , where is aconstant that depends solely on the loss function. At theheart of this algorithm, there is a remarkable similarity tothe mixture approach, or, more concretely, the notion ofexponential weighting that was discussed in Section III in thespecial case of the self-information loss.
Here is the idea: let be a given constant (to be chosenlater) and consider the weighted average of , i.e.,
where is the prediction of theth expert at time, andis theweightassigned to this expert at this time. The weights,at each time instant, are nonnegative numbers summing tounity. Intuitively, we would like to assign higher weightsto experts who were proven better in the past. Therefore, areasonable thing to do, following (10), is to assign to eachexpert a weight that is proportional to
where for the summation will be defined as zero (i.e.,uniform initial weighting). Now, if we are fortunate enoughthat there exists a strategysuch that for every
then it is easy to see that this strategy will serve our purpose.This is true because the above condition suggests the following
2140 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
conceptually simple algorithm:
0) Initialization: Set for andthen .
1) Prediction: Choose a prediction at time that satisfies(48).
2) Update: Upon receiving , update the weight functionaccording to
3) Iteration: Increment and go to 1).
It follows immediately from the definition of the algorithmthat the exponent of the cumulative loss associated withsatisfies
Thus the crucial question that remains to be addressed isregarding the conditions under which (48) is satisfied. Toput this question in perspective, first, observe that for theself-information loss function and , the functions
are probability measures of-tuples. Therefore, their weightedaverage (mixture) is itself a probability measure and as such,can be represented by
for a certain , which is the probability assignment cor-responding to the finite mixture. However, in general, thefunction may not be closed to convex combinations.Fortunately, it is shown that under fairly mild regularity con-ditions (see , , and  for details), it is guaranteedthat (48) always holds provided thatis chosen to be at most
and that , in which case the regret can be made assmall as . Many of the important loss functions, likethe self-information loss and the square-error loss, satisfy theseconditions. For example, if the function is concave
in for every (which is the case in linear prediction andsquared-error loss under some conditions ), namely,
then it is clear that the weighted average of the experts’predictions will be a suitable solution. Unfortunately, there arealso other important loss functions (like the loss function,
) for which . This means that for theseloss functions, the regret does not behave like ,but rather decays at a slower rate with, e.g., like .These cases should be handled separately.
What makes this algorithm even more interesting is thefact that it turns out to be minimax-optimal in the sense that
is also an asymptotic lower bound on the maximumregret. Unfortunately, the weak point of this lower bound isthat this maximum is taken not only over all sequences ,but also over all possible sets of experts! The algorithm is,therefore, asymptotically optimal in an extremely pessimisticsense, which is of special concern whenis large. What isleft to be desired then is a stronger bound that depends on therelationships among the experts. As an extreme example, if allexperts are identical then there is in fact only one expert, not
, and we would expect to obtain zero regret. Intuitively, wewould like the formal number of experts to be replaced bysome notion of an “effective” number of distinct experts, inanalogy and as an extension of the role played by capacityor by in the self-information loss case. To the best of ourknowledge, to date, there are no reported results of this kindin the literature except for Cesa-Bianchi and Lugosi  whocharacterized the minimax regret along with upper and lowerbounds for binary sequences and the Hamming loss function,but without any constructive algorithm yet.
Another drawback is associated with the algorithm itself. Touse this algorithm in practice, one should actually implementin parallel the prediction algorithms proposed by allexperts,which might be computationally demanding for large. Thisis in contrast to the situation in certain special cases, e.g.,when the experts correspond to all finite-state machines witha given number of states , , , . In these cases,there is no explicit implementation of all finite-state machinesin parallel.
In spite of these shortcomings, the problem of on-line pre-diction with expert advice has attracted fairly much attentionover the last few years and there are quite a few reportedextensions, modifications, and other variations on the theme(see, e.g.,  for a summary of recent work in on-linelearning). One extension that would be especially interesting isto tie it with the setting of the compound sequential decisionproblem in the sense that the predictor accesses only noisyobservations, whereas the loss function remains in terms of theclean outcomes. Clearly, the above weighting algorithm, in itspresent form, is not directly implementable since there is noperfect feedback on the loss associated with past expert advice.
B. Very Large Comparison Classes
We end this section with a natural analog to the case of verylarge classes of sources in the probabilistic setting, namely,very large comparison classes of predictors for which thereare normally no uniform redundancy rates.
In the general level, consider a nested infinite sequence ofindex sets , and their union .
MERHAV AND FEDER: UNIVERSAL PREDICTION 2141
Strictly speaking, is itself an index set, whose membersare of the form , where is the smallest integer suchthat . However, the basic property that makeshereindifferent than the index sets of Section IV-A is that it is so rich,that for every finite sequence , the minimum cumulative lossover all predictors indexed by is zero. In other words, there istoo much freedom within , and we are confronting again theundesirable overfitting effect discussed earlier. This happensin many important examples, e.g., whenconsists of the classof all finite-state predictors with an undetermined (but finite)number of states, or the class of all Markov predictors, oreven more specifically, all linear predictors with an unspecifiedfinite order, etc. Quite clearly, in all these situations, there areenough degrees of freedom to tailor a perfect predictor for anyfinite sequence , and thus our earlier definition (cf. SectionIV-A) of the target performance becomesmeaningless.
We are lead then to the conclusion that we must modifythe definition of the target performance. The key principle fordoing this is to keep an asymptotic regime of . To fixideas, consider an infinite sequence where
always designates the firstoutcomes of . First, similarlyas in Section IV-A, let us define
where it is assumed that each is an index set of the typediscussed in Section IV-A. As for asymptotics, we let firstgrow without bound, and define
where the operation manifests a worst case approach:since the sequence is not necessarily ergodic, i.e., the limitmay not exist, one must worry about the worst performancelevel obtained infinitely often along. Finally, we define ourtarget performance as
where now the limit clearly exists since is amonotonically nonincreasing sequence whose elements areobtained from minimizations over increasing sets of predictors.Since the limit is taken first, the asymptotic regimehere indeed meets the above mentioned requirement that
. The problem is now to devise a universal predictionalgorithm that asymptotically achieves .
One of the most popular applications of this general scenariois the one where consists of all strategies that are imple-mentable by finite-state machines, which means that each
corresponds to the class of finite-state machineswith no more than states. Specifically, each member ofis defined by two functions and . The function , referred toas thenext-state function, describes the evolution of the stateof the machine, , according to the recursion
where the initial state is fixed. The function describes thestrategy at time , which depends only on by
The idea behind this model is that the state variablerepre-sents the limited information that the machine can “memorize”from the past for the purpose of choosing the currentstrategy. An important special case of a finite-state machinewith states is that of ath-order Markov machine (alsocalled finite-memory machine), where .
Ziv and Lempel described, in their famous paper ,a target performance in this spirit in the context of datacompression of individual sequences using finite-state ma-chines. The best compression ratio obtained by finite-state encoders over infinitely long individual sequences (inthe above defined sense) has been referred to as thefinite-state compressibilityof , and the well-known Lempel–Zivalgorithm (LZ’78) has been shown to achieve the finite-statecompressibility for every sequence. In a later paper , Zivand Lempel extended this definition to compression of two-dimensional arrays (images), where the additional ingredientis in defining also a scanning strategy.
In , results in the same spirit have been obtained forsequential gambling over individual sequences, where againthe comparison class is that of gambling strategies that areimplementable by finite-state machines. Since the gamblingproblem is completely analogous to that of data compression,or more precisely, probability assignment under the self-information loss function (see also  discussed in SectionIV-A), the results therein are largely similar to those of Zivand Lempel . The formal setting of , however, issomewhat more compliant than  to our general definitionof cumulative loss minimization, where each loss term dependson one outcome only.
The results of  in turn provided the trigger to a laterwork , where the comparison class of finite-state predictorsfor binary sequences was studied under the Hamming lossfunction, defined as if , andotherwise. In other words, in this case, is simplyan estimate of the value of the next outcome, and theperformance measure is the relative frequency of predictionerrors. Analogously to , the quantity , in this specialcase, is called thefinite-state predictabilityof . Similarly,when is further confined to the class ofth-order Markovpredictors, then the correspondingly defined is called theMarkov predictabilityof . There are two main conclusionspointed out in .
The first is that the finite-state predictability and the Markovpredictability are always equivalent, which means that it issufficient to confine attention to Markov predictors in orderto achieve the finite-state predictability. It is worthwhile tonote that in the probabilistic setting, such a result would havebeen expected under certain mixing conditions because theeffect of the remote past fades away as time evolves, and onlythe immediate past (that is stored as the state of a Markovpredictor) should be essential. Yet, when it comes to individualsequences this finding is not at all trivial since the sequenceis arbitrary and there is no parallel assumption on mixingor fading memory. The proof of this result stems from pureinformation-theoretic considerations.
The second conclusion, which is largely based on the firstone, is on the algorithmic side. It turns out that a prediction
2142 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
strategy that corresponds to probability assignments based onthe incremental parsing procedure of the LZ algorithm (seealso  and ) asymptotically achieves the finite-statepredictability. The incremental parsing procedure sequentiallyparses a sequence into distinct phrases, where each new phraseis the shortest string that is not identical to any previouslyparsed phrase. The reason is that the incremental parsingprocedure works like a Markov predictor of time-varyingorder , where in the long run, is very large mostof the time because the phrases become longer and longer.Consequently, the Markov predictability, and hence also thefinite-state predictability, are eventually attained. But the deeppoint here lies in the simple fact that the incremental parsingalgorithm, which was originally developed as a buildingblock of a compression algorithm, serves also as the engineof a probability-assignment mechanism, which is useful forprediction.
This gives rise to the idea that this probability assignmentinduces a universal probability measure in the context ofindividual sequences. Loosely speaking, it means that theuniversal probability measure is proportional to ,where LZ is the LZ codeword length for , .This in turn can be thought of as an extension of Shtarkov’sML probability assignment because is well known to be an upper bound (within vanishingly small terms) of
, where the maximum is taken over all finite-statesources with a fixed number of states.
The problem of  was later extended  in severaldirections simultaneously: the alphabet of and the lossfunction were assumed to be more general. Also, classes ofpredictors other than that of deterministic finite-state predic-tors were considered, e.g., randomized finite-state predictors(where the next-state function is randomized), families oflinear predictors, etc. Many of the results of  turn outto carry over to this more general case.
Finally, one additional result of [75, Theorem 3] (see,also ) relates the individual-sequence setting backto the probabilistic setting. It tells us that under suitableregularity conditions, for a stationary and ergodic process
the quantity definedwith respect to finite-state or Markov predictors, agreesalmost surely with the probabilistic performance measure
. One special case of thisresult  is that the finite-state compressibility is almostsurely equal to the entropy rate of a stationary and ergodicsource. Another important example corresponds to the casewhere is the class of all linear predictors of order,and hence is the linear predictability. In the stationaryand ergodic case, the above cited result suggests that withprobability one, coincides with the varianceof the innovation process (that is, the residual linear predictionerror) given by
where is the power spectral density of the process.While the duality between certain classes of sources and the
corresponding classes of predictors was quite straightforward
in relatively small indexed (parametric) classes, the aboveresult establishes a parallel duality between the very large classof stationary and ergodic sources and the very large class offinite-state predictors or Markov predictors.
V. HIERARCHICAL UNIVERSALITY
So far we have focused on two substantially differentsituations of universal prediction, both of which take placein the probabilistic setting as well as in the deterministicsetting: Universality with respect to an indexed class,1 whichis relatively “small,” as opposed to universality with respect toa very large class, where no uniform redundancy rates exist.These two extreme situations reflect the interplay betweentwo conflicting goals, namely, fast decay of redundancy rateson the one hand, and universality with respect to classes aswide and general as possible, on the other. For example, theLempel–Ziv algorithm for data compression (or for predictiveprobability assignment) is universal for all stationary andergodic sources, but when a memoryless source is encountered,this algorithm gives a redundancy rate that might be muchslower than that of a universal scheme which is tailored to theclass of memoryless sources; see , , and .
Our basic assumption throughout this section is that thelarge class of sources (in the probabilistic setting) orpredictors (in the deterministic setting) can be representedas a countable union of a sequence of index sets ,which may, but not necessarily, have a certain structure, suchas nestedness . In the probabilistic setting,perhaps the first example that naturally comes into one’s mindis where each is the class of discreteth-order Markovsources, and hence the unionis the large class of all finite-order Markov sources. Furthermore, in the finite-alphabet case,if we slightly extend this class and take its “closure” withrespect to the information divergence “distance” measure, itwould include the class of all stationary sources. This isbecause every stationary source can be approximated, in thedivergence sense, by a sequence of Markov sources of growingorder [44, Theorem 3.5.1, p. 57], [47, Theorem 2.6.2, p. 52]. Afew other examples of hierarchical probabilistic models are thefollowing: i) finite-state sources with deterministic/randomizednext-state functions, ii) tree sources (FSMX), iii) noisy ver-sions of signals that are representable by countable families ofbasis functions, iv) arbitrarily varying sources , v) sourceswith countable alphabets (referred to as sequences of classes ofgrowing alphabets), and vi) piecewise-stationary memorylesssources. Most of these examples have dual comparison classesin the deterministic setting.
In view of the discussion in the above two paragraphs, anatural question that arises, at this point, is the following: canone devise a universal predictor that enjoys both the benefitsof a small indexed class and a large class? In other words,we would like to have, if possible, a universal predictor withrespect to the large class, but with the additional property that italso performs essentially as well as the best universal predictor
1Since this refers to both the probabilistic and the deterministic setting, theterm “class” here corresponds both to a class of sources in the probabilisticsetting, and a comparison class of predictors in the deterministic setting.
MERHAV AND FEDER: UNIVERSAL PREDICTION 2143
within every given indexed subclass of . In the probabilis-tic setting, this means that if we are so fortunate that the sourcehappens to be a member of a relatively small indexed class(e.g., a memoryless source), then the redundancy, or the regret,would be essentially the same as that of the best universalpredictor for this smaller class. In the analog deterministicsetting, we would like the universal predictor of this largeclass to behave similarly as the best universal predictor withina certain indexed comparison subclass. Note that the abovequestion is meaningful even if is merely a finite (rather thana countably infinite) union of . The reason is that theuniform redundancy rate of, that is, the redundancy-capacity,denoted by in the self-information loss case, might stillbe larger than that of any subset . Therefore, even inthis case, treating just as one big class might not be thebest thing to do.
In the probabilistic setting, Ryabko  was the first toaddress this interesting question for the above describednested sequence of classes of Markov sources, and for theself-information loss (universal coding). Generally speaking,Ryabko’s idea is to apply the following conceptually simpletwo-part code, referred to as atwice-universalcode. The firstpart of the code is a codeword for an integerwhose lengthis , and the second part is auniversal code with respect to , where is chosen so as tominimize the total codeword length. Clearly, this code attainsredundancy of
which obviously never exceeds for thetrue value of . Since behaves like inthe Markov case, the additional term does not affectthe rate of convergence within each . Thus although therecannot be uniform redundancy rates simultaneously over theentire class of Markov sources there is still asymptoticallyoptimal behavior within every .
An alternative to this two-part code, which cannot betransformed easily into a prediction scheme, is the mixtureapproach. Specifically, for the problem of prediction withself-information loss, the suggested solution is based on aprobability assignment formed by two-stage mixture, firstwithin each , and then over the integers .The first observation is that the mixture approach, with appro-priately chosen weight functions, is no worse than the abovetwo-part scheme. To see this, let us assume thatsatisfy Kraft’s inequality with equality (otherwise, they can beimproved), and consider the two-stage mixture
where is the capacity-achieving prior of . Then
where the left-most side corresponds to the performance ofthe mixture approach and the right-most side corresponds tothe performance of the two-part scheme with an optimummixture within each class. The message here is that for everyindividual sequence, the mixture approach is no worse than thetwo-part approach. In  this point is further explored anddeveloped for several examples of hierarchical classes (finite-state machines and others) in view of the fact that the firstterm of the right-most side above is also a lower bound for“most” sequences in a fairly strong sense (cf. Section III). Ofcourse, the last chain of inequalities continues to hold aftertaking expectations in the probabilistic setting.
It turns out though, that in the probabilistic setting themixture approach is not only no worse than the two-partapproach, but moreover, it is an optimal approach in a muchsharper and deeper sense. As an extension to the result of
-almost everywhere optimality of (cf. Section II), thefollowing holds for hierarchies of classes [39, Theorem 3]: thetwo-stage mixture with arbitrary weight functionswithin the classes, and , , overthe positive integers, simultaneously minimizes in essenceredundancy for -most points in of -most classes .If, in addition, is the capacity-achieving prior for all, then this minimum redundancy can be decomposed into a
sum of two terms, the first of which is , the capacitywithin the underlying class , and the second is an extraredundancy term that reflects the additional cost of universalitywith respect to the unknown. The latter term is always upper-bounded by . However, if we furtherassume that the classes are “easily distinguishable” in the sensethat there exists a good (model order) estimator forwithsmall average error probability [39, Theorem 4], thenis an asymptotically tight bound. This means that in the caseof distinguishable classes, is the optimalperformance even at the level of the higher order term ,which might be considerably larger for large. However, ifthe classes are not easily distinguishable, the mixture approachyields a smaller second-order redundancy term whereas thetwo-part coding approach continues to give . Someguidelines regarding the choice of(or, equivalently, )are given in . It should be noted that for any monotonenonincreasing sequence of probabilities, for all, namely, , and so is
optimum redundancy in the distinguishable case, as it can beasymptotically attained by a universal code for the integers.
From the viewpoint of sequential predictive probability as-signment, however, both the two-part method and the methodof mixtures are not directly implementable because in theformer, the minimizing depends on the entire , and in thelatter, may depend on . A possible alternative to thenonsequential minimization overcould be on-line estimationof and plug-in. An algorithm in this spirit has been proposedby Weinberger, Rissanen, and Feder  for hierarchies oftree sources in the probabilistic setting, where the estimator of
(which is associated the context, in this case) was basedon algorithm Context. Fortunately, the probability of errorin estimating decays sufficiently rapidly, so as to leavethe leading redundancy term unaffected. In the deterministic
2144 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
setting, however, it can be shown  that the method basedon the plug-in estimate of does not work, i.e., there aresequences for which the resulting “redundancy” is higher thanachieved when the class is known in advance.
The mixture approach, however, is useful in both theprobabilistic setting and the deterministic setting, giving usyet another reason to prefer it. To overcome the problemmentioned above, namely, the fact that the weights of themixture over the index depend on the horizon, we use fixed-weight functions. Fortunately, as mentioned in Section II, inmany cases are replaceable by mixture weights that do notdepend on and yet asymptotically achieve capacity.
At this point it is necessary to address a major practicalconcern: is it computationally feasible to implement the two-stage mixture probability assignment? More specifically, wehave seen (Sections III and IV) that in some importantexamples the mixture within a single indexed class is easilyimplementable, but is it still reasonably easy to implementthe second-stage mixture among (possibly infinitely) manyclasses. Unfortunately, there is no positive answer to thisquestion in the general level. Nonetheless, Willems, Shtarkov,and Tjalkens, in their award-winning paper  provided apositive answer to this question for finite hierarchies of classesof tree sources, using an efficient recursive method, referred toas context-tree weighting.Their method is optimal for everyindividual sequence in the sense of (60). For hierarchies ofcountably infinitely many classes, however, the implemen-tation issue is still unresolved. In  several examplesare demonstrated where the countably infinite mixture over
actually collapses to a finite one. This happens becausethe contributions of mixtures corresponding to allbeyonda certain threshold turn out to be identical and thencan be merged with the combined weight . Theproblem is, though, that normally grows with , and so,the computational burden of computing mixtures at everytime instant becomes explosively large as time elapses.
So far, we have discussed hierarchical universal predictionsolely under the self-information loss function. What can besaid about other loss functions? Apparently, we can deducefrom the self-information loss function to other loss functionsin the same way that this has been done in Sections III andIV. Beyond that, we are not aware of much reported workon this topic. We will mention only two directions that havebeen pursued explicitly. The first one is by Helmbold andSchapire , who have combined the exponential weightingmechanism of on-line prediction using expert advice (with respect to the absolute error loss function) together withthe context-tree weighting algorithm of Willems, Shtarkov,and Tjalkens  for competing with the best pruning ofa decision tree.
Other recent work is in hierarchical linear prediction forindividual sequences under the square error loss function, . In these papers, the linear prediction problem istransformed into a Gaussian sequential probability assignmentproblem. The universal assignment is obtained by a two-stagemixture, over the linear prediction coefficients and over themodel order. For the mixture over the parameters, a Gaussianprior is used, and the mixture can be evaluated analytically.
The probability assignment attained by the mixture does notcorrespond directly to a universal predictor, but fortunately,such correspondence can be made for a certain range ofvalues of the predicted sequence. Thus by a proper choiceof prior, the predictor can be scaled to any finite range ofthe sequence values. In addition, the mixture over the modelorder is performed in a computationally efficient way, sinceusing lattice filters, all possible linear predictors with modelorder up to some largest order can be weighted in anefficient recursive procedure whose complexity is not largerthan that for a conventional linear predictor of the modelorder . It was also noted, following , that a plug-inestimator of the parameter (resulting from the recursive leastsquares (RLS) algorithm) leads to universal prediction albeitat a slower rate than the mixture approach. The resultinguniversal linear predictor has been implemented and testedexperimentally in several practical communication and signalprocessing problems .
VI. CONCLUSION AND FUTURE DIRECTIONS
In this paper, an attempt has been made to provide anoverview on the current state-of-the-art in the problem area ofuniversal prediction. As explained in Section I, it is definitelynot, and not meant to be, a full encyclopedic survey of allscientific work that has ever been done on this topic. The aimwas to mention several important concepts from the authors’point of view. Let us summarize some of these concepts verybriefly.
We have seen that the problem of universal prediction hasbeen studied extensively both in the probabilistic and thedeterministic setting. There are many common features sharedby these two settings. First of all, in both of them the self-information loss case plays a central role, which stems fromseveral facts. i) It is an important loss function on its ownright for reasons that were explained in Section II. One of themain reasons is that we view the prediction problem as oneof probability assignment, and as such, the self-informationloss function arises in a very natural manner. ii) In the self-information loss case the theory is fairly mature and wellunderstood. iii) Results (both lower bounds and algorithms) forother loss functions can be obtained from the self-informationloss function. The second common feature of the proba-bilistic and the deterministic settings is in the large degreeof parallelism between the theories of universal prediction:universality with respect to small indexed classes, universalitywith respect to very large classes, and hierarchical universality,which actually bridges them. There is also a remarkable degreeof analogy between the quantitative results obtained in bothsettings in some cases. One of the fundamental connections isthat for stationary and ergodic sequences, the best attainableperformance level of the deterministic definition agrees almostsurely with its probabilistic counterpart.
However, there are a few differences as well: sometimesminimax redundancy rates of the deterministic setting aredifferent from those of the probabilistic setting. The plug-inapproach for predictive probability assignment works well inmany instances of the probabilistic setting, but it is normally
MERHAV AND FEDER: UNIVERSAL PREDICTION 2145
not a good approach in the deterministic setting. The minimaxredundancy of the deterministic setting is different from thatof the probabilistic setting. Randomization is sometimes nec-essary in the deterministic setting, but not in the probabilisticsetting.
Perhaps one of the interesting messages is that although theterm “probability assignment” originally comes from the prob-abilistic world, it is still meaningful in the pure deterministicsetting as well. This fact is far from being trivial. Moreover,there are very efficient algorithmic tools for obtaining goodprobability assignments, and one of them is the incrementalparsing procedure of the Lempel–Ziv algorithm.
We also see a few more theoretical problems which mightbe interesting to consider for future research. Some of themhave been mentioned in the body of the paper.
• Develop a more solid and general theory of universalprediction for general loss functions, in parallel and ex-tension of the theory of the self-information loss function.Derive tighter and stronger lower bounds for generalloss functions both in the probabilistic setting and in thedeterministic setting. For example, in the framework ofprediction using expert advice, take into account relationsamong the experts rather than assuming the worst set ofexperts.
• Extend results on universal prediction with respect to thecomparison class of finite-state machines to the case noisyobservations.
• Impose limitations on the resources of the universalsequential predictor. For example, if the comparison classis that of finite-state predictors, how many states shouldthe universal predictor have to guarantee redundancybelow a certain level?
Some of these challenges have defied the best efforts of manyresearchers so far. Others are yet to be explored.
 R. Agrawal, D. Teneketzis, and V. Anantharam, “Asymptotic adaptiveallocation schemes for controlled i.i.d. processes: Finite parameter case,”IEEE Trans. Automat. Contr., vol. 34, pp. 258–267, Mar. 1989.
 , “Asymptotic adaptive allocation schemes for controlled Markovchains: Finite parameter case,”IEEE Trans. Automat. Contr., vol. 34,pp. 1249–1259, Mar. 1989.
 P. H. Algoet, “Universal schemes for prediction, gambling and portfolioselection,”Ann. Probab., vol. 20, pp. 901–941, Apr. 1992.
 , “The strong law of large numbers for sequential decision underuncertainty,” IEEE Trans. Inform. Theory, vol. 40, pp. 609–633, May1994.
 P. H. Algoet and T. M. Cover, “Asymptotic optimality and asymptoticequipartition properties of log-optimal investment,”Ann. Probab., vol.16, no. 2, pp. 876–898, 1988.
 D. H. Bailey, “Sequential schemes for classifying and predicting ergodicprocesses,” Ph.D. dissertation, Stanford Univ., Stanford, CA, 1976.
 J. M. Bernardo, “Reference posterior distributions for Bayesian infer-ence,”J. Roy. Statist. Soc. B, vol. 41, no. 2, pp. 113–147, 1979.
 D. Blackwell, “An analog to the minimax theorem for vector payoffs,”Pac. J. Math., vol. 6, pp. 1–8, 1956.
 , “Controlled random walks,” inProc. Int. Congress Math., vol.3. Amsterdam, The Netherlands: North Holland, 1956, pp. 336–338.
 A. Blum, “On-line algorithms in machine learning.” [Online.] AvailableWWW: http://www.cs.cmu.edu/afs/cs.cmu.edu/user/ avrim/www/Papers/pubs.html.
 A. C. Blumer, “Minimax universal noiseless coding for unifilar andMarkov sources,”IEEE Trans. Inform. Theory, vol. IT-33, pp. 925–930,Nov. 1987.
 N. Cesa-Bianchi, Y. Freund, D. P. Helmbold, D. Haussler, R. E.Schapire, and M. K. Warmuth, “How to use expert advice,” inAnnu.ACM Symp. Theory of Computing, 1993, pp. 382–391.
 N. Cesa-Bianchi, Y. Freund, D. P. Helmbold, and M. K. Warmuth,“On-line prediction and conversion strategies,” inProc. EUROCOLT’93,(Oxford, U.K., 1993), pp. 205–216.
 N. Cesa-Bianchi and G. Lugosi, “On sequential prediction of individualsequences relative to a set of experts,” 1998, preprint.
 B. S. Clarke and A. R. Barron, “Information-theoretic asymptotics ofBayesian methods,”IEEE Trans. Inform. Theory, vol. 36, pp. 453–471,May 1990.
 B. S. Clarke and A. R. Barron, “Jeffreys’ prior is asymptotically leastfavorable under entropy risk,”J. Statist. Plan. Inform., vol. 41, pp.37–60, Aug. 1994.
 T. M. Cover, “Universal gambling schemes and the complexity measuresof Kolmogorov and Chaitin,” Tech. Rep. 12, Dept. Statist., StanfordUniv., Stanford, CA, Oct. 1974.
 , “Open problems in information theory,” inProc. MoscowInformation Theory Workshop.New York: IEEE Press, 1975, pp.35–36.
 T. M. Cover and R. King, “A convergent gambling estimate of theentropy of English,” IEEE Trans. Inform. Theory, vol. IT-24, pp.413–421, July 1978.
 T. M. Cover, “On the competitive optimality of Huffman code,”IEEETrans. Inform. Theory, vol. 37, pp. 172–174, Jan. 1991.
 , “Universal portfolios,”Math. Finance, vol. 1, no. 1, pp. 1–29,Jan. 1991.
 T. M. Cover and E. Ordentlich, “Universal portfolios with side in-formation,” IEEE Trans. Inform. Theory, vol. 42, pp. 348–363, Mar.1996.
 T. M. Cover and J. A. Thomas,Elements of Information Theory.NewYork: Wiley, 1991.
 I. Csiszar and J. Korner, Information Theory–Coding Theorems forDiscrete Memoryless Systems.New York: Academic, 1981.
 I. Csiszar and P. C. Shields, “Redundancy rates for renewal and otherprocesses,”IEEE Trans. Inform. Theory, vol. 42, pp. 2065–2072, Nov.1996.
 L. D. Davisson, “The prediction error of stationary Gaussian time seriesof unknown covariance,”IEEE Trans. Inform. Theory, vol. IT-11, pp.527–532, Oct. 1965.
 , “Universal noiseless coding,”IEEE Trans. Inform. Theory, vol.IT-19, pp. 783–795, Nov. 1973.
 , “Minimax noiseless universal coding for Markov sources,”IEEETrans. Inform. Theory, vol. IT-29, pp. 211–215, Mar. 1983.
 L. D. Davisson and A. L. Garcia, “A source matching approach tofinding minimax codes,”IEEE Trans. Inform. Theory, vol. IT-26, pp.166–174, Mar. 1980.
 L. D. Davisson, R. J. McEliece, M. B. Pursley, and M. S. Wallace, “Ef-ficient universal noiseless source codes,”IEEE Trans. Inform. Theory,vol. 27, pp. 269–278, May 1981.
 A. P. Dawid, “Present position and potential developments: Somepersonal views on statistical theory the prequential approach (withdiscussion),”J. Roy. Statist. Soc. A, vol. 147, pt. 2, pp. 278–292, 1984.
 , “Fisherian inference in likelihood and prequential frames ofreference (with discussion),”J. Roy. Statist. Soc. B, vol. 53, pp. 79–109,1991.
 , “Prequential data analysis,” inCurrent Issues in StatisticalInference(IMS Lecture Notes—Monograph Series 17), M. Ghosh andP. K. Pathak, Eds., 1992, pp. 113–126.
 A. P. Dawid and V. G. Vovk, “Prequential probability: Principles andproperties.” [Online.] Available WWW: http://www-stat.wharton.upenn.edu/Seq96/members/vovk/index. html.
 M. H. DeGroot,Optimal Statistical Decisions. New York: McGraw-Hill, 1970.
 A. DeSantis, G. Markowsky, and M. Wegman, “Learning probabilis-tic prediction functions,” inProc. 29th IEEE Symp. Foundations ofComputer Science, 1988, pp. 110–119.
 P. Elias, “Minimax optimal universal codeword sets,”IEEE Trans.Inform. Theory, vol. IT-29, pp. 491–502, July 1983.
 M. Feder, “Gambling using a finite state machine,”IEEE Trans. Inform.Theory, vol. 37, pp. 1459–1465, Sept. 1991.
 M. Feder and N. Merhav, “Hierarchical universal coding,”IEEE Trans.Inform. Theory, vol. 42, pp. 1354–1364, Sept. 1996.
 M. Feder, N. Merhav, and M. Gutman, “Universal prediction of indi-vidual sequences,”IEEE Trans. Inform. Theory, vol. 38, pp. 1258–1270,July 1992.
 M. Feder and A. Singer, “Universal data compression and linearprediction,” in Proc. DCC’98, 1998, pp. 511–520.
2146 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 44, NO. 6, OCTOBER 1998
 B. Fittingoff, “Universal methods of coding for the case of unknownstatistics,” in Proc. 5th Symp. Information Theory(Moscow/Gorky,USSR, 1972), pp. 129–135.
 Y. Freund and R. Schapire, “Game theory, on-line prediction andboosting,” inProc. 9th Ann. Workshop Computational Learning Theory,1996, pp. 89–98.
 R. G. Gallager, Information Theory and Reliable Communications.New York: Wiley, 1968.
 , “Source coding with side information and universal coding,”unpublished manuscript; also presented at theInt. Symp. InformationTheory, Oct. 1974.
 D. C. Gilliland, “Sequential compound estimation,”Ann. Math. Statist.,vol. 39, no. 6, pp. 1890–1904, 1968.
 R. M. Gray, Entropy and Information Theory.New York: Springer-Verlag, 1990.
 L. Gyorfi, I. Pali, and E. C. van der Meulen, “There is no universalsource code for an infinite source alphabet,”IEEE Trans. Inform. Theory,vol. 40, pp. 267–271, Jan. 1994.
 J. F. Hannan, “Approximation to Bayes risk in repeated plays,” inContributions to the Theory of Games, Ann. Math. Studies(PrincetonUniv., Princeton, NJ), vol. 3, no. 39, pp. 97–139, 1957.
 J. F. Hannan and H. Robbins, “Asymptotic solutions of the compounddecision problem for two completely specified distributions,”Ann. Math.Statist., vol. 26, pp. 37–51, 1955.
 D. Haussler, “A general minimax result for relative entropy,”IEEETrans. Inform. Theory, vol. 43, pp. 1276–1280, July 1997.
 D. Haussler, J. Kivinen, and M. K. Warmuth, “Sequential prediction ofindividual sequences under general loss functions,”IEEE Trans. Inform.Theory, to be published.
 D. Haussler and M. Opper, “Mutual information, metric entropy andcumulative relative entropy risk,”Ann. Statist., vol. 25, no. 6, Dec.1997, to be published.
 , “General bounds on the mutual information between a pa-rameter andn conditionally independent observations,” inProc. 8thAnnu. Workshop Computational Learning Theory (COLT’95), 1995, pp.402–411.
 D. P. Helmbold and R. E. Schapire, “Predicting nearly as well as thebest pruning of a decision tree,”Mach. Learn., vol. 27, pp. 51–68, 1997.
 Y. Hershkovitz and J. Ziv, “On fixed-database universal data compres-sion with limited memory,”IEEE Trans. Inform. Theory, vol. 43, pp.1966–1976, Nov. 1997.
 H. Jeffreys, “An invariant form of for the prior probability in estimationproblems,”Proc. Roy. Soc. London, vol. 186, pt. A, pp. 453–461, 1946.
 D. Kazakos, “Robust noiseless source coding through a game theoreticapproach,”IEEE Trans. Inform. Theory, vol. IT-29, pp. 576–583, July1983.
 J. L. Kelly, Jr., “A new interpretation of information rate,”Bell Syst.Tech. J., vol. 35, pp. 917–926, 1956.
 J. C. Kieffer, “A unified approach to weak universal source coding,”IEEE Trans. Inform. Theory, vol. IT-24, pp. 674–682, Nov. 1978.
 , “An ergodic theorem for constrained sequences of functions,”Bull. Amer. Math. Soc., vol. 21, pp. 249–253, 1989.
 R. E. Krichevskiy, “Laplace’s law of succession and universal encod-ing,” IEEE Trans. Inform. Theory, vol. 44, pp. 296–303, Jan. 1998.
 R. E. Krichevski and V. E. Trofimov, “The performance of universalencoding,”IEEE Trans. Inform. Theory, vol. IT-27, pp. 199–207, Mar.1981.
 T. L. Lai and H. Robbins, “Asymptotically efficient adaptive allocationrules,” Adv. Appl. Math., vol. 6, pp. 4–22, 1985.
 G. G. Langdon, “A note on the Lempel–Ziv model for compressingindividual sequences,”IEEE Trans. Inform. Theory, vol. IT-29, pp.284–287, 1983.
 P. S. Laplace, “Memoire sur la probabilite des causes par les evene-mens,”Memoires de l’Academie Royale del Sciences, no. 6, pp. 612–656,1774, reprinted inLaplace Complete Work, vol. 8. Paris, France:Gauthier-Villars, pp. 27–65. English translation by S. M. Stigler, 1986.
 , “Memoire sur les approximations des formulas qui sont func-tions de tres grands nombres et sur leur application aux probabilies,”Memoires de l’Academie des Sciences de Paris, pp. 353–415, 559–565,1810; reprinted inLaplace Complete Work,vol. 12. Paris, France:Gauthier-Villars,, pp. 301–353; English translation by S. M. Stigler,1986.
 A. Lempel and J. Ziv, “On the complexity of finite sequences,”IEEETrans. Inform. Theory, vol. IT-22, pp. 75–81, Jan. 1976.
 N. Littlestone, P. Long, and M. K. Warmuth, “On-line learning of linearfunctions,” inProc. 23rd Annu. ACM Symp. Theory of Computing, 1991,pp. 382–391.
 N. Littlestone and M. K. Warmuth, “The weighted majority algorithm,”Inform. Comput., vol. 108, no. 2, pp. 212–261, 1994.
 G. Louchard and W. Szpankowski, “On the average redundancy rate ofthe Lempel–Ziv code,”IEEE Trans. Inform. Theory, vol. 43, pp. 2–8,Jan. 1997.
 T. Matsushima, H. Inazumi, and S. Hirawasa, “A class of distortionlesscodes designed by Bayes decision theory,”IEEE Trans. Inform. Theory,vol. 37, pp. 1288–1293, Sept. 1991.
 R. Meir, “Performance bounds for nonlinear time-series prediction,”1997, preprint.
 R. Meir and N. Merhav, “On the stochastic complexity of learningrealizable and unrealizable rules,”Mach. Learn., vol. 19, no. 3, pp.241–261, 1995.
 N. Merhav and M. Feder, “Universal schemes for sequential decisionfrom individual sequences,”IEEE Trans. Inform. Theory, vol. 39, pp.1280–1292, July 1993.
 , “A strong version of the redundancy-capacity theorem of uni-versal coding,”IEEE Trans. Inform. Theory, vol. 41, pp. 714–722, May1995.
 N. Merhav, M. Feder, and M. Gutman, “Some properties of sequentialpredictors of binary Markov sources,”IEEE Trans. Inform. Theory, vol.39, pp. 887–892, May 1993.
 J. W. Miller, R. Goodman, and P. Smyth, “On loss functions whichminimize to conditional expected values and posterior probabilities,”IEEE Trans. Inform. Theory, vol. 39, pp. 1404–1408, July 1993.
 D. S. Modha and E. Masry, “Universal prediction of stationary randomprocesses,” 1996, preprint.
 G. Morvai, S. J. Yakowitz, and L. Gy¨orfi, “Nonparametric inference forergodic, stationary time series,”Ann. Statist., vol. 24, pp. 370–379, 1996.
 G. Morvai, S. J. Yakowitz, and P. H. Algoet, “Weakly convergent non-parametric forecasting of stationary time series,”IEEE Trans. Inform.Theory, vol. 43, pp. 483–497, Mar. 1997.
 Y. Nogami, “The k-extended set-compound estimation problem innonregular family of distributions,”Ann. Inst. Stat. Math., vol. 31A,pp. 169–176, 1979.
 M. Opper and D. Haussler, “Bounds for predictive errors in thestatistical mechanics of supervised learning,”Phys. Rev. Lett., vol. 75,pp. 3772–3775, 1995.
 , “Worst case prediction over sequences under log loss,” inTheMathematics of Information Coding, Extraction and Distribution, G.Cybenko, D. O’Leary, and J. Rissanen, Eds. New York: Springer-Verlag, 1997.
 D. S. Orenstein, “Guessing the next output of a stationary process,”Israel J. Math., vol. 30, pp. 292–296, 1978.
 A. Papoulis,Probability, Random Variables, and Stochastic Processes,3rd ed. (McGraw-Hill Series in Electrical Engineering). New York:McGraw-Hill, 1991.
 E. Plotnik, M. J. Weinberger, and J. Ziv, “Upper bounds on theprobability of sequences emitted by finite-state sources and on theredundancy of the Lempel–Ziv algorithm,”IEEE Trans. Inform. Theory,vol. 38, pp. 66–72, Jan. 1992.
 J. Rissanen, “Generalized Kraft’s inequality and arithmetic coding,”IBMJ. Res. Develop., vol. 20, no. 3, pp. 198–203, 1976.
 , “Modeling by shortest data description,”Automatica, vol. 14,pp. 465–471, 1978.
 , “Universal coding, information, prediction, and estimation,”IEEE Trans. Inform. Theory, vol. IT-30, pp. 629–636, July 1984.
 , “Complexity of strings in the class of Markov sources,”IEEETrans. Inform. Theory, vol. IT-32, pp. 526–532, 1986.
 , “Fisher information and stochastic complexity,”IEEE Trans.Inform. Theory, vol. 42, pp. 40–47, Jan. 1996.
 J. Rissanen and G. G. Langdon, “Universal modeling and coding,”IEEETrans. Inform. Theory, vol. IT-27, pp. 12–23, Jan. 1984.
 H. Robbins, “Asymptotically subminimax solutions of compound statis-tical decision problems,” inProc. 2nd Berkeley Symp. Math. Stat. Prob.,1951, pp. 131–148.
 R. T. Rockafeller,Convex Analysis. Princeton, NJ: Princeton Univ.Press, 1970.
 B. Ya. Ryabko, “Encoding a source with unknown but ordered proba-bilities,” Probl. Inform. Transm., pp. 134–138, Oct. 1979.
 , “Twice-universal coding,”Probl. Inform. Transm., vol. 20, no.3, pp. 173–177, July/Sept. 1984.
 , “Prediction of random sequences and universal coding,”Probl.Inform. Transm., vol. 24, no. 2, pp. 87–96, Apr./June 1988.
 E. Samuel, “Asymptotic solution of the sequential compound decisionproblem,” Ann. Math. Statist., pp. 1079–1095, 1963.
 , “Convergence of the losses of certain decision rules for thesequential compound decision problem,”Ann. Math. Statist., pp.1606–1621, 1964.
 S. A. Savari, “Redundancy of the Lempel–Ziv incremental parsing rule,”IEEE Trans. Inform. Theory, vol. 43, pp. 9–21, Jan. 1997.
MERHAV AND FEDER: UNIVERSAL PREDICTION 2147
 B. Scarpellini, “Conditional expectations of stationary processes,”Z.Wahrscheinlichkeitstheorie Verw. Gebiete, vol. 56, pp. 427–441, 1981.
 G. Schwarz, “Estimating the dimension of a model,”Ann. Statist., vol.6, no. 2, pp. 461–464, 1978.
 C. E. Shannon, “Prediction and entropy of printed English,”Bell Syst.Tech. J., vol. 30, pp. 5–64, 1951.
 , “The mind reading machine,” inShannon’s Collected Papers,A. D. Wyner and N. J. A. Sloane, Eds. New York: IEEE Press, 1993,pp. 688–689.
 P. C. Shields, “Uniform redundancy rates do not exist,”IEEE Trans.Inform. Theory, vol. 39, pp. 520–524, Mar. 1993.
 P. C. Shields and B. Weiss, “Universal redundancy rates for the classof B-processes do not exist,”IEEE Trans. Inform. Theory, vol. 41, pp.508–512, Mar. 1995.
 N. Shimkin, “Dynamic decision problems in multi-user systems,” Ph.D.dissertation, Technion–Israel Inst. Technol., Haifa, Isreal, Nov. 1991.
 Y. M. Shtar’kov, “Universal sequential coding of single messages,”Problems of Inform. Trans.,vol. 23, no. 3, pp. 175–186, July/Sept. 1987.
 A. Singer and M. Feder, “Universal linear prediction over parametersand model orders,”IEEE Trans. Signal Processing, to be published.
 J.-I. Takeuchi and A. R. Barron, “Asymptotically minimax regret forexponential and curved exponential families,” 1998, preprint.
 J. van Ryzin, “The sequential compound decision problem withm� n
finite loss matrix,”Ann. Math. Statist., vol. 37, pp. 954–975, 1966. S. B. Vardeman, “Admissible solutions ofk-extended finite state set and
the sequence compound decision problems,”J. Multiv. Anal., vol. 10,pp. 426–441, 1980.
 J. S. Vitter, “Optimal prefetching via data cxompression,” inProc. Conf.Foundations of Computer Science, 1991, pp. 121–130.
 V. G. Vovk, “Aggregating strategies,” inProc. 3rd Annu. WorkshopComputational Learning Theory(San Mateo, CA, 1990), pp. 371–383.
 , “A game of prediction with expert advice,” inProc. 3rd Annu.Workshop Computational Learning Theory(New York, NY, 1995), pp.51–60.
 M. J. Weinberger, N. Merhav, and M. Feder, “Optimal sequentialprobability assignment for individual sequences,”IEEE Trans. Inform.Theory, vol. 40, pp. 384–396, Mar. 1994.
 M. J. Weinberger, J. Rissanen, and M. Feder, “A universal finite memorysource,”IEEE Trans. Inform. Theory, vol. 41, pp. 643–652, May 1995.
 N. Wiener,Extrapolation, Interpolation, and Smoothing of StationaryTime Series. Cambridge, MA: MIT Press, 1949.
 F. M. J. Willems, Y. M. Shtarkov, and T. Tjalkens, “The context-treeweighting method: Basic properties,”IEEE Trans. Inform. Theory, vol.41, pp. 653–664, May 1995.
 Q. Xie and A. R. Barron, “Asymptotic minimax regret for data com-pression, gambling, and prediction,”IEEE Trans. Inform. Theory, 1996,submitted for publication.
 , “Minimax redundancy for the class of memoryless sources,”IEEE Trans. Inform. Theory, vol. 43, pp. 646–657, Mar. 1997.
 B. Yu, “Lower bounds on expected redundancy for nonparametricclasses,”IEEE Trans. Inform. Theory, vol. 42, pp. 272–275, Jan. 1996.
 J. Ziv, “Coding of sources with unknown statistics—Part I: Probabilityof encoding error,”IEEE Trans. Inform. Theory, vol. IT-18, pp. 384–394,May 1972.
 J. Ziv and A. Lempel, “A universal algorithm for sequential datacompression,”IEEE Trans. Inform. Theory, vol. IT-23, pp. 337–343,July 1977.
 , “Compression of individual sequences via variable-rate coding,”IEEE Trans. Inform. Theory, vol. IT-24, pp. 530–536, Sept. 1978.
 , “Universal coding of two-dimensional data,”IEEE Trans. In-form. Theory, vol. IT-32, pp. 2–8, Jan. 1986.