1

What is the Philosophy of Statistics? At one level of analysis at least, statisticians and philosophers of science ask many of the same questions:

What should be observed and what may justifiably be inferred from the resulting data? How well do data confirm or fit a model? What is a good test? Must predictions be “novel” in some sense? (selection effects, double counting, data mining) How can spurious relationships be distinguished from genuine regularities? from causal regularities?

How can we infer more accurate and reliable

observations from less accurate ones? When does a fitted model account for regularities in the

data?

That these very general questions are entwined with long standing debates in philosophy of science helps to explain why the field of statistics tends to cross over so often into philosophical territory.

2

Statistics Philosophy 3 ways statistical accounts are used in philosophy of science (1) Model Scientific Inference — to capture either the actual or rational ways to arrive at evidence and inference

(2) Resolve Philosophical Problems about scientific inference, observation, experiment;

(problem of induction, objectivity of observation, reliable evidence, Duhem's problem, underdetermination).

(3) Perform a Metamethodological Critique -scrutinize methodological rules, e.g., accord special weight to "novel" facts, avoid ad hoc hypotheses, avoid "data mining", require randomization. Philosophy Statistics

Central job to help resolve the conceptual, logical, and methodological discomforts of scientists as to: how to make reliable inferences despite uncertainties and errors?

Philosophy of statistics and the goal of a philosophy of science relevant for philosophical problems in scientific practice

3

Fresh methodological problems arise in practice surrounding a panoply of methods and models relied on to learn from incomplete, and often non-experimental, data.

Examples abound:

Disputes over hypothesis-testing in psychology (e.g., the recently proposed “significance test ban”);

Disputes over the proper uses of regression in applied statistics;

Disputes over dose-response curves in estimating risks;

Disputes about the use of computer simulations in “observational” sciences;

Disputes about “external validity” in experimental economics; and,

Across the huge landscape of fields using the latest, high-powered, computer methods, there are disputes about data-mining, algorithmic searches, and model validation.

Equally important are the methodological presuppositions that are not, but perhaps ought to be, disputed, debated, or at least laid out in the open — often, ironically, in the very fields in which philosophers of science immerse themselves.

4

I used to teach a course in this department: philosophy of science and economic methodology

We read how many economic methodologists questioned the value of philosophy of science

“If philosophers and others within science theory can’t agree about the constitution of the scientific method (or even whether asking about a scientific ‘method’ makes any sense), doesn’t it seem a little dubious for economists to continue blithely taking things off the shelf and attempting to apply them to economics?” (Hands, 2001, p. 6).

Deciding that it is, methodologists of economics increasingly look to sociology of science, rhetoric, evolutionary psychology.

The problem is not merely how this cuts philosophers of science out of being engaged in methodological practice; equally serious, is how it encourages practitioners to assume there are no deep epistemological problems with the ways they collect and base inferences on data.

5

“Professional agreement on statistical philosophy is not on the immediate horizon, but this should not stop us from agreeing on methodology”, as if “what is correct methodologically” does not depend on “what is correct philosophically” (Berger, 2003, p. 2).

In addition to the resurgence of the age-old controversies — significance test vs. confidence intervals, frequentist vs. Bayesian measures, — the latest statistical modeling techniques have introduced brand new methodological issues.

High-powered computer science packages offer a welter of algorithms for “automatically” selecting among this explosion of models, but as each boasts different, and incompatible, selection criteria, we are thrown back to the basic question of inductive inference: what is required, to severely discriminate among well-fitting models such that, when a claim (or hypotheses or model) survives a test the resulting data count as good evidence for the claim’s correctness or dependability or adequacy.

6

A romp through 4 "waves in philosophy of statistics" History and philosophy of statistics is a huge territory marked by 70 years of debates widely known for reaching unusual heights both of passion and of technical complexity. Wave I ~ 1930 –1955/60

Wave II~ 1955/60-1980

Wave III~1980-2005 & beyond

Wave IV ~ 2006 and beyond

7

A core question: What is the nature and role of

probabilistic concepts, methods, and models in making inferences in the face of limited data, uncertainty and error?

1. Two Roles For Probability:

Degrees of Confirmation and Degrees of Well-Testedness

a. To provide a post-data assignment of degree of probability, confirmation, support or belief in a hypothesis;

b. To assess the probativeness, reliability, trustworthiness, or severity of a test or inference procedure.

These two contrasting philosophies of the role of probability in statistical inference are very much at the heart of the central points of controversy in the “three waves” of philosophy of statistics…

8

Having conceded loss in the battle for justifying induction, philosophers appeal to logic to capture scientific method Inductive Logics Logic of falsification “Confirmation Theory” Rules to assign degrees of probability or confirmation to hypotheses given evidence e

Methodological falsification Rules to decide when to “prefer” or accept hypotheses

Carnap C(H,e)

Popper

Inductive Logicians we can build and try to justify “inductive logics” straight rule: Assign degrees of confirmation/credibility

Statistical affinity Bayesian (and likelihoodist) accounts

Deductive Testers we can reject induction and uphold the “rationality” of preferring or accepting H if it is “well tested”

Statistical affinity

Fisherian, Neyman-Pearson methods: probability enters to ensure reliability and severity of tests with these methods.

9

I. Philosophy of Statistics: “The First Wave”

WAVE I: circa 1930-1955: Fisher, Neyman, Pearson, Savage, and Jeffreys.

Statistical inference tools use data x0 to probe aspects of the data generating source:

In statistical testing, these aspects are in terms of statistical hypotheses about parameters governing a statistical distribution

H tells us the “probability of x under H”, written P(x;H)

(probabilistic assignments under a model)

Important to avoid confusion with conditional probabilities in Bayes’s theorem, P(x|H).

Testing model assumptions extremely important, though will not discuss.

10

Modern Statistics Begins with Fisher: “Simple” Significance Tests

Example. Let the sample be X = (X1, …,Xn), be IID from a

Normal distribution (NIID) with =1.

1. A null hypothesis H0: H0: = 0

e.g., 0 mean concentration of lead, no difference in mean survival in a given group, in mean risk, mean deflection of light.

2. A function of the sample, d(X), the test statistic: which reflects the difference between the data x0 = (x1, …,xn), and H0;

The larger d(x0) the further the outcome is from what is expected under H0, with respect to the particular question being asked.

3. The p-value is the probability of a difference larger than d(x0), under the assumption that H0 is true:

p(x0)=P(d(X) > d(x0); H0)

11

The observed significance level (p-value) with observed X = .1

p(x0)=P(d(X) > d(x0); H0).

The relevant test statistic d(X) is:

d(X) = ( X -0x,

where X is the sample mean with standard deviation x = (√n).

0Observed - Expected (under H )( )x

d

X

Since x n = 1/5 = .2, d(X) = .1 – 0 in units of x

yields

d(x0)=.1/.2 = .5

Under the null, d(X) is distributed as standard Normal, denoted by d(X) ~ N(0,1).

(Area to the right of .5) ~.3, i.e. not very significant.

12

Logic of Simple Significance Tests: Statistical Modus

Tollens

“Every experiment may be said to exist only in order to

give the facts a chance of disproving the null hypothesis”

(Fisher, 1956, p.160).

Statistical analogy to the deductively valid pattern modus

tollens:

If the hypothesis H0 is correct then, with high probability, 1-p, the data would not be statistically significant at level p.

x0 is statistically significant at level p. ____________________________

Thus, x0 is evidence against H0, or x0 indicates the falsity of H0.

Fisher described the significance test as a procedure for rejecting the null hypothesis and inferring that the phenomenon has been “experimentally demonstrated” once one is able to generate “at will” a statistically significant effect. (Fisher, 1935a, p. 14),

13

The Alternative or “Non-Null” Hypothesis

Evidence against H0 seems to indicate evidence for

some alternative.

Fisherian significance tests strictly consider only the

H0

Neyman and Pearson (N-P) tests introduce an

alternative H1 (even if only to serve as a direction of

departure).

Example. X = (X1, …,Xn), NIID with =1:

H0: = 0 vs. H1: > 0

Despite the bitter disputes with Fisher that were to

erupt soon after ~1935, Neyman and Pearson, at first saw

their work as merely placing Fisherian tests on firmer

logical footing.

Much of Fisher’s hostility toward N-P methods

reflects professional and personality conflicts more than

philosophical differences.

14

Neyman-Pearson (N-P) Tests

N-P hypothesis test: maps each outcome x = (x1, …,xn) into either the null hypothesis H0, or an alternative hypothesis H1 (where the two exhaust the parameter space) to ensure the probabilities of erroneous rejections (type I errors) and erroneous acceptances (type II errors) are controlled at prespecified values, e.g., 0.05 or 0.01, the significance level of the test.

Test T(: X = (X1, …,Xn), NIID with =1,

H0: = vs. H1: >

■ if d(x0) > c, "reject" H0, (or declare the result statistically significant at the level);

■ if d(x0) < c, "accept" H0, e.g. c=1.96 for =.025, i.e. “Accept/Reject” uninterpreted parts of the mathematical apparatus.

Type I error probability = P(d(x0) > c; H0) ≤ The Type II error probability:

P(Test T( does not reject H0 ; =1) =

= P(d(X) < c; H0) = ß(1), for any 1 > 0.

15

The "best" test at level at the same time minimizes the value of ß for all 1 > 0, or equivalently, maximizes the power:

POW(T(; 1)= P(d(X) > c; 1

T( is a Uniformly Most Powerful (UMP) level test

16

Inductive Behavior Philosophy Philosophical issues and debates arise once one begins to consider the interpretations of the formal apparatus ‘Accept/Reject’ are identified with deciding to take specific actions, e.g., publishing a result, announcing a new effect. The justification for optimal tests is that “it may often be proved that if we behave according to such a rule ... we shall reject H when it is true not more, say, than once in a hundred times, and in addition we may have evidence that we shall reject H sufficiently often when it is false.” Neyman: Tests are not rules of inductive inference but rules of behavior: The goal is not to adjust our beliefs but rather to “adjust our behavior” to limited amounts of data Is he just drawing a stark contrast between N-P tests and Fisherian as well as Bayesian methods? Or is the behavioral interpretation essential to the tests?

17

“Inductive behavior” vs. “Inductive inference”

battle commingles philosophical, statistical and personality

clashes. Fisher (1955) denounced the way that Neyman and

Pearson transformed ‘his’ significance tests into ‘acceptance procedures’.

They’ve turned my tests into mechanical rules or ‘recipes’ for ‘deciding’ to accept or reject statistical hypothesis H0,

The concern has more to do with speeding up production or making money than in learning about phenomena

18

N-P followers are like: “Russians (who) are made familiar with the ideal

that research in pure science can and should be geared to technological performance, in the comprehensive organized effort of a five-year plan for the nation.” (1955, 70)

“In the U.S. also the great importance of

organized technology has I think made it easy to confuse the process appropriate for drawing correct conclusions, with those aimed rather at…speeding production, or saving money”.

19

Pearson distanced himself from Neyman’s

“inductive behavior” jargon, calling it “Professor Neyman’s field rather than mine”.

But the most impressive mathematical results were in

the decision-theoretic framework of Neyman-Pearson-Wald.

Many of the qualifications by Neyman and Pearson

in the first wave are overlooked in the philosophy of statistics literature.

Admittedly, these “evidential” practices were not

made explicit *. (Had they been, the subsequent waves of philosophy of statistics might have looked very different).

*Mayo’s goal in ~ 1978

20

The Second Wave: ~1955/60 -1980

“Post-data criticisms of N-P methods”: Ian Hacking (1965), framed the main lines of criticism by philosophers “Neyman-Pearson tests as suitable for before-trial betting, but not for after-trial evaluation.” (p. 99):

Battles: “initial precision vs. final precision”,

“before-data vs. after data”

After the data, he claimed, the relevant measure of support is the (relative) likelihood

Two data sets x and y may afford the same "support" to H, yet warrant different inferences [on significance test reasoning] because x and y arose from tests with different error probabilities. o This is just what error statisticians want!

21

o But (at least early on) Hacking (1965) held to the

“Law of Likelihood”: x0 support hypotheses H1 more than H2 if,

P(x0;H1) > P(x0;H2).

Yet, as Barnard notes, “there always is such a rival hypothesis: That things just had to turn out the way they actually did” . Since such a maximally likelihood alternative H2 can always be constructed, H1 may always be found less well supported, even if H1 is true—no error control Hacking soon rejected the likelihood approach on such grounds, likelihoodist accounts are advocated by others.

22

Perhaps THE key issue of controversy in the

philosophy of statistics battles The (strong) likelihood principle, likelihoods suffice to convey “all that the data have to say” —

According to Bayes’s theorem, P(x|µ) ... constitutes the entire evidence of the experiment, that is, it tells all that the experiment has to tell. More fully and more precisely, if y is the datum of some other experiment, and if it happens that P(x|µ) and P(y|µ) are proportional functions of µ (that is, constant multiples of each other), then each of the two data x and y have exactly the same thing to say about the values of µ… (Savage 1962, p. 17.)

—the error probabilist needs to consider, in addition, the sampling distribution of the likelihoods. —significance levels and other error probabilities all violate the likelihood principle (Savage 1962).

23

Paradox of Optional Stopping

Instead of fixing the same size n in advance, in some tests, n is determined by a stopping rule:

In Normal testing, 2-sided H0: = 0 vs. H1: ≠ 0

Keep sampling until H is rejected at the .05 level

(i.e., keep sampling until | X | 1.96 / n ). Nominal vs. Actual significance levels: with n fixed the type 1 error probability is .05. With this stopping rule the actual significance level differs from, and will be greater than .05. By contrast, since likelihoods are unaffected by the stopping rule, the LP follower denies there really is an evidential difference between the two cases (i.e., n fixed and n determined by the stopping rule). Should it matter if I decided to toss the coin 100 times and happened to get 60% heads, or if I decided to keep tossing until I could reject at the .05 level (2-sided) and this happened to occur on trial 100? Should it matter if I kept going until I found statistical significance? Error statistical principles: Yes! — penalty for perseverance! The LP says NO!

24

Savage Forum 1959: Savage audaciously declares that the lesson to draw from the optional stopping effect is that “optional stopping is no sin” so the problem must lie with the use of significance levels. But why accept the likelihood principle (LP)? (simplicity and freedom?)

The likelihood principle emphasized in Bayesian statistics implies, … that the rules governing when data collection stops are irrelevant to data interpretation. It is entirely appropriate to collect data until a point has been proved or disproved (p. 193)…This irrelevance of stopping rules to statistical inference restores a simplicity and freedom to experimental design that had been lost by classical emphasis on significance levels (in the sense of Neyman and Pearson) (Edwards, Lindman, Savage 1963, p. 239).

For frequentists this only underscores the point raised years before by Pearson and Neyman:

A likelihood ratio (LR) may be a criterion of relative fit but it “is still necessary to determine its sampling distribution in order to control the error involved in rejecting a true hypothesis, because a knowledge of [LR] alone is not adequate to insure control of this error (Pearson and Neyman, 1930, p. 106).

25

The key difference: likelihood fixes the actual outcome, i.e., just d(x), while error statistics considers outcomes other than the one observed in order to assess the error properties

LP irrelevance of, and no control over, error probabilities.

("why you cannot be just a little bit Bayesian" EGEK 1996)

Update: A famous argument (1962, Birnbaum) purports to show that plausible error statistical principles entails the LP!

"Radical!" "Breakthrough!" (since the LP entails the irrelevance of error probabilities!

But the "proof" is flawed! (Mayo 2010 See blog).

26

The Statistical Significance Test Controversy

(Morrison and Henkel, 1970) – contributors chastise social

scientists for slavish use of significance tests

o Focus on simple Fisherian significance tests o Philosophers direct criticisms mostly to N-P tests.

Fallacies of Rejection: Statistical vs. Substantive Significance

(i) take statistical significance as evidence of substantive theory that explains the effect

(ii) Infer a discrepancy from the null beyond what the test warrants

(i) Paul Meehl: It is fallacious to go from a statistically

significant result, e.g., at the .001 level, to infer that “one’s substantive theory T, which entails the [statistical] alternative H1, has received .. quantitative support of magnitude around .999”

A statistically significant difference (e.g., in child rearing) is not automatically evidence for a Freudian theory.

T is subjected to only “a feeble risk”, violating Popper.

27

Fallacies of rejection: (i) Take statistical significance as evidence of

substantive theory that explains the effect

(ii) Infer a discrepancy from the null beyond what the test warrants.

Finding a statistically significant effect, d(x0) > c (cut-off for rejection) need not be indicative of large or meaningful effect sizes — test too sensitive

Large n Problem: an significant rejection of H0 can be

very probable, even with a substantively trivial discrepancy from H0 can

This is often taken as a criticism because it is assumed that statistical significance at a given level is more evidence against the null the larger the sample size (n) — fallacy!

"The thesis implicit in the [NP] approach [is] that a hypothesis may be rejected with increasing confidence or reasonableness as the power of the test increases (Howson and Urbach 1989 and later editions)

In fact, it is indicative of less of a discrepancy from the null than if it resulted from a smaller sample size.

28

(analogy with smoke detector: an alarm from one that often goes off from merely burnt toast (overly powerful or sensitive), vs. alarm from one that rarely goes off unless the house is ablaze)

Comes also in the form of the “Jeffrey-Good-Lindley” paradox

Even a highly statistically significant result can, with n sufficiently large, correspond to a high posterior probability to a null hypothesis.

29

Fallacy of Non-Statistically Significant Results

Test T() fails to reject the null, when the test statistic

fails to reach the cut-off point for rejection, i.e., d(x0) ≤ c .

A classic fallacy is to construe such a “negative” result as evidence FOR the correctness of the null hypothesis (common in risk assessment contexts).

“No evidence against” is not “evidence for” Merely surviving the statistical test is too easy, occurs too frequently, even when the null is false.

— results from tests lacking sufficient sensitivity or

power.

The Power Analytic Movement of the 60’s in psychology Jacob Cohen: By considering ahead of time the Power of

the test, select a test capable of detecting discrepancies of interest.

– pre-data use of power (for planning).

30

A multitude of tables were supplied (Cohen, 1988), but

until his death he bemoaned their all-to-rare use.

(Power is a feature of N-P tests, but apparently the prevalence of Fisherian tests in the social sciences, coupled, perhaps, with the difficulty in calculating power, resulted in ignoring power. There was also the fact that they were not able to get decent power in psychology; they turned to meta-analysis)

31

Post-data use of power to avoid fallacies of insensitive tests

If there's a low probability of a statistically significant

result, even if a non-trivial discrepancy non-trivial is present (low

power against non-trivial) ) then a non-significant difference is not good evidence that a non-trivial discrepancy is absent.

Still too course: power is always calculated relative to the cut-off point c for rejecting H0.

Consider test T() , = 1, n = 25, and let

non-trivial = .2

No matter what the non-significant outcome, power to detect

non-trivial is only .16!

So we’d have to deny the data were good evidence that < .2

This suggested to me (in writing my dissertation around 1978) that rather than calculating

(1) P(d(X) > c; =.2) Power

one should calculate

(2) P(d(X) > d(x0); =.2). observed power (severity)

Even if (1) is low, (2) may be high. We return to this in

the developments of Wave III.

32

III. The Third Wave: Relativism, Reformulations,

Reconciliations ~1980-2005+

(skip) Rational Reconstruction and Relativism in Philosophy of Science

Fighting Kuhnian battles to the very idea of a unified method of scientific inference, statistical inference less prominent in philosophy — largely used rational reconstructions of scientific episodes, — in appraising methodological rules, — in classic philosophical problems e.g., Duhem’s

problem—reconstruct a given assignment of blame so as to be “warranted” by Bayesian probability assignments.

no normative force. The recognition that science involves subjective judgments and values, reconstructions often appeal to a subjective Bayesian account (Salmon’s “Tom Kuhn Meets Tom Bayes”). (Kuhn thought this was confused: no reason to suppose an algorithm remains through theory change) Naturalisms, HPS

33

Wave III in Scientific Practice — Statisticians turn to eclecticism.

— Non-statistician practitioners (e.g., in psychology,

ecology, medicine), bemoan “unholy hybrids” a mixture of ideas from N-P methods, Fisherian tests, and

Bayesian accounts that is “inconsistent from both perspectives and burdened with conceptual confusion”. (Gigerenzer, 1993, p. 323).

Faced with foundational questions, non statistician practitioners raise anew the questions from the first and second waves.

Finding the automaticity and fallacies still rampant, most, if they are not calling for an outright “ban” on significance tests in research, insist on reforms and reformulations of statistical tests.

Task Force to consider Test Ban in Psychology: 1990s

34

Reforms and Reinterpretations Within Error Probability Statistics

Any adequate reformulation must:

(i) Show how to avoid classic fallacies (of acceptance and of rejection) —on principled grounds,

(ii) Show that it provides an account of inductive inference

35

Avoiding Fallacies

To quickly note my own recommendation (for test T(a)): Move away from coarse accept/reject rule; use specific result (significant or insignificant) to infer those discrepancies from the null that are well ruled-out, and those which are not. e.g., Interpretation of Non-Significant results:

If d(x) is not statistically significant, and the test had a very high probability of a more statistically significant difference if µ > µ0 + , then d(x) is good grounds for inferring µ ≤ µ0 + .

Use specific outcome to infer an upper bound µ ≤ µ* (values beyond are ruled out by given severity.) If d(x) is not statistically significant, but the test had a very low probability chance of a more

statistically significant difference if µ > µ0 + , then d(x) is poor evidence for inferring µ ≤ µ0 +

. The test had too little probative power to have detected such discrepancies even if they existed!

36

Takes us back to the post-data version of power:

Rather than construe “a miss as good as a mile”, parity of logic suggests that the post-data power assessment should

replace the usual calculation of power against :

POW(T(), ) = P(d(X) > c; =),

with what might be called the power actually attained or, to have a distinct term, the severity (SEV):

SEV(T(), ) = P(d(X) > d(x0); =),

where d(x0) is the observed (non-statistically significant) result.

37

Figure 1 compares power and severity for different

outcomes

Figure 1. POW(T(), =.2) =.168, irrespective of the value of d(x0) ; solid curve, the severity evaluations are data-specific: The severity for the inference: <

Both X = .39, and X = -.2, fail to reject H0, but

But with X = .39, SEV( < is low (.17)

But with X = -.2, SEV( < is high (.97)

38

Fallacies of Rejection: The Large n-Problem

While with a nonsignificant result, the concern is erroneously inferring that a discrepancy from µ0 is absent; With a significant result x0, the concern is erroneously inferring that it is present.

Utilizing the severity assessment an -significant difference with n1 passes µ > µ1 less severely than with n2 where n1 > n2.

Figure 2 compares test T(with three different sample sizes:

n = 25, n = 100, n = 400, denoted by T(n; where in each case d(x0) = 1.96 – reject at the cut-off

point.

In this way we solve the problems of tests too sensitive or not sensitive enough, but there’s one more thing ... showing how it supplies an account of inductive inference

Many argue in wave III that error statistical methods cannot supply an account of inductive inference because error probabilities conflict with posterior probabilities.

39

Figure 2 compares test T(with three different sample sizes:

n =25, n =100, n =400, denoted by T(n; in each case d(x0) = 1.96 – reject at the cut-off point.

Figure 2. In test T( (H0: < 0 against H1: > 0, and = 1), , c = 1.96 and d(x0) = 1.96. The severity for the inference: >

n = 25, SEV( > is .93 n = 100, SEV( > is .83 n = 400, SEV( > is .5

40

P-values vs. Bayesian Posteriors

A statistically significant difference from H0 can correspond to large posteriors in H0 . From the Bayesian perspective, it follows that p-values come up short as a measure of inductive evidence,

the significance testers balk at the recommended priors resulting in highly significant results being construed as no evidence against the null — or even evidence for it!

The conflict often considers the two sided T(2 test

H0: = vs. H1: ≠ .

(The difference between p-values and posteriors are far less marked with one-sided tests).

“Assuming a prior of .5 to H0, with n = 50 one can classically ‘reject H0 at significance level p = .05,’ although P(H0|x) = .52 (which would actually indicate that the evidence favors H0).” This is taken as a criticism of p-values, only because, it is assumed the .51 posterior is the appropriate measure of the beliefworthiness.

41

As the sample size increases, the conflict becomes

more noteworthy.

If n = 1000, a result statistically significant at the .05 level leads to a posterior to the null of .82!

SEV (H1) = .95 while the corresponding posterior has gone from .5 to .82. What warrants such a prior?

n (sample size) ______________________________________________________ p t n=10 n=20 n=50 n=100 n=1000 .10 1.645 .47 .56 .65 .72 .89 .05 1.960 .37 .42 .52 .60 .82 .01 2.576 .14 .16 .22 .27 .53 .001 3.291 .024 .026 .034 .045 .124

(1) Some claim the prior of .5 is a warranted frequentist assignment:

H0 was randomly selected from an urn in which 50% are true

(*) Therefore P(H0) = p

42

H0 may be 0 change in extinction rates, 0 lead concentration, etc. What should go in the urn of hypotheses?

For the frequentist: either H0 is true or false the probability in (*) is fallacious and results from an unsound instantiation. We are very interested in how false it might be, which is what we can do by means of a severity assessment.

(2) Subjective degree of belief assignments will not ensure the error probability, and thus the severity assessments we need.

(3) Some suggest an “impartial” or “uninformative” Bayesian prior gives .5 to H0, the remaining .5 probability being spread out over the alternative parameter space, Jeffreys. This “spiked concentration of belief in the null” is at odds with the prevailing view “we know all nulls are false”. The “Bayesian” recently co-opts 'error probability' to describe a posterior, but it is not a frequentist error probability which is measuring something very different.

43

Fisher: The Function of the p-Value Is Not Capable of Finding Expression

Faced with conflicts between error probabilities and Bayesian posterior probabilities, the error probabilist would conclude that the flaw lies with the latter measure.

Fisher: Discussing a test of the hypothesis that the stars are distributed at random, Fisher takes the low p-value (about 1 in 33,000) to “exclude at a high level of significance any theory involving a random distribution” (Fisher, 1956, page 42).

Even if one were to imagine that H0 had an extremely high prior probability, Fisher continues — never minding “what such a statement of probability a priori could possibly mean”— the resulting high posteriori probability to H0, he thinks, would only show that “reluctance to accept a hypothesis strongly contradicted by a test of significance” (ibid, page 44) . . . “is not capable of finding expression in any calculation of probability a posteriori” (ibid, page 43).

44

Wave IV? 2006+ The Reference Bayesians Abandon Coherence, the LP, and strive to match frequentist error probabilities! Contemporary “Impersonal” Bayesianism Because of the difficulty of eliciting subjective priors, and because of the reluctance among scientists to allow subjective beliefs to be conflated with the information provided by data, much current Bayesian work in practice favors conventional “default”, “uninformative,” or “reference”, priors . 1. What do reference posteriors measure?

A classic conundrum: there is no unique “noninformative” prior. (Supposing there is one leads to inconsistencies in calculating posterior marginal probabilities).

Any representation of ignorance or lack of information that succeeds for one parameterization will, under a different parameterization, entail having knowledge.

Contemporary “reference” Bayesians seeks priors that are simply conventions to serve as weights for reference posteriors.

45

not to be considered expressions of uncertainty, ignorance, or degree of belief.

may not even be probabilities; flat priors may not sum to one (improper prior). If priors are not probabilities, what then is the interpretation of a posterior? (a serious problem I would like to see Bayesian philosophers tackle).

2. Priors for the same hypothesis changes according to what experiment is to be done! Bayesian incoherence

If the prior is to represent information why should it be influenced by the sample space of a contemplated experiment? Violates the likelihood principle — the cornerstone of Bayesian coherency Reference Bayesians: it is “the price” of objectivity.

— seems to wreck havoc with basic Bayesian foundations, but without the payoff of an objective, interpretable output — even subjective Bayesians object

46

3. Reference posteriors with good frequentist properties Reference priors are touted as having some good frequentist properties, at least in one-dimensional problems. They are deliberately designed to match frequentist error probabilities.

If you want error probabilities, why not use techniques that provide them directly?

Note: using conditional probability — which is part and parcel of probability theory, as in “Bayes nets” does not make one a Bayesian —no priors to hypotheses…