# Conditional & Joint Probability

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23-Feb-2016Category

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Conditional & Joint ProbabilityA brief digression back to joint probability: i.e. both events O and H occurAgain, we can express joint probability in terms of their separate conditional and unconditional probabilitiesThis key result turns out to be exceedingly useful!Conditional ProbabilityConverting expressions of joint probabilityWe can therefore express everything only in terms of reciprocal conditional and unconditional probabilities:The intersection operator makes no assertion regarding order:This is usually expressed in a slightly rearranged formConditional ProbabilityBayes theorem expresses the essence of inference We can think of this as allowing us to compute the probability of some hidden event H given that some observable event O has occurred, provided we know the probability of the observed event O assuming that hidden event H has occurred Bayes theorem is a recipe for problems involving conditional probabilityConditional ProbabilityNormalizing the probabilitiesFor convenience, we often replace the probability of the observed event O with the sum over all possible values of H of the joint probabilities of O and H. Whew! But consider that if we now calculated Pr{H|O} for every H, the sum of these would be one, which is just as a probability should behave This summing of the expression in numerator normalizes the probabilities Conditional ProbabilityBayes theorem is so important that each part of this recipe has a special nameBayes theorem as a recipe for inference The posteriorThink of this perhaps as the evidence for some specific model H given the set of observations O.We are making an inference about H on the basis of O The priorOur best guess about H before any observation is made. Often we will make neutral assumptions, resulting in an uninformative prior. But priors can also come from the posterior results from some earlier experiment. How best to choose priors is an essential element of Bayesian analysis The likelihood modelWeve seen already that the probability of an observation given a hidden parameter is really a likelihood. Choosing a likelihood model is akin to proposing some process, H, by which the observation, O, might have come about The observable probabilityGenerally, care must be taken to ensure that our observables have no uncertainty, otherwise they are really hidden!!! Many paths give rise to the same sequence XThe Backward Algorithm P(x) = We would often like to know the total probability of some sequence:But wait! Didnt we already solve this problem using the forward algorithm!? Sometimes, the trip isnt about the destination. Stick with me!Well, yes, but were going to solve it again by iterating backwards through the sequence instead of forwardsDefining the backward variableThe Backward Algorithm Since we are effectively stepping backwards through the event sequence, this is formulated as a statement of conditional probability rather than in terms of joint probability as are forward variables bk(i) = P(xi+1 xL | pi = k) The backward variable for state k at position i the probability of the sequence from the end to the symbol at position i, given that the path at position i is k What if we had in our possession all of the backward variables for 1, the first position of the sequence?The Backward Algorithm Well obtain these initial position backward variables in a manner directly analogous to the method in the forward algorithm To get the overall probability of the sequence, we would need only sum the backward variables for each state (after correcting for the probability of the initial transition from Start) P(X) = A recursive definition for backward variablesThe Backward Algorithm As always with a dynamic programming algorithm, we recursively define the variables.. But this time in terms of their own values at later positions in the sequence The termination condition is satisfied and the basis case provided by virtue of the fact that sequences are finite and in this case we must eventually (albeit implicitly) come to the End state bk(i-1) = bk(i) = P(xi+1 xL | pi = k) 9If you understand forward, you already understand backward!The Backward Algorithm Initialization: bk(L) = 1 for all states k We usually dont need the termination step, except to check that the result is the same as from forward So why bother with the backward algorithm in the first place?Recursion (i = L-1,,1):Termination: P(X) = bk(i) = 10The Backward Algorithm The backward algorithm takes its name from its backwards iteration through the sequence S0.10.9State +State -A: 0.30C: 0.25G: 0.15T: 0.300.50.40.50.6A: 0.20C: 0.35G: 0.25T: 0.20The probability of a sequence summed across all possible state paths EndACG _ 110.20.6 * 0.15 * 10.4 * 0.25 * 1S = 0.190.6 * 0.25 * 0.20.4 * 0.35 * 0.19S = 0.05660.5 * 0.25 * 0.20.5 * 0.35 * 0.19S = 0.058250.1 * 0.3 * 0.058250.9 * 0.2 * 0.0566S = 0.0119355P(x) = 0.01193550.5 * 0.25 * 10.5 * 0.15 * 1S = 0.211The most probable state pathThe Decoding Problem The Viterbi algorithm will calculate the most probable state path, thereby allowing us to decode the state path in cases where the true state path is hidden The Viterbi algorithm generally does a good job of recovering the state pathp*= argmax P(x, p)p_64622311445316146343245452316262524425613233524355442454134246666215124646526536662616666426446665162436615266416215651SFFFFFFFFFFFFFLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLFFFFFFFFFSFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLLSequence

True state path

Viterbi patha.k.a MPSPp*= argmax P(x, p | )p12Limitations of the most probable state pathThe Decoding Problem but the most probable state path might not always be the best choice for further inference on the sequence This is the posterior probability of state k at position i when sequence x is knownP(pi = k|x)There may be other paths, sometimes several, that result in probabilities nearly as good as the MPSPThe MPSP tells us about the probability of the entire path, but it doesnt actually tell us what the most probable state might be for some particular observation xiMore specifically, we might want to know: the probability that observation xi resulted from being in state k, given the observed sequence x13The approach is a little bit indirect.Calculating the Posterior ProbabilitiesWe can sometimes ask a slightly different or related question, and see if that gets us closer to the result we seek Does anything here look like something we may have seen before??P(x,pi = k)Maybe we can first say something about the probability of producing the entire observed sequence with observation xi resulting from having been in state k.. P(x1xi, pi = k)P(xi+1xL|x1xi, pi = k) ==P(x1xi, pi = k)P(xi+1xL|pi = k) 14Calculating the Posterior Probabilities These terms are exactly our now familiar forward and backward variables!P(x,pi = k)=P(x1xi, pi = k)P(xi+1xL|pi = k) Limitations of the most probable state path=fk(i)bk(i) 15Calculating the Posterior ProbabilitiesOK, but what can we really do with these posterior probabilities?P(x,pi = k)Putting it all together using Bayes theorem fk(i)bk(i) P(pi = k|x) = P(x)We now have all of the necessary ingredients required to apply Bayes theorem We can now therefore find the posterior probability of being in state k for each position i in the sequence! We need only run both the forward and the backward algorithms to generate the values we need. Python: we probably want to store our forward and backward values as instance variables rather than method variables Remember, we can get P(x) directly from either the forward or backward algorithm!P(x|pi = k)P(pi = k)16Making use of the posterior probabilities Posterior Decoding Two primary applications of the posterior state path probabilitiesIn some scenarios, the overall path might not even be a permitted path through the model! We can define an alternative state path to the most probable state path:pi = argmax P(pi = k | x)kThis alternative to Viterbi decoding is most useful when we are most interested in the what the state might be at some particular point or points. Its possible that the overall path suggested by this might not be particularly likely17Plotting the posterior probabilitiesPosterior Decoding Key: true path / Viterbi path / posterior path _44263143366636346665525516166566666224264513226235262443416SFFFFFFFFLLLLLLLLLLLFFLLLLLFFLLLLLLLLFFFFFFFFFFFFFFFFFFFLLLLSFFFFFFFFFLLLLLLLLLLLLLLLLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFLLLLLLLLLLLLFFFFLLLLLLLLLLLLFFFFFFFFFFFFFFFFFFFFFFF

Since we know individual probabilities at each position we can easily plot the posterior probabilitiesmyHMM.generate(60,143456789)18Plotting the posterior probabilities with matplotlibPosterior Decoding Assuming that we have list variables self._x containing the range of the sequence and self._y containing the posterior probabilities Note: you may need to convert the log_float probabilities back to normal floats I found it more convenient to just define the __float__() method in log_float from pylab import * # this line at the beginning of the file . . . class HMM(object): . . . # all your other stuff def show_posterior(self): if self._x and self._y: plot (self._x, self._y) show() return19