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Bayesian interpretation of pseudocount regularizers

 

We can use Bayesian probability techniques to interpret the pseudocount regularizers. To apply these methods we have to view amino acids as being generated by a two-stage random process. First, a 20-dimensional density vector tex2html_wrap_inline2232 over the amino acids is chosen randomly, then amino acids are chosen randomly with probabilities tex2html_wrap_inline2234 . The probability of amino acid i given a sample s is the integral over all possible vectors tex2html_wrap_inline2232 of the probability of choosing that vector times the probability of choosing i given that vector:

displaymath2244

Computing the probability tex2html_wrap_inline2246 requires applying Bayes' rule:

displaymath2248

giving us a new formula for the probability of amino acid i:

displaymath2252

The probability tex2html_wrap_inline2254 is easily computed for any density vector tex2html_wrap_inline2232 , but we need to know the prior distribution of tex2html_wrap_inline2232 in order to compute the integral. The computation for tex2html_wrap_inline2254 is the same as in Section 2.1:

displaymath2262

There is an obvious generalization to non-integer s(j) values by replacing the factorial function with the equivalent expression using the Gamma function:

displaymath2266

In order to compute the integral, we must choose a model for the the prior distribution of tex2html_wrap_inline2232 . One choice that allows us to compute the integral is to model the prior as a Dirichlet distribution, that is

displaymath2270

for some parameter vector z, where C is a constant chosen so that tex2html_wrap_inline2276 P tex2html_wrap_inline2278 .

Showing in detail how to compute the integral is beyond the scope of this paper, but the answer can be derived from the standard definition of the Beta function [GR65, p. 948,]

eqnarray604

and the combining formula [GR65, p. 285,]:

displaymath2280

By writing the integral over all tex2html_wrap_inline2232 vectors as a multiple integral over the 20 dimensions of the vector and doing some rearrangement, we can get the solution

eqnarray614

where we have introduced the tex2html_wrap_inline2284 notation as an simple generalization of tex2html_wrap_inline2286 to the vector argument z.

With this choice of prior distribution for tex2html_wrap_inline2232 , we can compute

eqnarray619

We can now compute the estimated probability of the sample

eqnarray627

The integral for estimating the conditional probability of amino acid i given sample s is then

eqnarray644

Notation: tex2html_wrap_inline2320 is used above to mean the vector consisting of a one in the ith position and a zero elsewhere. tex2html_wrap_inline2324 is one if i=j and zero otherwise.

This rather involved computation finally ends up with the pseudocount method for estimating the probability of an amino acid given a sample of amino acids. The regularizer parameters z can be interpreted as assuming a Dirichlet distribution for the prior probabilities tex2html_wrap_inline2330 . Previous work with pseudocounts has relied heavily on this Bayesian interpretation of the parameters, going so far as to assign tex2html_wrap_inline2332 , which does indeed provide the optimal estimates for tex2html_wrap_inline1482 , but which we have seen in Section 3.2 is not the best setting of the parameters for |s|>0.

The posterior distribution of tex2html_wrap_inline2232 after seeing a sample s is tex2html_wrap_inline2342 P tex2html_wrap_inline2344 P tex2html_wrap_inline2346 P tex2html_wrap_inline2348 . As we can see from the above computations, this posterior distribution is again a Dirichlet distribution, with parameters s(j)+z(j), instead of the prior distribution's parameters z(j). This interpretation of tex2html_wrap_inline2354 as the parameters of the posterior distribution is what inspired naming them the posterior counts. The scaling of tex2html_wrap_inline1632 does matter for this interpretation, and so not all the posterior counts produced by regularizers can be automatically interpreted as Dirichlet posterior distributions on tex2html_wrap_inline2232 .

We can extend the Bayesian analysis to compute the posterior distribution of tex2html_wrap_inline2232 given that we have seen several independent samples: tex2html_wrap_inline2342 P tex2html_wrap_inline2364 . The computation is fairly straightforward. First we apply Bayes rule:

eqnarray667

Repeating the mathematics for a single sample would be tedious, but we can take a shortcut. Since the posterior distribution after seeing a sample is again a Dirichlet distribution, we can treat it as the prior distribution for adding the next sample. Using this trick, we can see that the final posterior distribution after seeing all n samples is a Dirichlet distribution with parameters tex2html_wrap_inline2382 . In other words, we get the same result from observing n independent samples as we would get from adding all the samples together and using the resulting counts as a single sample.


next up previous contents
Next: Bayesian interpretation of Dirichlet Up: Regularizers for Estimating Distributions Previous: Partial derivatives for Dirichlet

Rey Rivera
Thu Aug 1 17:59:45 PDT 1996