Ilya Nemenman, F. Shafee, William Bialek
We study properties of popular near–uniform (Dirichlet) priors for learn- ing undersampled probability distributions on discrete nonmetric spaces and show that they lead to disastrous results. However, an Occam–style phase space argument expands the priors into their inﬁnite mixture and resolves most of the observed problems. This leads to a surprisingly good estimator of entropies of discrete distributions.
Learning a probability distribution from examples is one of the basic problems in data analysis. Common practical approaches introduce a family of parametric models, leading to questions about model selection. In Bayesian inference, computing the total probability of the data arising from a model involves an integration over parameter space, and the resulting “phase space volume” automatically discriminates against models with larger numbers of parameters—hence the description of these volume terms as Occam factors [1, 2]. As we move from ﬁnite parameterizations to models that are described by smooth functions, the integrals over parameter space become functional integrals and methods from quantum ﬁeld theory allow us to do these integrals asymptotically; again the volume in model space consistent with the data is larger for models that are smoother and hence less complex . Further, at least under some conditions the relevant degree of smoothness can be determined self–consistently from the data, so that we approach something like a model independent method for learning a distribution .
describe as the number of times ni each possibility is observed in a set of N =PK
The results emphasizing the importance of phase space factors in learning prompt us to look back at a seemingly much simpler problem, namely learning a distribution on a dis- crete, nonmetric space. Here the probability distribution is just a list of numbers fqig, i = 1; 2;(cid:1)(cid:1)(cid:1) ; K, where K is the number of bins or possibilities. We do not assume any metric on the space, so that a priori there is no reason to believe that any qi and qj should be similar. The task is to learn this distribution from a set of examples, which we can i=1 ni samples. This problem arises in the context of language, where the index i might label words or phrases, so that there is no natural way to place a metric on the space, nor is it even clear that our intuitions about similarity are consistent with the constraints of a met- ric space. Similarly, in bioinformatics the index i might label n–mers of the the DNA or amino acid sequence, and although most work in the ﬁeld is based on metrics for sequence comparison one might like an alternative approach that does not rest on such assumptions. In the analysis of neural responses, once we ﬁx our time resolution the response becomes a set of discrete “words,” and estimates of the information content in the response are de-
termined by the probability distribution on this discrete space. What all of these examples have in common is that we often need to draw some conclusions with data sets that are not in the asymptotic limit N (cid:29) K. Thus, while we might use a large corpus to sample the distribution of words in English by brute force (reaching N (cid:29) K with K the size of the vocabulary), we can hardly do the same for three or four word phrases.
In models described by continuous functions, the inﬁnite number of “possibilities” can never be overwhelmed by examples; one is saved by the notion of smoothness. Is there some nonmetric analog of this notion that we can apply in the discrete case? Our intuition is that information theoretic quantities may play this role. If we have a joint distribution of two variables, the analog of a smooth distribution would be one which does not have too much mutual information between these variables. Even more simply, we might say that smooth distributions have large entropy. While the idea of “maximum entropy inference” is common , the interplay between constraints on the entropy and the volume in the space of models seems not to have been considered. As we shall explain, phase space factors alone imply that seemingly sensible, more or less uniform priors on the space of discrete probability distributions correspond to disastrously singular prior hypotheses about the entropy of the underlying distribution. We argue that reliable inference outside the asymptotic regime N (cid:29) K requires a more uniform prior on the entropy, and we offer one way of doing this. While many distributions are consistent with the data when N (cid:20) K, we provide empirical evidence that this ﬂattening of the entropic prior allows us to make surprisingly reliable statements about the entropy itself in this regime.
At the risk of being pedantic, we state very explicitly what we mean by uniform or nearly uniform priors on the space of distributions. The natural “uniform” prior is given by