Dependent Multinomial Models Made Easy: Stick Breaking with the Pólya-Gamma Augmentation
Many practical modeling problems involve discrete data that are best represented as draws from multinomial or categorical distributions. For example, nucleotides in a DNA sequence, children's names in a given state and year, and text documents are all commonly modeled with multinomial distributions. In all of these cases, we expect some form of dependency between the draws: the nucleotide at one position in the DNA strand may depend on the preceding nucleotides, children's names are highly correlated from year to year, and topics in text may be correlated and dynamic. These dependencies are not naturally captured by the typical Dirichlet-multinomial formulation. Here, we leverage a logistic stick-breaking representation and recent innovations in Pólya-gamma augmentation to reformulate the multinomial distribution in terms of latent variables with jointly Gaussian likelihoods, enabling us to take advantage of a host of Bayesian inference techniques for Gaussian models with minimal overhead.
Summary: The paper presents a data-augmentation scheme for performing closed-form Gibbs sampling in models with Gaussian priors and multinomial likelihoods. The proposed scheme is based on introducing Polya-gamma random variables (one per observation) which allows Gaussianifying the multinomial likelihood and leading to closed-form Gibbs sampling updates. The proposed scheme is evaluated on a number of models with Gaussian priors and multinomial likelihood and is shown to demonstrate superior performance when compared to several baselines.Comments:Models with Gaussian priors and multinomial likelihoods are known to be troublesome when doing inference due to non-conjugacy. This paper presents a neat trick to "Gaussianify" the multinomial likelihood using an auxiliary variable construction based on the recently proposed Polya-gamma augmentation [3] by Polson and Scott (2012). This allows closed-form Gibbs samplingThe idea of using the Polya-gamma trick has been used in several papers recently for binary data (using logistic link) and count data (using negative binomial link). However, the muiltinomial case hasn't been explored much. I should also mention that the case of multinomial distribution was also considered in the original Polya-gamma paper by Polson and Scott (please see the appendix S6.3: http://arxiv.org/pdf/1205.0310v3.pdf) and used recently in [8]. This aspect should be discussed well before the proposed method is described.That said, this paper takes a somewhat different approach for handling the multinomial likelihood than the one presented in the appendix of [3]. Although the resulting equations here appear very similar to those obtained in [3], the proposed approach in this paper is more elegant in its construction (using a stick-breaking construction). In particular, this also enables block sampling whereas [8] which used the trick presented in [3] requires component-wise sampling. The experimental results corroborate the advantage of doing the block sampling. This makes the proposed construction very appealing.I also like the range of applications considered to demonstrate the usefulness and advantage of the proposed inference method over the alternatives.A minor point: I would be curious to see how much of a computational overhead does sampling the Polya-gamma variables introduces. At least, some discussion of this will be nice to have. Overall, although built on some standard pieces (Polya-gamma sampling, stick-breaking construction), the proposed inference method provides an easy way to do Gibbs sampling in non-conjugate models and is expected to be useful for a wide range of non-conjugate models with Gaussian priors and multinomial likelihood (as shown in some of the models presented in the paper).