Dependent Multinomial Models Made Easy: Stick Breaking with the P\'olya-Gamma Augmentation

TL;DR

This paper introduces a novel Bayesian approach for modeling dependent multinomial data using a logistic stick-breaking representation combined with Pólya-Gamma augmentation, simplifying inference in complex discrete data scenarios.

Contribution

It presents a new reformulation of multinomial models that captures dependencies more naturally and facilitates Bayesian inference with Gaussian likelihood techniques.

Findings

Enables efficient Bayesian inference for dependent multinomial data.

Provides a flexible framework for modeling correlated categorical data.

Simplifies the computational process using Pólya-Gamma augmentation.

Abstract

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 P\'olya-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.