# Spatio-Temporal Models for Big Multinomial Data using the Conditional Multivariate Logit-Beta Distribution

## TL;DR

This paper presents a Bayesian spatio-temporal model for high-dimensional multinomial data using the conditional multivariate logit-beta distribution, enabling flexible covariance structures and efficient inference.

## Contribution

It introduces the MN-STM, a novel Bayesian model that handles nonstationary, asymmetric covariances in space and time for multinomial data, with conjugate inference via the logit-beta distribution.

## Key findings

- Model effectively captures complex spatio-temporal dependencies.
- Demonstrates superior performance on simulated and real data.
- Provides a scalable inference algorithm for high-dimensional multinomial data.

## Abstract

We introduce a Bayesian approach for analyzing high-dimensional multinomial data that are referenced over space and time. In particular, the proportions associated with multinomial data are assumed to have a logit link to a latent spatio-temporal mixed effects model. This strategy allows for covariances that are nonstationarity in both space and time, asymmetric, and parsimonious. We also introduce the use of the conditional multivariate logit-beta distribution into the dependent multinomial data setting, which leads to conjugate full-conditional distributions for use in a collapsed Gibbs sampler. We refer to this model as the multinomial spatio-temporal mixed effects model (MN-STM). Additionally, we provide methodological developments including: the derivation of the associated full-conditional distributions, a relationship with a latent Gaussian process model, and the stability of the non-stationary vector autoregressive model. We illustrate the MN-STM through simulations and through a demonstration with public-use Quarterly Workforce Indicators (QWI) data from the Longitudinal Employer Household Dynamics (LEHD) program of the U.S. Census Bureau.