Bayesian Modeling of Temporal Dependence in Large Sparse Contingency Tables

Abstract

In many applications, it is of interest to study trends over time in relationships among categorical variables, such as age group, ethnicity, religious affiliation, political party and preference for particular policies. At each time point, a sample of individuals provide responses to a set of questions, with different individuals sampled at each time. In such settings, there tends to be abundant missing data and the variables being measured may change over time. At each time point, one obtains a large sparse contingency table, with the number of cells often much larger than the number of individuals being surveyed. To borrow information across time in modeling large sparse contingency tables, we propose a Bayesian autoregressive tensor factorization approach. The proposed model relies on a probabilistic Parafac factorization of the joint pmf characterizing the categorical data distribution at each time point, with autocorrelation included across times. Efficient computational methods are developed relying on MCMC. The methods are evaluated through simulation examples and applied to social survey data.

Bayesian inference on group differences in multivariate categorical data

“…This factorization induces a provably flexible statistical model, which incorporates dimensionality reduction, and borrows information between the cell probabilities in sparse tables to provide efficient inference on the entire joint probability mass function. Notable recent generalizations of the model proposed by Dunson and Xing (2009) incorporate additional sparsity , dynamic patterns (Kunihama and Dunson, 2013), classification of univariate outcomes (Yang and Dunson, 2016), data imputation (Fosdick et al, 2016;Murray and Reiter, 2016), and inference in case-control studies with several categorical predictors . Refer to Johndrow et al (2017) for a theoretical justification, and connections with log-linear models.…”

Nonparametric Bayes Modeling for Case Control Studies with Many Predictors

“…Potentially log‐linear models can be used, but unless the vast majority of the interactions are discarded a priori, one obtains an unmanageably enormous number of terms to estimate, store, and process. These bottlenecks are freed by the use of Bayesian low rank tensor factorizations, which have had promising performance in practice ((Dunson and Xing, ); (Bhattacharya and Dunson, ); (Kunihama and Dunson, ); (Zhou et al, )). Johndrow, Bhattacharya, and Dunson (2014) recently showed that a large subclass of sparse log‐linear models have low rank tensor factorizations, providing support for the use of tensor factorizations as a computationally convenient alternative.…”

Switching Poisson Gamma Dynamical Systems

“…Sequentially observed count tensors present unique statistical challenges because they tend to be bursty [12], high-dimensional, and sparse [13,14]. There are few models that are tailored to the challenging properties of both time series and count tensors.…”