Data Augmentation, Frequentist Estimation, and the Bayesian Analysis of Multinomial Logit Models

This article introduces a generalization of Tanner and Wong’s data augmentation algorithm which can be used when the complete data posterior distribution cannot be directly sampled. The algorithm proposes parameter values based on complete data sampling distributions of convenient frequentist estimators which ignore some information in the complete data likelihood. The proposals are filtered using a Metropolis-Hastings probability to produce draws from the Bayesian posterior distribution of interest. The result is a method of deriving closed form Metropolis-Hastings proposals which do not need to be tuned. The method is used to sample the parameters of a multinomial logit model from their posterior distribution in a manner similar to Albert and Chib’s (1993) algorithm for probit regression. The algorithm converges geometrically ergodically, and its convergence rate can be accelerated through the use of working parameter methods developed for standard data augmentation algorithms. As with standard data augmentation, the method remains useful when multinomial logit models are embedded in more complicated settings, such as hierarchical models. The algorithm’s utility in complicated settings is illustrated on partial credit models describing ordinal item response data from the 1998 National Assessment of Educational Progress.