Inference for Bayesian Nonparametric Models with Binary Response Data via Permutation Counting

Abstract

Since the beginning of Bayesian nonparametrics in the early 1970s, there has been a wide interest in constructing models for binary response data. Such data arise naturally in problems dealing with bioassay, current status data and sensitivity testing, and are equivalent to left and right censored observations if the inputs are one-dimensional. For models based on the Dirichlet process, inference is possible via Markov chain Monte Carlo (MCMC) simulations. However, there exist multiple processes based on different principles, for which such MCMC-based methods fail. Examples include logistic Gaussian processes and quantile pyramids. These require MCMC for posterior inference given exact observations, and thus become intractable when the data comprise both left and right censored observations. Here we present a new importance sampling algorithm for nonparametric models given exchangeable binary response data. It can be applied to any model from which samples can be generated, or even only approximately generated. The main idea behind the algorithm is to exploit the symmetries introduced by exchangeability. Calculating the importance weights turns out to be equivalent to evaluating the permanent of a certain class of $(0,1)$-matrix, which we prove can be done in polynomial time by deriving an explicit algorithm.