Structured priors for sparse probability vectors with application to model selection in Markov chains

We develop two prior distributions for probability vectors which, in contrast to the popular Dirichlet distribution, retain sparsity properties in the presence of data. Our models are appropriate for count data with many categories, most of which are expected to have negligible probability. Both models are tractable, allowing for efficient posterior sampling and marginalization. Consequently, they can replace the Dirichlet prior in hierarchical models without sacrificing convenient Gibbs sampling schemes. We derive both models and demonstrate their properties. We then illustrate their use for model-based selection with a hierarchical model in which we infer the active lag from time-series data. Using a squared-error loss, we demonstrate the utility of the models for data simulated from a nearly deterministic dynamical system. We also apply the prior models to an ecological time series of Chinook salmon abundance, demonstrating their ability to extract insights into the lag dependence.