| Summary: | We develop a mixture model and diagnostic for Bayesian estimation and selection in high-order, discrete-state Markov chains. Both extend the mixture transition distribution, which constructs a transition probability tensor by aggregating probabilities from a set of single-lag transition matrices, through inclusion of mixture components dependent on multiple lags. We demonstrate two uses for the proposed model: identification of relevant lags through over-specification and shrinkage via priors for sparse probability vectors, and parsimonious approximation of multi-lag dynamics by mixing low-order transition models. The diagnostic yields a general and interpretable mixture decomposition for transition probability tensors estimated by any means. We demonstrate the utility of the model and diagnostic with simulation studies, and further apply the methodology to a data analysis from the high-order Markov chain literature, and to a time series of pink salmon abundance in Alaska, United States. Supplemental files for this article are available online.
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