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Markov chains are a natural and well understood tool for describingone-dimensional patterns in time or space. We show how to infer $k$-th orderMarkov chains, for arbitrary $k$, from finite data by applying Bayesian methodsto both parameter estimation and model-order selection. Extending existingresults for multinomial models of discrete data, we connect inference tostatistical mechanics through information-theoretic (type theory) techniques.We establish a direct relationship between Bayesian evidence and the partitionfunction which allows for straightforward calculation of the expectation andvariance of the conditional relative entropy and the source entropy rate.Finally, we introduce a novel method that uses finite data-size scaling withmodel-order comparison to infer the structure of out-of-class processes.

Inferring Markov chains: Bayesian estimation, model comparison, entropy rate, and out-of-class modeling