Recursive estimation of high-order Markov chains: Approximation by finite mixtures

• High-order Markov chain is approximated by mixture of low order Markov chains.
• Corresponding Bayesian approximate recursive estimator is proposed.
• Forgetting counteracting error accumulation is tailored to this case.
• A simple example related to Ultimatum game indicates the power of the approach.

A high-order Markov chain is a universal model of stochastic relations between discrete-valued variables. The exact estimation of its transition probabilities suffers from the curse of dimensionality. It requires an excessive amount of informative observations as well as an extreme memory for storing the corresponding sufficient statistic. The paper bypasses this problem by considering a rich subset of Markov-chain models, namely, mixtures of low dimensional Markov chains, possibly with external variables. It uses Bayesian approximate estimation suitable for a subsequent decision making under uncertainty. The proposed recursive (sequential, one-pass) estimator updates a product of Dirichlet probability densities (pds) used as an approximate posterior pd, projects the result back to this class of pds and applies an improved data-dependent stabilised forgetting, which counteracts the dangerous accumulation of approximation errors.