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Abstract:
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Whereas the number of parameters in a general higher-order Markov model is exponential in the order of dependence and the model has limited flexibility, sparse Markov models help with these problems. A sparse Markov model is a higher-order Markov model for which conditioning histories are grouped into classes such that the conditional probability distribution given any history of the class is the same. This paper introduces a model where variables following a sparse Markov structure are latent, and all inference over the latent states is conditioned on observed data. Then several tasks are considered in this sparse Markov setting: determining an appropriate model and parameter estimation, methodology for efficient computation of conditional distributions of statistics of the hidden states, determining the likelihood of the observations, and obtaining the most likely hidden state at each time point and the most likely hidden state sequence, given the observations. An application is given to modeling the fluctuations in price of the S&P 500.
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