On Markov chain Monte Carlo for sparse and filamentary distributions
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A novel strategy that combines a given collection of reversible Markov kernels is proposed. It consists in a Markov chain that moves, at each iteration, according to one of the available Markov kernels selected via a state-dependent probability distribution which is thus dubbed locally informed. In contrast to random-scan approaches that assume a constant selection probability distribution, the state-dependent distribution is typically specified so as to privilege moving according to a kernel which is relevant for the local topology of the target distribution. The second contribution is to characterize situations where a locally informed strategy should be preferred to its random-scan counterpart. We find that for a specific class of target distribution, referred to as sparse and filamentary, that exhibits a strong correlation between some variables and/or which concentrates its probability mass on some low dimensional linear subspaces or on thinned curved manifolds, a locally informed strategy converges substantially faster and yields smaller asymptotic variances than an equivalent random-scan algorithm. The research is at this stage essentially speculative: this paper combines a series of observations on this topic, both theoretical and empirical, that could serve as a groundwork for further investigations.
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