Posterior Expectation of Regularly Paved Random Histograms

<jats:p>We present a novel method for averaging a sequence of histogram states visited by a Metropolis-Hastings Markov chain whose stationary distribution is the posterior distribution over a dense space of tree-based histograms. The computational efficiency of our posterior mean histogram estimate relies on a statistical data-structure that is sufficient for nonparametric density estimation of massive, multidimensional metric data. This data-structure is formalized as statistical regular paving (SRP). A regular paving (RP) is a binary tree obtained by selectively bisecting boxes along their first widest side. SRP augments RP by mutably caching the recursively computable sufficient statistics of the data. The base Markov chain used to propose moves for the Metropolis-Hastings chain is a random walk that data-adaptively prunes and grows the SRP histogram tree. We use a prior distribution based on Catalan numbers and detect convergence heuristically. The performance of our posterior mean SRP histogram is empirically assessed for large sample sizes simulated from several multivariate distributions that belong to the space of SRP histograms.</jats:p>