Richard Everitt tweetted yesterday about a recent publication in JCGS by Rajib Paul, Steve MacEachern and Mark Berliner on convergence assessment via stratification. (The paper is free-access.) Since this is another clear interest of mine’s, I had a look at the paper in the train to Besançon. (And wrote this post as a result.)
The idea therein is to compare the common empirical average with a weighted average relying on a partition of the parameter space: restricted means are computed for each element of the partition and then weighted by the probability of the element. Of course, those probabilities are generally unknown and need to be estimated simultaneously. If applied as is, this idea reproduces the original empirical average! So the authors use instead batches of simulations and corresponding estimates, weighted by the overall estimates of the probabilities, in which case the estimator differs from the original one. The convergence assessment is then to check both estimates are comparable. Using for instance Galin Jone’s batch method since they have the same limiting variance. (I thought we mentioned this damning feature in Monte Carlo Statistical Methods, but cannot find a trace of it except in my lecture slides…)
The difference between both estimates is the addition of weights p_in/q_ijn, made of the ratio of the estimates of the probability of the ith element of the partition. This addition thus introduces an extra element of randomness in the estimate and this is the crux of the convergence assessment. I was slightly worried though by the fact that the weight is in essence an harmonic mean, i.e. 1/q_ijn/Σ q_imn… Could it be that this estimate has no finite variance for a finite sample size? (The proofs in the paper all consider the asymptotic variance using the delta method.) However, having the weights adding up to K alleviates my concerns. Of course, as with other convergence assessments, the method is not fool-proof in that tiny, isolated, and unsuspected spikes not (yet) visited by the Markov chain cannot be detected via this comparison of averages.
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 non-parametric density estimation of massive, multi-dimensional 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 L1-consistency of the the initializing strategy over SRP histograms using a data-driven randomized priority queue based on a generalized statistically equivalent blocks principle is proved by bounding the Vapnik-Chervonenkis shatter coefficients of the class of SRP histogram partitions. 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.
Xi'an's Og 