intrinsic quantity for a Markov chain?

I was attending a lecture this morning at CREST by Patrice Bertail where he was using estimated renewal parameters on a Markov chain to build (asymptotically) convergent bootstrap procedures. Estimating renewal parameters is obviously of interest in MCMC algorithms as they can be used to assess the convergence of the associated Markov chain: That is, if the estimation does not induce a significant bias. Another question that came to me during the talk is that; since those convergence assessments techniques are formally holding for any small set, choosing the small set in order to maximise the renewal rate also maximises the number of renewal events and hence the number of terms in the control sequence: Thus, the maximal renewal rate þ is definitely a quantity of interest: Now, is this quantity þ an intrinsic parameter of the chain, i.e. a quantity that drives its mixing and/or converging behaviour(s)? For instance; an iid sequence has a renewal rate of 1; because the whole set is a “small” set. Informally, the time between two consecutive renewal events is akin to the time between two simulations from the target and stationary distribution, according to the Kac’s representation we used in our AAP paper with Jim Hobert. So it could be that þ is directly related with the effective sample size of the chain, hence the autocorrelation. (A quick web search did not produce anything relevant:) Too bad this question did not pop up last week when I had the opportunity to discuss it with Sean Meyn in Gainesville!

MCMC convergence assessment

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.