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!
Archive for renewal process
We (Kerrie Mengersen, Pierre Pudlo, and myself) have now revised our ABC with empirical likelihood paper and resubmitted both to arXiv and to PNAS as “Approximate Bayesian computation via empirical likelihood“. The main issue raised by the referees was that the potential use of the empirical likelihood (EL) approximation is much less widespread than the possibility of simulating pseudo-data, because EL essentially relies on an iid sample structure, plus the availability of parameter defining moments. This is indeed the case to some extent and also the reason why we used a compound likelihood for our population genetic model. There are in fact many instances where we simply cannot come up with a regular EL approximation… However, the range of applications of straight EL remains wide enough to be of interest, as it includes most dynamical models like hidden Markov models. To illustrate this point further, we added (in this revision) an example borrowed from the recent Biometrika paper by David Cox and Christiana Kartsonaki (which proposes a frequentist alternative to ABC based on fractional design). This model ended up being fairly appealing wrt our perspective: while the observed data is dependent in a convoluted way, being a superposition of N renewal processes with gamma waiting times, it is possible to recover an iid structure at the same cost as a regular ABC algorithm by using the pseudo-data to recover an iid process (the sequence of renewal processes indicators)…The outcome is quite favourable to ABCel in this particular case, as shown by the graph below (top: ABCel, bottom: ABC, red line:truth):
This revision (started while visiting Kerrie in Brisbane) was thus quite beneficial to our perception of ABC in that (a) it is indeed not as universal as regular ABC and this restriction should be spelled out (the advantage being that, when it can be implemented, it usually runs much much faster!), and (b) in cases where the pseudo-data must be simulated, EL provides a reference/benchmark for the ABC output that comes for free… Now I hope to manage to get soon out of the “initial quality check” barrage to reach the Editorial Board!
“Bayes Theorem is a simple consequence of the axioms of probability, and is therefore accepted by all as valid. However, some who challenge the use of personal probability reject certain applications of Bayes Theorem.” J. Kadane, p.44
Principles of uncertainty by Joseph (“Jay”) Kadane (Carnegie Mellon University, Pittsburgh) is a profound and mesmerising book on the foundations and principles of subjectivist or behaviouristic Bayesian analysis. Jay Kadane wrote Principles of uncertainty over a period of several years and, more or less in his own words, it represents the legacy he wants to leave for the future. The book starts with a large section on Jay’s definition of a probability model, with rigorous mathematical derivations all the way to Lebesgue measure (or more exactly the McShane-Stieltjes measure). This section contains many side derivations that pertain to mathematical analysis, in order to explain the subtleties of infinite countable and uncountable sets, and the distinction between finitely additive and countably additive (probability) measures. Unsurprisingly, the role of utility is emphasized in this book that keeps stressing the personalistic entry to Bayesian statistics. Principles of uncertainty also contains a formal development on the validity of Markov chain Monte Carlo methods that is superb and missing in most equivalent textbooks. Overall, the book is a pleasure to read. And highly recommended for teaching as it can be used at many different levels. Continue reading