null recurrent = zero utility?

The stability result that the ratio

converges holds for a Harris π-null-recurrent Markov chain for all functions f,g in L¹(π) [Meyn & Tweedie, 1993, Theorem 17.3.2] is rather fascinating. However, it is unclear it can be useful in simulation environments, as for the integral priors we have been studying over the years with Juan Antonio Cano and Diego Salmeron Martinez. Above, the result of an experiment where I simulated a Markov chain as a Normal random walk in dimension one, hence a Harris π-null-recurrent Markov chain for the Lebesgue measure λ, and monitored the stabilisation of the ratio (1) when using two densities for f and g,  to its expected value (1, shown by a red horizontal line). There is quite a variability in the outcome (repeated 100 times),  but the most intriguing is the quick stabilisation of most cumulated averages to values different from 1. Even longer runs display this feature

which I would blame on the excursions of the random walk far away from the central regions for both f and g, that is on long sequences where zeroes keep being added to numerator and denominators in (1). As far as integral approximation is concerned, this is not very helpful!

Juan Antonio Cano Sanchez (1956-2018)

I have just learned the very sad news that Juan Antonio Cano, from Universidad de Murcia, with whom Diego Salmerón and I wrote two papers on integral priors, has passed away, after a long fight against a kidney disease. Having communicated with him recently, I am quite shocked by him passing away as I was not aware of his poor health. The last time we met was at the O’Bayes 2015 meeting in Valencià, with a long chat in the botanical gardens of the Universitat de Valencià. Juan Antonio was a very kind and unassuming person, open and friendly, with a continued flow of research in Objective Bayes methodology and in particular on integral priors. Hasta luego, Juan Antonio!

Objective Bayesian hypothesis testing

Our paper with Diego Salmerón and Juan Cano using integral priors for binomial regression and objective Bayesian hypothesis testing (one of my topics of interest, see yesterday’s talk!) eventually appeared in Statistica Sinica. This is Volume 25,  Number 3, of July 2015 and the table of contents shows an impressively diverse range of topics.

Workshop Métodos Bayesianos 11

The special day on Bayesian methods (this was the second edition) organised by Miguel Gómez Villegas at Universidad Complutense de Madrid was quite intense (ending up after 8pm!) but definitely worth the trip. First, I was not the only one to talk about ABC since Miguel González Velasco presented a work on ABC and MCMC applied to branching processes. Gonzalo García-Donato also argued through a very convincing talk about the desideratas on priors used in model selection: I had of course though about those during Anabel’s thesis last February, but this was very well-defended. Antonio Cano Sánchez and Diego Salmerón gave a two-men show on their current integral prior research, which reminded me of the work we did together a few years ago, opening new vistas for models with covariates. Julián de la Horra Navarro talked about robust losses I had never seen before, Rosario Susi Garcia about prior sensitivity for Bayesian networks [again something I had not seen previously], and Juan José Egozcue introduced us to a more mathematical topic of representing priors and posteriors within a new algebra, linked with compositional data analysis, although its overall impact on Bayesian grounds somehow escaped me. Lizbetz Naranjo Albarrán discussed the MCMC estimation of a misspecified logistic regression model, with the surprising feature that the misspecification was not directly observed. In addition, I met several friends and had the opportunity of discussing with them about our on-going research. In particular, they set me thinking on the extension of our work to more than two competing models. Rekindling earlier thoughts about the choice of statistic, tolerance, and distance one model at the time (but not beyond my earlier reticences on comparing models based on different statistics). And a possible postvalidation of published works on ABC model choice according to our criterion.

The day ended up in typical Spanish time, i.e. with a superb if fairly late dinner, right next to the beautiful Teatro Real. (I still managed to get a few hours sleep, though, unlike the kids I saw walking the streets after a long night out, when taking the metro to the airport in the early morning..)