## Robert’s paradox [reading in Reading]

**O**n Wednesday afternoon, Richard Everitt and Dennis Prangle organised an RSS workshop in Reading on Bayesian Computation. And invited me to give a talk there, along with John Hemmings, Christophe Andrieu, Marcelo Pereyra, and themselves. Given the proximity between Oxford and Reading, this felt like a neighbourly visit, especially when I realised I could take my bike on the train! John Hemmings gave a presentation on synthetic models for climate change and their evaluation, which could have some connection with Tony O’Hagan’s recent talk in Warwick, Dennis told us about “the lazier ABC” version in connection with his “lazy ABC” paper, [from my very personal view] Marcelo expanded on the Moreau-Yoshida expansion he had presented in Bristol about six months ago, with the notion that using a Gaussian tail regularisation of a super-Gaussian target in a Langevin algorithm could produce better convergence guarantees than the competition, including Hamiltonian Monte Carlo, Luke Kelly spoke about an extension of phylogenetic trees using a notion of lateral transfer, and Richard introduced a notion of biased approximation to Metropolis-Hasting acceptance ratios, notion that I found quite attractive if not completely formalised, as there should be a Monte Carlo equivalent to the improvement brought by biased Bayes estimators over unbiased classical counterparts. (Repeating a remark by Persi Diaconis made more than 20 years ago.) Christophe Andrieu also exposed some recent developments of his on exact approximations à la Andrieu and Roberts (2009).

Since those developments are not yet finalised into an archived document, I will not delve into the details, but I found the results quite impressive and worth exploring, so I am looking forward to the incoming publication. One aspect of the talk which I can comment on is related to the exchange algorithm of Murray et al. (2006). Let me recall that this algorithm handles double intractable problems (i.e., likelihoods with intractable normalising constants like the Ising model), by introducing auxiliary variables with the same distribution as the data given the new value of the parameter and computing an augmented acceptance ratio which expectation is the targeted acceptance ratio and which conveniently removes the unknown normalising constants. This auxiliary scheme produces a random acceptance ratio and hence differs from the exact-approximation MCMC approach, which target directly the intractable likelihood. It somewhat replaces the unknown constant with the density taken at a plausible realisation, hence providing a proper scale. At least for the new value. I wonder if a comparison has been conducted between both versions, the naïve intuition being that the ratio of estimates should be more variable than the estimate of the ratio. More generally, it seemed to me [during the introductory part of Christophe’s talk] that those different methods always faced a harmonic mean danger when being phrased as expectations of ratios, since those ratios were not necessarily squared integrable. And not necessarily bounded. Hence my rather gratuitous suggestion of using other tools than the expectation, like maybe a median, thus circling back to the biased estimators of Richard. (And later cycling back, unscathed, to Reading station!)

On top of the six talks in the afternoon, there was a small poster session during the tea break, where I met Garth Holloway, working in agricultural economics, who happened to be a (unsuspected) fan of mine!, to the point of entitling his poster “Robert’s paradox”!!! The problem covered by this undeserved denomination connected to the bias in Chib’s approximation of the evidence in mixture estimation, a phenomenon that I related to the exchangeability of the component parameters in an earlier paper or set of slides. So “my” paradox is essentially label (un)switching and its consequences. For which I cannot claim any fame! Still, I am looking forward the completed version of this poster to discuss Garth’s solution, but we had a beer together after the talks, drinking to the health of our mutual friend John Deely.

January 28, 2015 at 11:30 am

Just a quick one – I think you meant super-Gaussian rather than sub-exponential tails (i.e. light rather than heavy). I remember the focus of Marcelo’s talk was log-concave targets.

January 28, 2015 at 12:01 pm

Yes indeed, sorry about this confusion!