dominating measure

Yet another question on X validated reminded me of a discussion I had once  with Jay Kadane when visiting Carnegie Mellon in Pittsburgh. Namely the fundamentally ill-posed nature of conjugate priors. Indeed, when considering the definition of a conjugate family as being a parameterised family Þ of distributions over the parameter space Θ stable under transform to the posterior distribution, this property is completely dependent (if there is such a notion as completely dependent!) on the dominating measure adopted on the parameter space Θ. Adopted is the word as there is no default, reference, natural, &tc. measure that promotes one specific measure on Θ as being the dominating measure. This is a well-known difficulty that also sticks out in most “objective Bayes” problems, as well as with maximum entropy priors. This means for instance that, while the Gamma distributions constitute a conjugate family for a Poisson likelihood, so do the truncated Gamma distributions. And so do the distributions which density (against a Lebesgue measure over an arbitrary subset of (0,∞)) is the product of a Gamma density by an arbitrary function of θ. I readily acknowledge that the standard conjugate priors as introduced in every Bayesian textbook are standard because they facilitate (to a certain extent) posterior computations. But, just like there exist an infinity of MaxEnt priors associated with an infinity of dominating measures, there exist an infinity of conjugate families, once more associated with an infinity of dominating measures. And the fundamental reason is that the sampling model (which induces the shape of the conjugate family) does not provide a measure on the parameter space Θ.

Chateau du Lucquet

nor

BayesComp 20: call for contributed sessions!

Just to remind readers of the incoming deadline for BayesComp sessions:

The deadline for providing a title and brief abstract that the session is April 1, 2019. Please provide the names and affiliations of the organizer and the three speakers (the organizer can be one of them). Each session lasts 90 minutes and each talk should be 30 minutes long including Q&A. Contributed sessions can also consist of tutorials on the use of novel software. Decisions will be made by April 15, 2019. Please send your proposals to Christian Robert, co-chair of the scientific committee. We look forward to seeing you at BayesComp 20!

In case you do not feel like organising a whole session by yourself, contact the ISBA section you feel affinity with and suggest it helps building this session together!

ERC panel [step #1]

Although this post was written ages ago, regulations of the European Research Council (ERC) prevented me from posting it until now, for confidentiality reasons. I was indeed nominated as an expert member of the ERC panel on starting grants for mathematics [a denomination including statistics, obviously, but also quantum physics or some aspects of it], which means evaluating a hundred-ish applications of young researchers (five years from PhD) to select about ten of them to be richly funded for the coming five years. The reason for secrecy is that the panel members have to be protected from pursuits from the candidates (or, more likely, from their senior mentors). While this is a pretty heavy commitment, above 20 days total, the evaluation process gets quite interesting and the most annoying part is to have to reject proposals that should be funded, were more funds available. (For obvious reasons, I cannot get into the details of individual proposals, but let me just bemoan that there were too few proposals connected to statistics!) I may however get into my appreciation of the collective work of the panel during the first step selection process. I actually knew no other member prior to my joining the panel and was impressed at how smoothly we managed to work together and incorporate different opinions in a joint perspective. When I re-read these sentences, it feels like langue de bois (double talk), really!, but they truly represent my feelings at the end of the meeting. Making me (almost) looking forward the second step of interviewing the selected candidates in another week-long meeting, again in Brussels, for the interviews and final selection and ranking. (Which is when anonymity falls apart.)

 

“one of New Zealand’s darkest days”

asymptotics of synthetic likelihood

David Nott, Chris Drovandi and Robert Kohn just arXived a paper on a comparison between ABC and synthetic likelihood, which is both interesting and timely given that synthetic likelihood seems to be lacking behind in terms of theoretical evaluation. I am however as puzzled by the results therein as I was by the earlier paper by Price et al. on the same topic. Maybe due to the Cambodia jetlag, which is where and when I read the paper.

My puzzlement, thus, comes from the difficulty in comparing both approaches on a strictly common ground. The paper first establishes convergence and asymptotic normality for synthetic likelihood, based on the 2003 MCMC paper of Chernozukov and Hong [which I never studied in details but that appears like the MCMC reference in the econometrics literature]. The results are similar to recent ABC convergence results, unsurprisingly when assuming a CLT on the summary statistic vector. One additional dimension of the paper is to consider convergence for a misspecified covariance matrix in the synthetic likelihood [and it will come back with a revenge]. And asymptotic normality of the synthetic score function. Which is obviously unavailable in intractable models.

The first point I have difficulty with is how the computing time required for approximating mean and variance in the synthetic likelihood, by Monte Carlo means, is not accounted for in the comparison between ABC and synthetic likelihood versions. Remember that ABC only requires one (or at most two) pseudo-samples per parameter simulation. The latter requires M, which is later constrained to increase to infinity with the sample size. Simulations that are usually the costliest in the algorithms. If ABC were to use M simulated samples as well, since it already relies on a kernel, it could as well construct [at least on principle] a similar estimator of the [summary statistic] density. Or else produce M times more pairs (parameter x pseudo-sample). The authors pointed out (once this post out) that they do account for the factor M when computing the effective sample size (before Lemma 4, page 12), but I still miss why the ESS converging to N=MN/M when M goes to infinity is such a positive feature.

Another point deals with the use of multiple approximate posteriors in the comparison. Since the approximations differ, it is unclear that convergence to a given approximation is all that should matter, if the approximation is less efficient [when compared with the original and out-of-reach posterior distribution]. Especially for a finite sample size n. This chasm in the targets becomes more evident when the authors discuss the use of a constrained synthetic likelihood covariance matrix towards requiring less pseudo-samples, i.e. lower values of M, because of a smaller number of parameters to estimate. This should be balanced against the loss in concentration of the synthetic approximation, as exemplified by the realistic examples in the paper. (It is also hard to see why M could be not of order √n for Monte Carlo reasons.)

The last section in the paper is revolving around diverse issues for misspecified models, from wrong covariance matrix to wrong generating model. As we just submitted a paper on ABC for misspecified models, I will not engage into a debate on this point but find the proposed strategy that goes through an approximation of the log-likelihood surface by a Gaussian process and a derivation of the covariance matrix of the score function apparently greedy in both calibration and computing. And not so clearly validated when the generating model is misspecified.

30 women who cannot attend the international women’s day…