I just arXived a survey entitled Bayesian computational tools in connection with a chapter the editors of the Annual Review of Statistics and Its Application asked me to write. (A puzzling title, I would have used Applications, not Application. Puzzling journal too: endowed with a prestigious editorial board, I wonder at the long-term perspectives of the review, once “all” topics have been addressed. At least, the “non-profit” aspect is respected: $100 for personal subscriptions and $250 for libraries, plus a one-year complimentary online access to volume 1.) Nothing terribly novel in my review, which illustrates some computational tool in some Bayesian settings, missing five or six pages to cover particle filters and sequential Monte Carlo. I however had fun with a double-exponential (or Laplace) example. This distribution indeed allows for a closed-form posterior distribution on the location parameter under a normal prior, which can be expressed as a mixture of truncated normal distributions. A mixture of (n+1) normal distributions for a sample of size n. We actually noticed this fact (which may already be well-known) when looking at our leading example in the consistent ABC choice paper, but it vanished from the appendix in the later versions. As detailed in the previous post, I also fought programming issues induced by this mixture, due to round-up errors in the most extreme components, until all approaches provided similar answers.
Archive for consistency
“We congratulate the authors for this very pleasant overview of the type of problems that are currently tackled by Bayesian nonparametric inference and for demonstrating how prolific this field has become. We do share the authors viewpoint that many Bayesian nonparametric models allow for more flexible modelling than parametric models and thus capture finer details of the data. BNP can be a good alternative to complex parametric models in the sense that the computations are not necessarily more difficult in Bayesian nonparametric models. However we would like to mitigate the enthusiasm of the authors since, although we believe that Bayesian nonparametric has proved extremely useful and interesting, we think they oversell the “nonparametric side of the Force”! Our main point is that by definition, Bayesian nonparametric is based on prior probabilities that live on infinite dimensional spaces and thus are never completely swamped by the data. It is therefore crucial to understand which (or why!) aspects of the model are strongly influenced by the prior and how.
As an illustration, when looking at Example 1 with the censored zeroth cell, our reaction is that this is a problem with no proper solution, because it is lacking too much information. In other words, unless some parametric structure of the model is known, in which case the zeroth cell is related with the other cells, we see no way to infer about the size of this cell. The outcome produced by the authors is therefore unconvincing to us in that it seems to only reflect upon the prior modelling (α,G*) and not upon the information contained in the data. Now, this prior modelling may be to some extent justified based on side information about the medical phenomenon under study, however its impact on the resulting inference is palatable.
Recently (and even less recently) a few theoretical results have pointed out this very issue. E.g., Diaconis and Freedman (1986) showed that some priors could surprisingly lead to inconsistent posteriors, even though it was later shown that many priors lead to consistent posteriors and often even to optimal asymptotic frequentist estimators, see for instance van der Vaart and van Zanten (2009) and Kruijer et al. (2010). The worry about Bayesian nonparametrics truly appeared when considering (1) asymptotic frequentist properties of semi-parametric procedures; and (2) interpretation of inferential aspects of Bayesian nonparametric procedures. It was shown in various instances that some nonparametric priors which behaved very nicely for the estimation of the whole parameter could have disturbingly suboptimal behaviour for some specific functionals of interest, see for instance Arbel et al. (2013) and Rivoirard and Rousseau (2012). We do not claim here that asymptotics is the answer to everything however bad asymptotic behaviour shows that something wrong is going on and this helps understanding the impact of the prior. These disturbing bad results are an illustration that in these infinite dimensional models the impact of the prior modelling is difficult to evaluate and that although the prior looks very flexible it can in fact be highly informative and/or restrictive for some aspects of the parameter. It would thus be wrong to conclude that every aspect of the parameter is well-recovered because some are. It has been a well-known fact for Bayesian parametric models, leading to extensive research on reference and other types of objective priors. It is even more crucial in the nonparametric world. No (nonparametric) prior can be suited for every inferential aspect and it is important to understand which aspects of the parameter are well-recovered and which ones are not.
We also concur with the authors that Dirichlet mixture priors provide natural clustering mechanisms, but one may question the “natural” label as the resulting clustering is quite unstructured, growing in the number of clusters as the number of observations increases and not incorporating any prior constraint on the “definition” of a cluster, except the one implicit and well-hidden behind the non-parametric prior. In short, it is delicate to assess what is eventually estimated by this clustering methods.
These remarks are not to be taken criticisms of the overall Bayesian nonparametric approach, just the contrary. We simply emphasize (or recall) that there is no such thing as a free lunch and that we need to post the price to pay for potential customers. In these models, this is far from easy and just as far from being completed.”
Arbel, J., Gayraud, G., and Rousseau, J. (2013). Bayesian adaptive optimal estimation using a sieve prior. Scandinavian Journal of Statistics, to appear.
Diaconis, P. and Freedman, D. (1986). On the consistency of Bayes estimates. Ann. Statist., 14:1-26.
Kruijer, W., Rousseau, J., and van der Vaart, A. (2010). Adaptive Bayesian density estimation with location-scale mixtures. Electron. J. Stat., 4:1225-1257.
Rivoirard, V. and Rousseau, J. (2012). On the Bernstein Von Mises theorem for linear functionals of the density. Ann. Statist., 40:1489-1523.
van der Vaart, A. and van Zanten, J. H. (2009). Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth. Ann. Statist., 37:2655-2675.
As I attended Jamie Robins’ session in Varanasi and did not have a clear enough idea of the Robbins and Wasserman paradox to discuss it viva vocce, here are my thoughts after reading Larry’s summary. My first reaction was to question whether or not this was a Bayesian statistical problem (meaning why should I be concered with the problem). Just as the normalising constant problem was not a statistical problem. We are estimating an integral given some censored realisations of a binomial depending on a covariate through an unknown function θ(x). There is not much of a parameter. However, the way Jamie presented it thru clinical trials made the problem sound definitely statistical. So end of the silly objection. My second step is to consider the very point of estimating the entire function (or infinite dimensional parameter) θ(x) when only the integral ψ is of interest. This is presumably the reason why the Bayesian approach fails as it sounds difficult to consistently estimate θ(x) under censored binomial observations, while ψ can be. Of course, if we want to estimate the probability of a success like ψ going through functional estimation this sounds like overshooting. But the Bayesian modelling of the problem appears to require considering all unknowns at once, including the function θ(x) and cannot forget about it. We encountered a somewhat similar problem with Jean-Michel Marin when working on the k-nearest neighbour classification problem. Considering all the points in the testing sample altogether as unknowns would dwarf the training sample and its information content to produce very poor inference. And so we ended up dealing with one point at a time after harsh and intense discussions! Now, back to the Robins and Wasserman paradox, I see no problem in acknowledging a classical Bayesian approach cannot produce a convergent estimate of the integral ψ. Simply because the classical Bayesian approach is an holistic system that cannot remove information to process a subset of the original problem. Call it the curse of marginalisation. Now, on a practical basis, would there be ways of running simulations of the missing Y’s when π(x) is known in order to achieve estimates of ψ? Presumably, but they would end up with a frequentist validation…
Stuart Russell (University of California, Berkeley, visiting Paris 6 this year) will give a seminar next week, Thursday November 22, 10am, LIP6, Université Paris 6, on Global Seismic Monitoring: A Bayesian Approach. Here is the link to the LIP6 webpage.
On Thursday November 15, 3pm, Institut Henri Poincaré, Jean-Michel Marin will give a talk at our big’MC seminar on the Consistency of Adaptive Multiple Importance Sampling (AMIS), following a long search of ours for this proof and a recent resolution of his along with Pierre Pudlo and Mohammed Sedki! Hopefully soon discussed on the ‘Og….
When I came back from LGM2012 in Trondheim, I found the latest issue of Series B on my desk. It is much thicker than in “my” days, with about 250 pages in this June 2012 issue! (One reason is that it contains two Read Papers with their discussions, amounting to 110 pages of the journal.) The first Read Paper was “Catching up faster by switching sooner” by van Erven, Grünwald and de Rooij, that we discussed with Nicolas Chopin. There are also comments (among others!) from Stephen Lauritzen, Iain Murray, and Aki Vehtari, who also spoke about Bayesian model evaluation tools at LGM2012. The second Read Paper is Fearnhead’s and Prangle’s semi-automatic ABC that I discussed last December. I have already posted about this Read Paper and used some of the discussion in preparing my ABC PhD class in Roma. However, the remark we made in our discussion with Jean-Michel Marin that the Bayes factor would not be a pertinent summary statistic for model choice is wrong, as shown by Dennis Prangle in his poster at the workshop in Bristol. And, when reading the reply by Paul Fearnhead and Dennis Prangle, I do not see a satisfactory answer to my demand of more formal conditions for Theorem 2 and its corollary, the convergence of the noisy ABC posterior to the true parameter (page 425), to apply. (Such results exist in indirect inference.)