approximate maximum likelihood estimation using data-cloning ABC

“By accepting of having obtained a poor approximation to the posterior, except for the location of its main mode, we switch to maximum likelihood estimation.”

Presumably the first paper ever quoting from the ‘Og! Indeed, Umberto Picchini arXived a paper about a technique merging ABC with prior feedback (rechristened data cloning by S. Lele), where a maximum likelihood estimate is produced by an ABC-MCMC algorithm. For state-space models. This relates to an earlier paper by Fabio Rubio and Adam Johansen (Warwick), who also suggested using ABC to approximate the maximum likelihood estimate. Here, the idea is to use an increasing number of replicates of the latent variables, as in our SAME algorithm, to spike the posterior around the maximum of the (observed) likelihood. An ABC version of this posterior returns a mean value as an approximate maximum likelihood estimate.

“This is a so-called “likelihood-free” approach [Sisson and Fan, 2011], meaning that knowledge of the complete expression for the likelihood function is not required.”

The above remark is sort of inappropriate in that it applies to a non-ABC setting where the latent variables are simulated from the exact marginal distributions, that is, unconditional on the data, and hence their density cancels in the Metropolis-Hastings ratio. This pre-dates ABC by a few years, since this was an early version of particle filter.

“In this work we are explicitly avoiding the most typical usage of ABC, where the posterior is conditional on summary statistics of data S(y), rather than y.”

Another point I find rather negative in that, for state-space models, using the entire time-series as a “summary statistic” is unlikely to produce a good approximation.

The discussion on the respective choices of the ABC tolerance δ and on the prior feedback number of copies K is quite interesting, in that Umberto Picchini suggests setting δ first before increasing the number of copies. However, since the posterior gets more and more peaked as K increases, the consequences on the acceptance rate of the related ABC algorithm are unclear. Another interesting feature is that the underlying MCMC proposal on the parameter θ is an independent proposal, tuned during the warm-up stage of the algorithm. Since the tuning is repeated at each temperature, there are some loose ends as to whether or not it is a genuine Markov chain method. The same question arises when considering that additional past replicas need to be simulated when K increases. (Although they can be considered as virtual components of a vector made of an infinite number of replicas, to be used when needed.)

The simulation study involves a regular regression with 101 observations, a stochastic Gompertz model studied by Sophie Donnet, Jean-Louis Foulley, and Adeline Samson in 2010. With 12 points. And a simple Markov model. Again with 12 points. While the ABC-DC solutions are close enough to the true MLEs whenever available, a comparison with the cheaper ABC Bayes estimates would have been of interest as well.

hierarchical models are not Bayesian models

When preparing my OxWaSP projects a few weeks ago, I came perchance on a set of slides, entitled “Hierarchical models are not Bayesian“, written by Brian Dennis (University of Idaho), where the author argues against Bayesian inference in hierarchical models in ecology, much in relation with the previously discussed paper of Subhash Lele. The argument is the same, namely a possibly major impact of the prior modelling on the resulting inference, in particular when some parameters are hardly identifiable, the more when the model is complex and when there are many parameters. And that “data cloning” being available since 2007, frequentist methods have “caught up” with Bayesian computational abilities.

Let me remind the reader that “data cloning” means constructing a sequence of Bayes estimators corresponding to the data being duplicated (or cloned) once, twice, &tc., until the point estimator stabilises. Since this corresponds to using increasing powers of the likelihood, the posteriors concentrate more and more around the maximum likelihood estimator. And even recover the Hessian matrix. This technique is actually older than 2007 since I proposed it in the early 1990’s under the name of prior feedback, with earlier occurrences in the literature like D’Epifanio (1989) and even the discussion of Aitkin (1991). A more efficient version of this approach is the SAME algorithm we developed in 2002 with Arnaud Doucet and Simon Godsill where the power of the likelihood is increased during iterations in a simulated annealing version (with a preliminary version found in Duflo, 1996).

I completely agree with the author that a hierarchical model does not have to be Bayesian: when the random parameters in the model are analysed as sources of additional variations, as for instance in animal breeding or ecology, and integrated out, the resulting model can be analysed by any statistical method. Even though one may wonder at the motivations for selecting this particular randomness structure in the model. And at an increasing blurring between what is prior modelling and what is sampling modelling as the number of levels in the hierarchy goes up. This rather amusing set of slides somewhat misses a few points, in particular the ability of data cloning to overcome identifiability and multimodality issues. Indeed, as with all simulated annealing techniques, there is a practical difficulty in avoiding the fatal attraction of a local mode using MCMC techniques. There are thus high chances data cloning ends up in the “wrong” mode. Moreover, when the likelihood is multimodal, it is a general issue to decide which of the modes is most relevant for inference. In which sense is the MLE more objective than a Bayes estimate, then? Further, the impact of a prior on some aspects of the posterior distribution can be tested by re-running a Bayesian analysis with different priors, including empirical Bayes versions or, why not?!, data cloning, in order to understand where and why huge discrepancies occur. This is part of model building, in the end.

Feedback on data cloning

Following some discussions I had last week at Banff about data cloning, I re-read the 2007 “Data cloning” paper published in Ecology Letters by Lele, Dennis, and Lutscher. Once again, I see a strong similarity with our 2002 Statistics and Computing SAME algorithm, as well as with the subsequent (and equally similar) “A multiple-imputation Metropolis version of the EM algorithm” published in Biometrika by Gaetan and Yao in 2003—Biometrika to which Arnaud and I had earlier and unsuccessfully submitted this unpublished technical report on the convergence of the SAME algorithm… (The SAME algorithm is also described in detail in the 2005 book Inference in Hidden Markov Models, Chapter 13.)

10w2170, Banff [2]

Over the two days of the Hierarchical Bayesian Methods in Ecology workshop, we managed to cover normal models, testing, regression, Gibbs sampling, generalised linear models, Metropolis-Hastings algorithms and of course a fair dose of hierarchical modelling. At the end of the Saturday marathon session, we spent one and half discussing some models studied by the participants, which were obviously too complex to be solved on the spot but well-defined so that we could work on MCMC implementation and analysis. And on Sunday morning, a good example of Poisson regression proposed by Devin Goodman led to an exciting on-line programming of a random effect generalised model, with the lucky occurrence of detectable identifiability issues that we could play with… I am impressed at the resilience of the audience given the gruesome pace I pursued over those two days, covering the five first chapters of Bayesian Core, all the way to the mixtures! In retrospect, I think I need to improve my coverage of testing as the noninformative case presumably sounded messy. And unconvincing. I also fear the material on hierarchical models was not sufficiently developed. But, overall, the workshop provided a wonderful opportunity to exchange with bright PhD students from Ecology and Forestry about their models and (hierarchical) Bayesian modelling.