unbiased product of expectations

While I was not involved in any way, or even aware of this research, Anthony Lee, Simone Tiberi, and Giacomo Zanella have an incoming paper in Biometrika, and which was partly written while all three authors were at the University of Warwick. The purpose is to design an efficient manner to approximate the product of n unidimensional expectations (or integrals) all computed against the same reference density. Which is not a real constraint. A neat remark that motivates the method in the paper is that an improved estimator can be connected with the permanent of the n x N matrix A made of the values of the n functions computed at N different simulations from the reference density. And involves N!/ (N-n)! terms rather than N to the power n. Since it is NP-hard to compute, a manageable alternative uses random draws from constrained permutations that are reasonably easy to simulate. Especially since, given that the estimator recycles most of the particles, it requires a much smaller version of N. Essentially N=O(n) with this scenario, instead of O(n²) with the basic Monte Carlo solution, towards a similar variance.

This framework offers many applications in latent variable models, including pseudo-marginal MCMC, of course, but also for ABC since the ABC posterior based on getting each simulated observation close enough from the corresponding actual observation fits this pattern (albeit the dependence on the chosen ordering of the data is an issue that can make the example somewhat artificial).

interdependent Gibbs samplers

Mark Kozdoba and Shie Mannor just arXived a paper on an approach to accelerate a Gibbs sampler. With title “interdependent Gibbs samplers“. In fact, it presents rather strong similarities with our SAME algorithm. More of the same, as Adam Johanssen (Warwick) entitled one of his papers! The paper indeed suggests multiplying replicas of latent variables (e.g., an hidden path for an HMM) in an artificial model. And as in our 2002 paper, with Arnaud Doucet and Simon Godsill, the focus here is on maximum likelihood estimation (of the genuine parameters, not of the latent variables). Along with argument that the resulting pseudo-posterior is akin to a posterior with a powered likelihood. And a link with the EM algorithm. And an HMM application.

“The generative model consist of simply sampling the parameters ,  and then sampling m independent copies of the paths”

If anything this proposal is less appealing than SAME because it aims directly at the powered likelihood, rather than utilising an annealed sequence of powers that allows for a primary exploration of the whole parameter space before entering the trapping vicinity of a mode. Which makes me fail to catch the argument from the authors that this improves Gibbs sampling, as a more acute mode has on the opposite the dangerous feature of preventing visits to other modes. Hence the relevance to resort to some form of annealing.

As already mused upon in earlier posts, I find it most amazing that this technique has been re-discovered so many times, both in statistics and in adjacent fields. The idea of powering the likelihood with independent copies of the latent variables is obviously natural (since a version pops up every other year, always under a different name), but earlier versions should eventually saturate the market!

running ABC when the likelihood is available

Today I refereed a paper where the authors used ABC to bypass convergence (and implementation) difficulties with their MCMC algorithm. And I am still pondering whether or not this strategy makes sense. If only because ABC needs to handle the same complexity and the same amount of parameters as an MCMC algorithm. While shooting “in the dark” by using the prior or a coarse substitute to the posterior. And I wonder at the relevance of simulating new data when the [true] likelihood value [at the observed data] can be computed. This would sound to me like the relevant and unique “statistics” worth considering…

rare events for ABC

Dennis Prangle, Richard G. Everitt and Theodore Kypraios just arXived a new paper on ABC, aiming at handling high dimensional data with latent variables, thanks to a cascading (or nested) approximation of the probability of a near coincidence between the observed data and the ABC simulated data. The approach amalgamates a rare event simulation method based on SMC, pseudo-marginal Metropolis-Hastings and of course ABC. The rare event is the near coincidence of the observed summary and of a simulated summary. This is so rare that regular ABC is forced to accept not so near coincidences. Especially as the dimension increases.  I mentioned nested above purposedly because I find that the rare event simulation method of Cérou et al. (2012) has a nested sampling flavour, in that each move of the particle system (in the sample space) is done according to a constrained MCMC move. Constraint derived from the distance between observed and simulated samples. Finding an efficient move of that kind may prove difficult or impossible. The authors opt for a slice sampler, proposed by Murray and Graham (2016), however they assume that the distribution of the latent variables is uniform over a unit hypercube, an assumption I do not fully understand. For the pseudo-marginal aspect, note that while the approach produces a better and faster evaluation of the likelihood, it remains an ABC likelihood and not the original likelihood. Because the estimate of the ABC likelihood is monotonic in the number of terms, a proposal can be terminated earlier without inducing a bias in the method.

This is certainly an innovative approach of clear interest and I hope we will discuss it at length at our BIRS ABC 15w5025 workshop next February. At this stage of light reading, I am slightly overwhelmed by the combination of so many computational techniques altogether towards a single algorithm. The authors argue there is very little calibration involved, but so many steps have to depend on as many configuration choices.

postdoc on missing data at École Polytechnique

Julie Josse contacted me for advertising a postdoc position at École Polytechnique, in Palaiseau, south of Paris. “The fellowship is focusing on missing data. Interested graduates should apply as early as possible since the position will be filled when a suitable candidate is found. The Centre for Applied Mathematics (CMAP) is  looking for highly motivated individuals able to develop a general multiple imputation method for multivariate continuous and categorical variables and its implementation in the free R software. The successful candidate will be part of research group in the statistical team on missing values. The postdoc will also have excellent opportunities to collaborate with researcher in public health with partners on the analysis of a large register from the Paris Hospital (APHP) to model the decisions and events when severe trauma patients are handled by emergency doctors. Candidates should contact Julie Josse at polytechnique.edu.”