Three interesting recent arXiv postings and not enough time to read them all and in the ‘Og bind them! (Of course, comments from readers welcome!)
Formulating a statistical inverse problem as one of inference in a Bayesian model has great appeal, notably for what this brings in terms of coherence, the interpretability of regularisation penalties, the integration of all uncertainties, and the principled way in which the set-up can be elaborated to encompass broader features of the context, such as measurement error, indirect observation, etc. The Bayesian formulation comes close to the way that most scientists intuitively regard the inferential task, and in principle allows the free use of subject knowledge in probabilistic model building. However, in some problems where the solution is not unique, for example in ill-posed inverse problems, it is important to understand the relationship between the chosen Bayesian model and the resulting solution. Taking emission tomography as a canonical example for study, we present results about consistency of the posterior distribution of the reconstruction, and a general method to study convergence of posterior distributions. To study efficiency of Bayesian inference for ill-posed linear inverse problems with constraint, we prove a version of the Bernstein-von Mises theorem for nonregular Bayesian models.
(Certainly unlikely to please the member of the audience in Zürich who questioned my Bayesian credentials for considering “true” models and consistency….)
Recently, Andrieu, Doucet and Holenstein (2010) introduced a general framework for using particle filters (PFs) to construct proposal kernels for Markov chain Monte Carlo (MCMC) methods. This framework, termed Particle Markov chain Monte Carlo (PMCMC), was shown to provide powerful methods for joint Bayesian state and parameter inference in nonlinear/non-Gaussian state-space models. However, the mixing of the resulting MCMC kernels can be quite sensitive, both to the number of particles used in the underlying PF and to the number of observations in the data. In this paper we suggest alternatives to the three PMCMC methods introduced in Andrieu et al. (2010), which are much more robust to a low number of particles as well as a large number of observations. We consider some challenging inference problems and show in a simulation study that, for problems where existing PMCMC methods require around 1000 particles, the proposed methods provide satisfactory results with as few as 5 particles.
(I have not read the paper enough in-depth to be critical, however “hard” figures like 5, or 10³, are always suspicious in that they cannot carry to the general case…)
In this paper we present an algorithm for rapid Bayesian analysis that combines the benefits of nested sampling and artificial neural networks. The blind accelerated multimodal Bayesian inference (BAMBI) algorithm implements the MultiNest package for nested sampling as well as the training of an artificial neural network (NN) to learn the likelihood function. In the case of computationally expensive likelihoods, this allows the substitution of a much more rapid approximation in order to increase significantly the speed of the analysis. We begin by demonstrating, with a few toy examples, the ability of a NN to learn complicated likelihood surfaces. BAMBI’s ability to decrease running time for Bayesian inference is then demonstrated in the context of estimating cosmological parameters from WMAP and other observations. We show that valuable speed increases are achieved in addition to obtaining NNs trained on the likelihood functions for the different model and data combinations. These NNs can then be used for an even faster follow-up analysis using the same likelihood and different priors. This is a fully general algorithm that can be applied, without any pre-processing, to other problems with computationally expensive likelihood functions.
(This is primarily an astronomy paper that uses a sample produced by the nested sampling algorithm MultiNest to build a neural network instead of the model likelihood. The algorithm thus requires the likelihood to be available at some stage.)