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Register a new userHigh-dimensional Bayesian model selection by proximal nested sampling
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Imaging methods often rely on Bayesian statistical inference strategies to solve difficult imaging problems. Applying Bayesian methodology to imaging requires the specification of a likelihood function and a prior distribution, which define the Bayesian statistical model from which the posterior distribution of the image is derived. Specifying a suitable model for a specific application can be very challenging, particularly when there is no reliable ground truth data available. Bayesian model selection provides a framework for selecting the most appropriate model directly from the observed data, without reference to ground truth data. However, Bayesian model selection requires the computation of the marginal likelihood (Bayesian evidence), which is computationally challenging, prohibiting its use in high-dimensional imaging problems. In this work we present the proximal nested sampling methodology to objectively compare alternative Bayesian imaging models, without reference to ground truth data. The methodology is based on nested sampling, a Monte Carlo approach specialised for model comparison, and exploits proximal Markov chain Monte Carlo techniques to scale efficiently to large problems and to tackle models that are log-concave and not necessarily smooth (e.g., involving L1 or total-variation priors). The proposed approach can be applied computationally to problems of dimension O(10^6) and beyond, making it suitable for high-dimensional inverse imaging problems. It is validated on large Gaussian models, for which the likelihood is available analytically, and subsequently illustrated on a range of imaging problems where it is used to analyse different choices for the sparsifying dictionary and measurement model.
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In this paper we review the concepts of Bayesian evidence and Bayes factors, also known as log odds ratios, and their application to model selection. The theory is presented along with a discussion of analytic, approximate and numerical techniques. Specific attention is paid to the Laplace approximation, variational Bayes, importance sampling, thermodynamic integration, and nested sampling and its recent variants. Analogies to statistical physics, from which many of these techniques originate, are discussed in order to provide readers with deeper insights that may lead to new techniques. The utility of Bayesian model testing in the domain sciences is demonstrated by presenting four specific practical examples considered within the context of signal processing in the areas of signal detection, sensor characterization, scientific model selection and molecular force characterization.
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Yang et al. (2016) proved that the symmetric random walk Metropolis--Hastings algorithm for Bayesian variable selection is rapidly mixing under mild high-dimensional assumptions. We propose a novel MCMC sampler using an informed proposal scheme, which we prove achieves a much faster mixing time that is independent of the number of covariates, under the same assumptions. To the best of our knowledge, this is the first high-dimensional result which rigorously shows that the mixing rate of informed MCMC methods can be fast enough to offset the computational cost of local posterior evaluation. Motivated by the theoretical analysis of our sampler, we further propose a new approach called two-stage drift condition to studying convergence rates of Markov chains on general state spaces, which can be useful for obtaining tight complexity bounds in high-dimensional settings. The practical advantages of our algorithm are illustrated by both simulation studies and real data analysis.
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