Marcelo Pereyra

Bayesian inference by convex optimisation: theory, methods, and algorithms.

Abstract:

Convex optimisation has become the main Bayesian computation methodology in many areas of data science such as mathematical imaging and machine learning, where high dimensionality is often addressed by using models that are log-concave and where maximum-a-posteriori (MAP) estimation can be performed efficiently by optimisation. The first part of this talk presents a new decision-theoretic derivation of MAP estimation and shows that, contrary to common belief, under log-concavity MAP estimators are proper Bayesian estimators. A main novelty is that the derivation is based on differential geometry. Following on from this, we establish universal theoretical guarantees for the estimation error involved and show estimation stability in high dimensions. Moreover, the second part of the talk describes a new general methodology for approximating Bayesian high-posterior-density regions in log-concave models. The approximations are derived by using recent concentration of measure results related to information theory, and can be computed very efficiently, even in large-scale problems, by using convex optimisation techniques. The approximations also have favourable theoretical properties, namely they outer-bound the true high-posterior-density credibility regions, and they are stable with respect to model dimension. The proposed methodology is finally illustrated on two high-dimensional imaging inverse problems related to tomographic reconstruction and sparse deconvolution, where they are used to explore the uncertainty about the solutions, and where convex-optimisation-empowered proximal Markov chain Monte Carlo algorithms are used as benchmark to compute exact credible regions and measure the approximation error.