Youssef Marzouk: Computational challenges in Bayesian inversion
Dr. Youssef Marzouk, Associate Professor in MIT's Department of Aeronautics and Astronautics, presesnts "Computational challenges in Bayesian inversion" at the MIT Earth Resources Laboratory on March 30, 2018.
""Bayesian inference provides a natural framework for quantifying uncertainty in parameter estimates and model predictions, and for combining heterogeneous sources of information. Characterizing the results of Bayesian inference---by simulating from the posterior distribution---often proceeds via Markov chain Monte Carlo (MCMC) sampling, but the associated computational expense is a major bottleneck for complex posteriors and large-scale forward models. We will discuss two recent efforts aimed at complementary aspects of this problem.
For the first problem, we introduce a new approach for accelerating posterior simulation, borrowing ideas from approximation theory and derivative-free optimization. Previous efforts at integrating approximate models into inference typically sacrifice either exactness or efficiency; our approach addresses these limitations by exploiting useful convergence characteristics of local approximations. We develop an asymptotically exact MCMC scheme using local approximations, and describe variations of the algorithm that exploit parallel computation. Results suggest that when the likelihood has some regularity, the number of model evaluations per MCMC step can be greatly reduced without biasing the Monte Carlo estimate.
Second, we will describe a broad framework for using transport in Bayesian computation. This framework seeks a deterministic coupling of the posterior measure with a tractable "reference" measure (e.g., a standard Gaussian). Such couplings are induced by transport maps, and enable direct simulation from the posterior simply by evaluating the transport map at samples from the reference. Approximate transports can also be used to "precondition" and accelerate standard Monte Carlo schemes. Within this framework, one can describe many useful notions of low-dimensional structure associated with inference: for instance, sparse or decomposable transports underpin modeling and computation with non-Gaussian Markov random fields, and low rank transports arise frequently in inverse problems. These structures also suggest efficient algorithms, which we will demonstrate on inference problems arising in spatial statistics and PDEs."