Page 156 - Contributed Paper Session (CPS) - Volume 2

CPS1488 Willem van den B. et al.

EP-IS: Combining expectation propagation and
importance sampling for bayesian
nonlinear inverse problems
Willem van den Boom, Alexandre H. Thiery
National University if Singapore, Singapore

Abstract
Bayesian analysis of inverse problems provides principled measures of
uncertainty quantification. However, Bayesian computation for inverse
problems is often challenging due to the high-dimensional or functional
nature of the parameter space. We consider a setting with a Gaussian process
prior in which exact computation of nonlinear inverse problems is too
expensive while linear problems are readily solved. Motivated by the latter, we
iteratively linearize nonlinear inverse problems. Doing this for data subsets
separately yields an expectation propagation (EP) algorithm. The EP cavity
distributions provide proposal distributions for an importance sampler that
refines the posterior approximation beyond what linearization yields. The
result is a hybrid between fast, linearization-based approaches, and sampling
based methods, which are more accurate but usually too slow.

Keywords
Bayesian computation; Gaussian process; Linearization; Monte Carlo

1. Introduction
Consider the inverse problem
~{ℎ(), } (1)
2

where y is an n-dimensional vector of observations, h : F → R a known
n
forward map for some function space , ∈ the unknown state space of
interest, and σ the error variance. An example of an inverse problem is when f
2
is the temperature distribution at time zero while y contains noisy temperature
measurements at a set of locations at a later point in time. Then, the heat
equation determines the map h. Our goal is to do Bayesian inference on (1).
That is, we consider a prior distribution () on f and aim to compute the
posterior on f,

()(|)
(|)= , (2)
()
where ( | ) is given by (1). This is intractable for general () and
forward maps h. Approximations are therefore used but deterministic
approximations like Laplace’s method can be too inaccurate while Monte Carlo
sampling from the posterior is too computationally expensive. We therefore

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