Multimodal Information Gain in Bayesian Design of Experiments

Abstract

One of the well-known challenges in optimal experimental design is how to efficiently estimate the nested integrations of the expected information gain. The Gaussian approximation and associated importance sampling have been shown to be effective at reducing the numerical costs. However, they may fail due to the non-negligible biases and the numerical instabilities. A new approach is developed to compute the expected information gain, when the posterior distribution is multimodal -a situation previously ignored by the methods aiming at accelerating the nested numerical integrations. Specifically, the posterior distribution is approximated using a mixture distribution constructed by multiple runs of global search for the modes and weighted local Laplace approximations. Under any given probability of capturing all the modes, we provide an estimation of the number of runs of searches, which is dimension independent. It is shown that the novel global-local multimodal approach can be significantly more accurate and more efficient than the other existing approaches, especially when the number of modes is large. The methods can be applied to the designs of experiments with both calibrated and uncalibrated observation noises.

Small-noise approximation for Bayesian optimal experimental design with nuisance uncertainty

“…Bayesian optimal experimental design aims to determine the experimental setup, as described by the design parameter ξ, for the Bayesian inference of θ t . We consider batch design similar to [10], where different experimental setups can be expressed by the parameter ξ, which may be a boolean-, integer-, real-, or complex-valued scalar or vector variable. This parameter depends on the particular physical experiment.…”

“…This method works if the nuisance covariance is sufficiently small compared to the covariance of the error to ensure positive definiteness of the resulting covariance matrix. A slightly different approach was used in [10], where the error covariance was considered an unknown parameter, also requiring marginalization. Another approach is followed in [26], where the authors propose to estimate hyperparameters beforehand and then consider them to be fixed to avoid high-dimensional quadrature for the marginalization.…”

Double-loop quasi-Monte Carlo estimator for nested integration

“…Specifically, in this setting, the nonlinear outer function f in ( 12) is the logarithm, and the inner function g is the likelihood function. Alternatively, we can construct an importance sampling distribution for ϑ to reduce the variance of (quasi) MC methods [7,8,9,2,3,10,13,15,16,17,18]. For a thorough discussion on the Bayesian OED formulation, we refer the reader to [7,2,4].…”