Strategic Monte Carlo Methods for State and Parameter Estimation in High Dimensional Nonlinear Problems

TL;DR

This paper introduces Strategic Monte Carlo (SMC) sampling, a method combining variational annealing and Monte Carlo techniques to efficiently estimate the full posterior distribution in high-dimensional nonlinear data assimilation problems.

Contribution

The paper presents a novel SMC sampling approach that enhances understanding of the posterior distribution structure near its maximum in complex models.

Findings

Accurately estimates mean, standard deviation, and higher moments.

Reveals multimodal structures in the posterior distribution.

Reduces computational time by focusing on high probability regions.

Abstract

In statistical data assimilation one seeks the largest maximum of the conditional probability distribution $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$ of model states, $\mathbf{X}$, and parameters,$\mathbf{p}$, conditioned on observations $\mathbf{Y}$ through minimizing the `action', $A(\mathbf{X}) = -\log P(\mathbf{X},\mathbf{p}|\mathbf{Y})$. This determines the dominant contribution to the expected values of functions of $\mathbf{X}$ but does not give information about the structure of $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$ away from the maximum. We introduce a Monte Carlo sampling method, called Strategic Monte Carlo (SMC) sampling, for estimating $P(\mathbf{X}, \mathbf{p}|\mathbf{Y})$ in the neighborhood of its largest maximum to remedy this limitation. SMC begins with a systematic variational annealing (VA) procedure for finding the smallest minimum of $A(\mathbf{X})$. SMC generates accurate estimates for the mean, standard deviation and other higher moments of $P(\mathbf{X},\mathbf{p}|\mathbf{Y})$. Additionally, the random search allows for an understanding of any multimodal structure that may underly the dynamics of the problem. SMC generates a gaussian probability control term based on the paths determined by VA to minimize a cost function $A(\mathbf{X},\mathbf{p})$. This probability is sampled during the Monte Carlo search of the cost function to constrain the search to high probability regions of the surface thus substantially reducing the time necessary to sufficiently explore the space.