Asymptotic optimality of myopic information-based strategies for Bayesian adaptive estimation

TL;DR

This paper establishes that greedy, information-based strategies for Bayesian adaptive estimation are asymptotically optimal under certain conditions, even when considering random costs of observations, ensuring minimal posterior uncertainty in the long run.

Contribution

It provides a general asymptotic theory proving the optimality of myopic information gain strategies, extending to settings with variable observation costs.

Findings

Greedy information gain maximization is asymptotically optimal.

Posterior entropy converges almost surely under certain conditions.

Optimality holds even with random observation costs.

Abstract

This paper presents a general asymptotic theory of sequential Bayesian estimation giving results for the strongest, almost sure convergence. We show that under certain smoothness conditions on the probability model, the greedy information gain maximization algorithm for adaptive Bayesian estimation is asymptotically optimal in the sense that the determinant of the posterior covariance in a certain neighborhood of the true parameter value is asymptotically minimal. Using this result, we also obtain an asymptotic expression for the posterior entropy based on a novel definition of almost sure convergence on "most trials" (meaning that the convergence holds on a fraction of trials that converges to one). Then, we extend the results to a recently published framework, which generalizes the usual adaptive estimation setting by allowing different trial placements to be associated with different, random costs of observation. For this setting, the author has proposed the heuristic of maximizing the expected information gain divided by the expected cost of that placement. In this paper, we show that this myopic strategy satisfies an analogous asymptotic optimality result when the convergence of the posterior distribution is considered as a function of the total cost (as opposed to the number of observations).