Bayesian recovery of the initial condition for the heat equation

TL;DR

This paper develops a Bayesian method for recovering the initial state of the heat equation from noisy data, analyzing how prior choices affect convergence rates and coverage properties, with numerical illustrations.

Contribution

It introduces a Bayesian framework with smoothness-dependent priors for initial condition recovery, achieving optimal rates and exploring coverage behavior.

Findings

Posterior contraction rates depend on true and prior smoothness.

Certain priors attain minimax optimal convergence rates.

Credible sets' coverage varies with prior smoothness, affecting their size.

Abstract

We study a Bayesian approach to recovering the initial condition for the heat equation from noisy observations of the solution at a later time. We consider a class of prior distributions indexed by a parameter quantifying "smoothness" and show that the corresponding posterior distributions contract around the true parameter at a rate that depends on the smoothness of the true initial condition and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the optimal minimax rate. One type of priors leads to a rate-adaptive Bayesian procedure. The frequentist coverage of credible sets is shown to depend on the combination of the prior and true parameter as well, with smoother priors leading to zero coverage and rougher priors to (extremely) conservative results. In the latter case credible sets are much larger than frequentist confidence sets, in that the ratio of diameters diverges to infinity. The results are numerically illustrated by a simulated data example.