  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.
