  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).
