  Consider a decision maker who is responsible to dynamically collect
observations so as to enhance his information about an underlying phenomena of
interest in a speedy manner while accounting for the penalty of wrong
declaration. Due to the sequential nature of the problem, the decision maker
relies on his current information state to adaptively select the most
``informative'' sensing action among the available ones. In this paper, using
results in dynamic programming, lower bounds for the optimal total cost are
established. The lower bounds characterize the fundamental limits on the
maximum achievable information acquisition rate and the optimal reliability.
Moreover, upper bounds are obtained via an analysis of two heuristic policies
for dynamic selection of actions. It is shown that the first proposed heuristic
achieves asymptotic optimality, where the notion of asymptotic optimality, due
to Chernoff, implies that the relative difference between the total cost
achieved by the proposed policy and the optimal total cost approaches zero as
the penalty of wrong declaration (hence the number of collected samples)
increases. The second heuristic is shown to achieve asymptotic optimality only
in a limited setting such as the problem of a noisy dynamic search. However, by
considering the dependency on the number of hypotheses, under a technical
condition, this second heuristic is shown to achieve a nonzero information
acquisition rate, establishing a lower bound for the maximum achievable rate
and error exponent. In the case of a noisy dynamic search with size-independent
noise, the obtained nonzero rate and error exponent are shown to be maximum.
