  Sequential decision problems are often approximately solvable by simulating
possible future action sequences. Metalevel decision procedures have been
developed for selecting which action sequences to simulate, based on estimating
the expected improvement in decision quality that would result from any
particular simulation; an example is the recent work on using bandit algorithms
to control Monte Carlo tree search in the game of Go. In this paper we develop
a theoretical basis for metalevel decisions in the statistical framework of
Bayesian selection problems, arguing (as others have done) that this is more
appropriate than the bandit framework. We derive a number of basic results
applicable to Monte Carlo selection problems, including the first finite
sampling bounds for optimal policies in certain cases; we also provide a simple
counterexample to the intuitive conjecture that an optimal policy will
necessarily reach a decision in all cases. We then derive heuristic
approximations in both Bayesian and distribution-free settings and demonstrate
their superiority to bandit-based heuristics in one-shot decision problems and
in Go.
