Bayesian adaptive estimation under a random cost of observation associated with each observable variable ∗

In this paper, we adopt a decision theoretic view to Bayesian adaptive estimation. We extend the framework to situations where each observable variable is associated with a certain random cost of observation and consider the goal of maximizing the expected utility of a sequential experiment that ends when the total cost overruns a given budget. For example, the cost could be defined as the random time taken by each trial in an experiment, and one might wish to maximize the expected total information gain over as many trials as can be completed in 15 minutes. We propose a trial placement rule that maximizes the expected immediate gain in utility divided by the expected cost of observation. This myopic rule is shown to be asymptotically optimal under certain conditions and it is expected to work well in the same situations where the greedy immediate gain maximization works in the absence of costs. However, by simple concerete examples, we also show that the ubiquitous greedy information gain maximization strategy can in fact be arbitrarily much worse than the optimal strategy for a certain number of trials. This research was supported by the Academy of Finland (grant number 121855) and by the European Commission’s FP6, Marie Curie Excellence Grants (MCEXT-CE-2004-014203). The author is grateful to Matti Vihola for comments. Address: Agora Center, University of Jyväskylä, P.O.Box 35, FI-40014 Jyväskylä, Finland. Email address: [email protected]. Fax: +358 14 2604400.