Obtaining the best value for money in adaptive sequential estimation

In Kujala, J. V., Richardson, U., &; Lyytinen, H. (2010). A Bayesian-optimal principle for learner-friendly adaptation in learning games. Journal of Mathematical Psychology, 54(2), 247–255], we considered an extension of the conventional Bayesian adaptive estimation framework to situations where each observable variable is associated with a certain random cost of observation. We proposed an algorithm that chooses each placement by maximizing the expected gain in utility divided by the expected cost. In this paper, we formally justify this placement rule as an asymptotically optimal solution to the problem of maximizing the expected utility of an experiment that terminates 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 min. A simple, analytically tractable example is considered.