Twenty Questions for Localizing Multiple Objects by Counting: Bayes Optimal Policies for Entropy Loss

We consider the problem of twenty questions with noiseless answers, in which we aim to locate multiple objects by querying the number of objects in each of a sequence of chosen sets. We assume a joint Bayesian prior density on the locations of the objects and seek to choose the sets queried to minimize the expected entropy of the Bayesian posterior distribution after a fixed number of questions. An optimal policy for accomplishing this task is characterized by the dynamic programming equations, but the curse of dimensionality prevents its tractable computation. We first derive a lower bound on the performance achievable by an optimal policy. We then provide explicit performance bounds relative to optimal for two computationally tractable policies: greedy, which maximizes the one-step expected reduction in entropy; and dyadic, which splits the search domain in successively finer partitions. We also show that greedy performs at least as well as the dyadic policy. This can help when choosing the policy most appropriate for a given application: the dyadic policy is easier to compute and nonadaptive, allowing its use in parallel settings or when questions are inexpensive relative to computation; while the greedy policy is more computationally intensive but also uses questions more efficiently, making it the better choice when robust sequential computation is possible. Numerical experiments demonstrate that both procedures outperform a divide-and-conquer benchmark policy from the literature, called sequential bifurcation. Finally, we further characterize performance under the dyadic policy by showing that the entropy of the posterior distribution is asymptotically normal.