Budgeted Distribution Learning of Belief Net Parameters

Most learning algorithms assume that a training dataset is given initially. We ad- dress the common situation where data is not available initially, but can be obtained, at a cost. We focus on learning Bayesian belief networks (BNs) over discrete variables. As such BNs are models of probabilistic distri- butions, we consider the "generative" chal- lenge of learning the parameters for a fixed structure, that best match the true distribu- tion. We focus on the budgeted learning set- ting, where there is a known fixed cost ci for acquiring the value of the i th feature for any specified instance, and a known total budget to spend acquiring all information. After for- mally defining this problem from a Bayesian perspective, we first consider non-sequential algorithms that must decide, before seeing any results, which features of which instances to probe. We show this is NP-hard, even if all variables are independent, then prove that the greedy allocation algorithm iga is opti- mal here when the costs are uniform, but can otherwise be sub-optimal. We then show that general (sequential) policies perform better than non-sequential, and explore the chal- lenges of learning the parameters for gen- eral belief networks in this sequential setting, describing conditions for when the obvious round-robin algorithm will, versus will not, work optimally. We also explore the eec- tiveness of this and various other heuristic algorithms.