Probing-Based Sequential Decision Making Under Cognitive Limitations

We study a variant of the active sequential hypothesis problem (SHT) with a decision-maker having a source probing capability, and a cognitive impairment in data processing. Here a data source outputs memoryless data samples, conditioned on the instantiated (binary) hypothesis and his current state. A potentially noisy version of the generated data is observed by a decision-maker that has cognitive limitations in processing the generated data; the cognitive impairment can, however, be managed by the decision-maker, albemt under some overall constraint. The decision-maker can also probe the data source, and induce a change in its state so as to generate a sample from a favorable source distribution from a collection known a-priori to all. However, similar to its choice of cognition level, the decision-maker has a probing-constraint (possibly due to practical limitations) that limits its long term choice of state selections. We first formalize the above problem as a constrained optimal stopping time problem in a sequential environment. We then present a systematic study of an optimal strategy for the decision maker comprising the stopping time, the pair of sequences of probing-state selection and cognition-levels, and the terminal decision rule. Using classical techniques of active SHT (involving dynamic programming), we show that the sequential probability ratio test (SPRT) is an optimal testing procedure for this problem, and then characterize the optimal stopping time. We also present an optimal choice for the probing state and cognition sequences.