# Inference on a selectively revealed sample

I think this question may be related to cryptography, so I may have the wrong stack exchange, but I am not really sure.

Suppose there are two people Sam and Pam. Suppose we have a distribution, a set of outcomes sampled from that distribution, and a reveal strategy (which I will explain more in a minute). Sam knows all three, but Pam only knows the distribution and the reveal strategy. Sam begins to reveal the outcomes sampled from the distribution one-by-one, based on the strategy. He cannot reveal a previously revealed outcome in the sample. The strategy he chose may be deterministic (in which case it is simply an order on the outcomes in the sample), or it may be probabilistic, in which case there is a function mapping each element of the power set of Sam's sample set (where repeated outcomes are considered unique) to a random variable. I think the deterministic case is straight-forward to understand (if not let me know), but, just to be clear, in the stochastic case, the function from the power set to discrete distributions on the sets provides Sam with a probability distribution over the sampled outcome set for every possible set of previous outcome revelations provided to Pam (which reduces the number of options he has to reveal in the future). Sampling from this probability distribution provides Sam with the next outcome to reveal.

Pam would like to provide a summary or best estimate of the remainder of Sam's sample set after every reveal given her knowledge of the distribution from which Sam's sample set was sampled, Sam's reveal strategy, and all the previously revealed outcomes in Sam's sample.

First, is this a well-characterized problem? If so, has it been studied? If so, is this kind of inference possible in any sort of generality with respect to reveal strategies? It seems to be possible if, for example, Sam were to reveal his outcomes in order from least to greatest, but maybe this is trivial (although even in this case I don't know exactly how to re-summarize the outcome set after every reveal, except maybe in the case where this distribution from which the outcome set was drawn is the uniform distribution). Obviously, if a stochastic strategy maps to a uniform distribution in every case, Pam can learn nothing more about Sam's sample set than what was provided by the distribution from which Sam's sample set was drawn.

• Welcome to CV, user023049! Neat question. Jan 11 at 23:17