On some testing of hypothesis problems with information constraints

A problem of hypothesis testing is considered where some part of the data cannot be directly observed. Our helper observes those data and can send us some limited amount of information about them. What kind of information allows us to make the best statistical inferences? In particular, what is the minimal information sufficient in order to get the same results as if we could directly observe all the data? Some upper bounds for that minimal amount of information and some related results are obtained. A problem of hypothesis testing is considered where some part of the data cannot be directlyobserved. Our helper observes those data and can send us some limited amount of information about them. What kind of information allows us to make the best statistical inferences? In particular, what is the minimal information sufficient in order to get the same results as if we could directly observe all the data? Some upper bounds for that minimal amount of information and some related results are obtained.