Testing Probability Distributions Underlying Aggregated Data

Abstract

In this paper, we analyze and study a hybrid model for testing and learning probability distributions. Here, in addition to samples, the testing algorithm is provided with one of two different types of oracles to the unknown distribution D over [n]. More precisely, we consider both the dual and cumulative dual access models, in which the algorithm A can both sample from D and respectively, for any i ∈ [n],

query the probability mass D(i) (query access); or

get the total mass of {1,…,i}, i.e. ∑[i over j=1] D(j) (cumulative access)

In these two models, we bypass the strong lower bounds established in both of the previously studied sampling and query oracle settings for a number of problems, giving constant-query algorithms for most of them. Finally, we show that while the testing algorithms can be in most cases strictly more efficient, some tasks remain hard even with this additional power.