Testing probability distributions underlying aggregated data
  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 define both the dual and cumulative dual access models, in which the algorithm $A$ can both sample from $D$ and respectively, for any $i\in[n]$,   - query the probability mass $D(i)$ (query access); or   - get the total mass of $\{1,\dots,i\}$, i.e. $\sum_{j=1}^i D(j)$ (cumulative access)   These two models, by generalizing the previously studied sampling and query oracle models, allow us to bypass the strong lower bounds established for a number of problems in these settings, while capturing several interesting aspects of these problems -- and providing new insight on the limitations of the models. 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. 
Cl\'ement Canonne and Ronitt Rubinfeld
cs.DS cs.DM math.PR math.ST stat.TH