\sup_{S\subseteq [n]} (p(S)-q(S)) > \varepsilon?
\mathcal{P}\subseteq \Delta([n])
# Frontiers in Distribution Testing

## FOCS 2017 Workshop: Saturday, October 14 (Berkeley)

The workshop included an Open Problems session *(see the schedule below)*, which resulted in a list of 12 open questions and directions to tackle and explore. Nine of those have been posted to the Sublinear.info website; the tenth was also posed at a previous workshop, in Baltimore; the last two, more open-ended, are described below.

- Problem 1: Rényi Entropy Estimation
- Problem 2: Beyond Identity Testing
- Problem 3: Instance-Specific Hellinger Testing
- Problem 4: Efficient Profile Maximum Likelihood Computation
- Problem 5: Sample Stretching (or “how to create samples out of thin air”)
- Problem 6: Equivalence Testing Lower Bound via Communication Complexity
- Problem 7: Equivalence Testing with Conditional Samples (or “how to remove that pesky quadratic gap”)
- Problem 8: Separating PDF and CDF Query Models (or “with more power comes... more power. Right?”)
- Problem 9: AM vs. NP for Proofs of Proximity in Distribution Testing (or “do we need that Merlin anyway?”)
- Problem 10: Correcting Independence of Distributions (or “how to create independence out of thin air”)
- Problem 11: There has recently been some work on testing Ising models, Bayes nets, and Markov chains under the viewpoint of distribution testing. What other models, assumptions, or families are natural to study distribution testing of high-dimensional probability distributions?
*(asked by Costis Daskalakis)*. - Problem 12: in Phylogenetics, the question of testing whether a set of species, given their genomes, can be organized as a family tree (or whether, instead, some cross-relations between species happened) is a crucial problem. How to formalize a relevant and natural version of this problem, and study it rigorously either under the model of distribution testing, or a suitable variant thereof?
*(asked by Costis Daskalakis)*.