  We study the role of interactivity in distributed statistical inference under
information constraints, e.g., communication constraints and local differential
privacy. We focus on the tasks of goodness-of-fit testing and estimation of
discrete distributions. From prior work, these tasks are well understood under
noninteractive protocols. Extending these approaches directly for interactive
protocols is difficult due to correlations that can build due to interactivity;
in fact, gaps can be found in prior claims of tight bounds of distribution
estimation using interactive protocols.
  We propose a new approach to handle this correlation and establish a unified
method to establish lower bounds for both tasks. As an application, we obtain
optimal bounds for both estimation and testing under local differential privacy
and communication constraints. We also provide an example of a natural testing
problem where interactivity helps.
