Efficient Parameter Estimation of Truncated Boolean Product Distributions

We study the problem of estimating parameters a Boolean product distribution in d dimensions, when samples are truncated by set $$S \subseteq \{0, 1\}^d$$ accessible through membership oracle. This is first time that computational and statistical complexity learning from considered discrete setting. introduce natural notion fatness truncation S, under which reveal enough information about true distribution. show if sufficiently fat, can be generated samples. A stunning consequence virtually any task (e.g., total variation distance, parameter estimation, uniformity or identity testing) performed efficiently for distributions, also samples, with small increase sample complexity. generalize our approach to ranking distributions over alternatives, where we how implies efficient estimation Mallows models Exploring limits identify three conditions necessary identifiability: (i) S should rich enough; (ii) queries; (iii) leave randomness all directions. By carefully adapting Stochastic Gradient Descent (Daskalakis et al., FOCS 2018), these sufficient distributions.