Learning Exponential Families from Truncated Samples

Missing data problems have many manifestations across many scientific fields. A fundamental type of missing data problem arises when samples are truncated, i.e., samples that lie in a subset of the support are not observed. Statistical estimation from truncated samples is a classical problem in statistics which dates back to Galton, Pearson, and Fisher. A recent line of work provides the first efficient estimation algorithms for the parameters of a Gaussian distribution [10] and for linear regression with Gaussian noise [11, 14, 37] . In this paper we generalize these results to log-concave exponential families. We provide an estimation algorithm that shows that extrapolation is possible for a much larger class of distributions while it maintains a polynomial sample and time complexity on average. Our algorithm is based on Projected Stochastic Gradient Descent and is not only applicable in a more general setting but is also simpler than the recent algorithms of [10, 26, 11, 14, 37] . Our work also has interesting implications for learning general log-concave distributions and sampling given only access to truncated data.