Motivated by causal inference and physical computing systems, we consider the problem of learning mixture of two bounded-support exponential-family distributions with a shared interaction parameter matrix and mixture component-specific linear parameter vectors. The unknown sequence of component labels belongs to a known structured class. We propose an estimator based on a convex loss that avoids the log-partition function; for each fixed label assignment, estimation is a convex optimization problem. We show that the conditional population loss uniquely identifies the interaction matrix and the linear parameter attached to each sample. For a global empirical minimizer, we establish finite-sample bounds on the interaction error in Frobenius norm and the average squared linear-parameter error. The bounds depend on the logarithm of the number of admissible label assignments, and, for fixed dimension and model bounds, imply consistency when this complexity grows sublinearly with sample size. Additional occupancy and separation conditions yield component and label recovery.
Phevos Paschalidis, Constantinos Daskalakis, Devavrat Shah
原文摘要
We study causal inference under outcome interference for sequential, observational settings. Specifically, we consider settings where the binary outcomes over N units are Markovian across T time steps. At each time step, the outcomes of N units have dependencies captured through an Ising model; each outcome is also impacted through an external field capturing the effects of its treatment as well as latent confounders. Similar to panel data literature, these latent confounders are modeled to have a low-rank factor structure. Our data is a single sample from this high-dimensional distribution. To estimate causal quantities of interest, we provide a computationally efficient method based on Maximum Pseudo-Likelihood Estimation (MPLE) for learning the model parameters. Under mild assumptions, we establish non-asymptotic consistency for parameter estimation and show this translates to faithful estimation of causal quantities of interest after sampling from the learned model. We demonstrate the efficacy of the method through synthetic experiments as well as a real-world case-study investigating causal effects of vaccine rates on COVID-19 death rates within US counties nationwide.