Gaussian Processes for Time-Varying Treatment Effects

Abstract

Gaussian Process (GP) models have gained popularity for their flexibility to handle correlation among data sampled from the Gaussian Distribution. The correlation frequently characterizes time-dependent data, such as step count data across different time-horizons. Unlike Gaussian Processes used in a regression setting, the outcomes $y$ are not continuous but are discrete, and the posterior distribution of the outcomes conditioned on our data cannot be solved analytically in closed-form.

This thesis surveys the theory and computation of Gaussian Processes in fitting time-varying treatment effects, drawing upon binary outcomes for its simplicity in illustrating the ideas behind fitting Gaussian Processes on non-Gaussian parameters. Binary outcomes are defined by a Bernoulli distribution, from which the probability parameter $\pi$ exists as both a function of input variables as well as a random variable described by a Gaussian Process.

Gaussian Processes fitted on changing values of $\pi$ over time provide greater flexibility in describing the effects of input variables on a target variable across time. This flexibility has certain desirable qualities in context of causal inference, decision theory, and bandit learning.