Bounds and Prediction Intervals for Individual Treatment Effects

Abstract

This dissertation investigates several problems related to bounds and prediction intervals for the individual treatment effect (ITE). While traditional causal inference has primarily focused on population-level parameters such as the average treatment effect (ATE) and the conditional average treatment effect (CATE), the ITE—often considered the ideal target for personalized decision-making -- has recently garnered increasing attention. However, the ITE is generally not identifiable from the observed data, even in the context of randomized experiments. As a result, we consider the problem of bounding the ITE using prediction intervals. In particular, when the marginal distributions of potential outcomes are identifiable from a large, well-conducted randomized experiment, we aim to answer the general question: what constraints exist on the joint distribution of potential outcomes, given these known marginals? Chapters 2 and 3 lay the theoretical foundation for addressing this question. In Chapter 2, we revisit a classical problem posed by Kolmogorov concerning the sharp upper and lower bounds for the cumulative distribution function (cdf) of the sum of two random variables with fixed marginals. Motivated in part by the challenges of bounding individual treatment effects, we focus on the \emph{achievability} of these bounds. Specifically, we distinguish between bounds that are achievable and those that although they provide an infimum or supremum -- and hence cannot be improved -- are not attained by any distribution. We contribute new results for the case of discrete random variables, and we also work to clarify, correct, and make more accessible several theorems in the existing literature. In Chapter 3, we apply the insights from Chapter 2 to the difference of two random variables, with an application on individual treatment effects. We identify and address logical gaps in some prior work and illustrate our results through an example. Then we connect the problem of characterizing joint distributions with fixed marginals to the theory of couplings of probability measures. We generalize a finite version of Strassen’s theorem using a max-flow/min-cut construction, which can be applied on prediction intervals (sets) for the ITE. Finally, we explore a natural extension: bounding the probability mass function (pmf) of the difference of two random variables. In Chapter 4, we build upon the results of the previous chapters and focus on prediction intervals for individual treatment effects (ITE). For a binary treatment, we consider all three types of outcomes: binary, ordinal, and continuous. We begin by examining how to construct valid prediction intervals given known marginal distributions. We then address the converse problem: what necessary conditions must hold for a joint distribution of potential outcomes to exist such that a given prediction interval is valid? We discuss scenarios in which certain points must necessarily be included in the interval. Finally, we compare and contrast the ITE with the average treatment effect (ATE), highlighting their differing implications for causal inference.