EVERY BIT AS GOOD: RESOLUTION-BASED INFERENCE, BINARY MODEL REPRESENTATIONS, AND A BOOLEAN FRAMEWORK FOR EXPLAINABLE DECISIONS

This work investigates the statistical properties and applications of binary decomposition, resolution-based inference, and representations based on binary variables. We begin by introducing a resolution-based nonparametric two-sample test designed with interpretation in mind. The test decomposes distributional differences into orthogonal signals of non-uniformity up to a user-selected resolution level using the principle of binary expansion. Broadening our scope to arbitrary forms of dependence, we explain how binary "linear" models exhibit Gaussian linear model parallels in estimation and inference. We examine these properties as an additional theoretical tool for understanding generalized linear models for a binary response, emphasizing the unintuitive properties of interaction terms. Next, we compare the multilinear form of estimators based on binary data, identifying structural and combinatorial properties with their continuous analogs. We go on to show that the naturally recursive structure of boolean functions generalizes certain classical results in causal inference. Relevant to modern machine learning applications, we explore bases for functions on the boolean hypercube as a way of understanding and approximating otherwise black-box models.