Javier González

Javier González, Aditya V. Nori

Recent advances in AI have been significantly driven by the capabilities of large language models (LLMs) to solve complex problems in ways that resemble human thinking. However, there is an ongoing debate about the extent to which LLMs are capable of actual reasoning. Central to this debate are two key probabilistic concepts that are essential for connecting causes to their effects: the probability of necessity (PN) and the probability of sufficiency (PS). This paper introduces a framework that is both theoretical and practical, aimed at assessing how effectively LLMs are able to replicate real-world reasoning mechanisms using these probabilistic measures. By viewing LLMs as abstract machines that process information through a natural language interface, we examine the conditions under which it is possible to compute suitable approximations of PN and PS. Our research marks an important step towards gaining a deeper understanding of when LLMs are capable of reasoning, as illustrated by a series of math examples.

Javier González, Aditya V. Nori

Large language models (LLMs) are powerful AI tools that can generate and comprehend natural language text and other complex information. However, the field lacks a mathematical framework to systematically describe, compare and improve LLMs. We propose Hex a framework that clarifies key terms and concepts in LLM research, such as hallucinations, alignment, self-verification and chain-of-thought reasoning. The Hex framework offers a precise and consistent way to characterize LLMs, identify their strengths and weaknesses, and integrate new findings. Using Hex, we differentiate chain-of-thought reasoning from chain-of-thought prompting and establish the conditions under which they are equivalent. This distinction clarifies the basic assumptions behind chain-of-thought prompting and its implications for methods that use it, such as self-verification and prompt programming. Our goal is to provide a formal framework for LLMs that can help both researchers and practitioners explore new possibilities for generative AI. We do not claim to have a definitive solution, but rather a tool for opening up new research avenues. We argue that our formal definitions and results are crucial for advancing the discussion on how to build generative AI systems that are safe, reliable, fair and robust, especially in domains like healthcare and software engineering.

Melanie F. Pradier, Javier González

Matching is a popular approach in causal inference to estimate treatment effects by pairing treated and control units that are most similar in terms of their covariate information. However, classic matching methods completely ignore the geometry of the data manifold, which is crucial to define a meaningful distance for matching, and struggle when covariates are noisy and high-dimensional. In this work, we propose GeoMatching, a matching method to estimate treatment effects that takes into account the intrinsic data geometry induced by existing causal mechanisms among the confounding variables. First, we learn a low-dimensional, latent Riemannian manifold that accounts for uncertainty and geometry of the original input data. Second, we estimate treatment effects via matching in the latent space based on the learned latent Riemannian metric. We provide theoretical insights and empirical results in synthetic and real-world scenarios, demonstrating that GeoMatching yields more effective treatment effect estimators, even as we increase input dimensionality, in the presence of outliers, or in semi-supervised scenarios.