Probabilistic Circuits as Reasoning Machines in Artificial Intelligence (Part I)

This cumulative habilitation thesis studies probabilistic circuits (PCs) as a powerful and tractable framework for reasoning and learning under uncertainty in artificial intelligence (AI). It first advocates for probability as a core language for AI, emphasizing its connections to logic and information theory; the conceptual simplicity of probabilistic reasoning---based primarily on the sum and product rules; the parallels between probabilistic inference and human cognition; and the role of probability in optimal decision making. However, probability also faces significant computational challenges, as probabilistic inference is NP-hard in almost all probabilistic models. PCs address these challenges through structural constraints that ensure exact computation of a wide range of inference queries in polynomial time, such as marginals, conditionals, most probable explanations, expectations, and more advanced inference tasks. This thesis synthesizes a decade of research across foundations, algorithmic developments, and empirical validation of PCs. Key contributions highlighted in this work are foundational theory of PCs, Bayesian approaches for learning PCs, scalable implementations and integration with deep learning, hybrid models that combine PCs with intractable models, and connections with symbolic machine learning paradigms. This is the first part of my Habilitation Thesis. The second part is omitted, as it comprises the cumulative part of the thesis and has been published at various venues (see Chapter 5).