# A Compositional Atlas of Tractable Circuit Operations for Probabilistic Inference

Antonio Vergari, YooJung Choi, Anji Liu, Stefano Teso, and Guy Van den Broeck.
In Advances in Neural Information Processing Systems 35 (NeurIPS), 2021

### TL;DR

We derive a unified framework for tractable inference on probabilistic circuits by characterizing the tractability of atomic operations that can be composed to form complex queries.

### Abstract

Circuit representations are becoming the lingua franca to express and reason about tractable generative and discriminative models. In this paper, we show how complex inference scenarios for these models that commonly arise in machine learning—from computing the expectations of decision tree ensembles to information-theoretic divergences of sum-product networks—can be represented in terms of tractable modular operations over circuits. Specifically, we characterize the tractability of simple transformations—sums, products, quotients, powers, logarithms, and exponentials—in terms of sufficient structural constraints of the circuits they operate on, and present novel hardness results for the cases in which these properties are not satisfied. Building on these operations, we derive a unified framework for reasoning about tractable models that generalizes several results in the literature and opens up novel tractable inference scenarios.

### Citation

@inproceedings{VergariNeurIPS21,
author 	= {Vergari, Antonio and Choi, YooJung and Liu, Anji and Teso, Stefano and Van den Broeck, Guy},
title     = {A Compositional Atlas of Tractable Circuit Operations for Probabilistic Inference},
booktitle = {Advances in Neural Information Processing Systems 35 (NeurIPS)},
month     = {dec},
year      = {2021},
}


Preliminary version appeared in the UAI 2021 Workshop on Tractable Probabilistic Modeling (TPM).