Parameterized Complexity Results for Bayesian Inference

TL;DR

This paper establishes complexity classifications for Bayesian inference based on variable count and directed graph structure, revealing inherent computational challenges and the need for exponential resources in certain cases.

Contribution

It introduces new parameterized complexity classes and completeness results, advancing understanding of Bayesian inference's computational complexity.

Findings

Inference is complete for certain parameters, indicating high complexity.

Deterministic algorithms may require exponential space in terms of pathwidth.

Results relate Bayesian inference complexity to graph parameters like pathwidth.

Abstract

We present completeness results for inference in Bayesian networks with respect to two different parameterizations, namely the number of variables and the topological vertex separation number. For this we introduce the parameterized complexity classes $\mathsf{W[1]PP}$ and $\mathsf{XLPP}$, which relate to $\mathsf{W[1]}$ and $\mathsf{XNLP}$ respectively as $\mathsf{PP}$ does to $\mathsf{NP}$. The second parameter is intended as a natural translation of the notion of pathwidth to the case of directed acyclic graphs, and as such it is a stronger parameter than the more commonly considered treewidth. Based on a recent conjecture, the completeness results for this parameter suggest that deterministic algorithms for inference require exponential space in terms of pathwidth and by extension treewidth. These results are intended to contribute towards a more precise understanding of the parameterized complexity of Bayesian inference and thus of its required computational resources in terms of both time and space.