Approximate Counting of Graphical Models Via MCMC Revisited

TL;DR

This paper extends previous MCMC-based methods to estimate ratios of various classes of graphical models up to 31 nodes, providing new insights into their asymptotic properties and implications for DAG learning.

Contribution

It advances the estimation of ratios of different graphical model classes using MCMC from 20 to 31 nodes and proves asymptotic equivalence of certain ratios.

Findings

Extended ratio estimation to 31 nodes

Computed ratios of connected and general graphs

Proved asymptotic ratio of 1 for connected models

Abstract

In Pe\~na (2007), MCMC sampling is applied to approximately calculate the ratio of essential graphs (EGs) to directed acyclic graphs (DAGs) for up to 20 nodes. In the present paper, we extend that work from 20 to 31 nodes. We also extend that work by computing the approximate ratio of connected EGs to connected DAGs, of connected EGs to EGs, and of connected DAGs to DAGs. Furthermore, we prove that the latter ratio is asymptotically 1. We also discuss the implications of these results for learning DAGs from data.