Proof Supplement - Learning Sparse Causal Models is not NP-hard (UAI2013)

TL;DR

This paper provides detailed proofs and introduces the FCI+ algorithm, a sound and complete method for causal discovery in complex models with latent confounders, with polynomial worst-case complexity for sparse graphs.

Contribution

It presents the FCI+ algorithm, improving upon FCI by achieving polynomial worst-case complexity for sparse causal graphs with latent confounders.

Findings

FCI+ is sound and complete for causal discovery.

FCI+ has polynomial worst-case complexity in sparse graphs.

The algorithm extends the FCI method with improved computational efficiency.

Abstract

This article contains detailed proofs and additional examples related to the UAI-2013 submission `Learning Sparse Causal Models is not NP-hard'. It describes the FCI+ algorithm: a method for sound and complete causal model discovery in the presence of latent confounders and/or selection bias, that has worst case polynomial complexity of order $N^{2(k+1)}$ in the number of independence tests, for sparse graphs over $N$ nodes, bounded by node degree $k$. The algorithm is an adaptation of the well-known FCI algorithm by (Spirtes et al., 2000) that is also sound and complete, but has worst case complexity exponential in $N$.