Counting Markov Blanket Structures

TL;DR

This paper introduces a formula to efficiently count Markov blanket structures, revealing that their number grows exponentially but remains significantly fewer than Bayesian network structures, especially as variables increase.

Contribution

The paper provides a novel formula for counting MB structures and quantitatively compares their growth to BN structures, highlighting their relative scarcity.

Findings

Number of MB structures grows exponentially with variables.

Fewer MB structures contain the target than BN structures.

The ratio of BN to MB structures increases exponentially.

Abstract

Learning Markov blanket (MB) structures has proven useful in performing feature selection, learning Bayesian networks (BNs), and discovering causal relationships. We present a formula for efficiently determining the number of MB structures given a target variable and a set of other variables. As expected, the number of MB structures grows exponentially. However, we show quantitatively that there are many fewer MB structures that contain the target variable than there are BN structures that contain it. In particular, the ratio of BN structures to MB structures appears to increase exponentially in the number of variables.