Markov blanket comparison¶

Definition¶

The Markov blanket of a vertex \(X\) in a DAG \(G\) is the set

\[\mathrm{MB}_G(X) = \mathrm{Pa}_G(X) \cup \mathrm{Ch}_G(X) \cup \mathrm{Sp}_G(X),\]

where \(\mathrm{Sp}_G(X)\) denotes the spouses of \(X\) — the parents of \(X\)’s children, excluding \(X\) itself. Under the causal Markov assumption (Pearl, 1988), \(\mathrm{MB}_G(X)\) is the smallest set \(\mathbf{S} \subseteq V\) for which

\[X \perp\!\!\!\perp V \setminus (\mathbf{S} \cup \{X\}) \mid \mathbf{S}.\]

That is, conditioning on \(\mathrm{MB}_G(X)\) renders \(X\) independent of every other variable in the graph. In a Bayesian-network sense \(\mathrm{MB}_G(X)\) contains exactly the information needed to predict \(X\), and no more.

Why scope a comparison to the Markov blanket?¶

Many downstream uses of a recovered graph depend on \(\hat{G}\) only through the per-vertex Markov blankets, not through the global structure:

Feature selection / prediction of \(X\). The Bayes-optimal predictor of \(X\) from observational data depends only on \(\mathrm{MB}(X)\). Errors in \(\mathrm{MB}_{\hat{G}}(X)\) translate directly into prediction bias for \(X\).
Conditional independence testing. A CI test \(X \perp\!\!\!\perp Y \mid \mathbf{S}\) is consistent with \(G\) only if \(\mathbf{S}\) separates \(X\) and \(Y\) in \(G\); the relevant separating sets are subsets of \(\mathrm{MB}_G(X)\) or \(\mathrm{MB}_G(Y)\).
Local-search learners. Algorithms in the Max-Min Parents and Children / IAMB family (Tsamardinos, Brown, & Aliferis, 2003) recover one Markov blanket at a time and never assemble the global structure.

For these workflows the per-vertex MB quality of \(\hat{G}\) is more informative than a single global SHD. The latter aggregates errors across the whole graph; the former localises them to the variables they affect.

Two notions of Markov-blanket error¶

A pair \((\hat{G}, G^*)\) can disagree about the Markov blanket of \(X\) in two distinct ways:

Structural error¶

The two Markov-blanket sub-graphs differ — same vertices, but different edges or orientations. Quantified by applying any standard comparative metric to the sub-graphs:

\[\mathrm{SHD}_{\mathrm{MB}(X)}(\hat{G}, G^*) = \mathrm{SHD}\bigl(\mathrm{MB}_{\hat{G}}(X),\, \mathrm{MB}_{G^*}(X)\bigr).\]

This is the natural local analogue of the global SHD. By construction \(\mathrm{SHD}_{\mathrm{MB}(X)} \le \mathrm{SHD}\), and the gap quantifies the share of structural error that lies outside \(\mathrm{MB}(X)\).

Membership error¶

The two blankets differ as sets of vertices, irrespective of how the within-blanket edges are oriented. Quantified by the Jaccard distance

\[d_{\mathrm{J}}\bigl(\mathrm{MB}_{\hat{G}}(X),\, \mathrm{MB}_{G^*}(X)\bigr) = 1 - \frac{\lvert \mathrm{MB}_{\hat{G}}(X) \cap \mathrm{MB}_{G^*}(X) \rvert} {\lvert \mathrm{MB}_{\hat{G}}(X) \cup \mathrm{MB}_{G^*}(X) \rvert}.\]

Membership error captures the question “which variables enter the blanket?” — relevant when the downstream task is feature selection rather than structure learning. The two notions are complementary: the same MB membership can support very different sub-graph structures, and conversely two identical sub-graphs have zero Jaccard distance trivially.

API surface in `bnmetrics`¶

call	returns	use
`bnmetrics.markov_blanket_indices(g, var)`	tuple of int indices for \(\{X\} \cup \mathrm{MB}_G(X)\)	feed into a custom membership metric (e.g. Jaccard)
`bnmetrics.markov_blanket(g, var)`	sub-`GraphLike` over \(\{X\} \cup \mathrm{MB}_G(X)\), with every endpoint mark incident to those vertices preserved	pass to any standard comparative metric for the structural error
`bnmetrics.compare(g1, g2, per_node=True)`	a `Comparison` whose `.per_node` field maps each vertex name to its descriptive + comparative metrics, computed on the MB sub-graph	survey MB-scoped error across all vertices in one call

Jaccard distance over MB membership is not currently a built-in metric; it can be computed in two lines from markov_blanket_indices if needed.

References¶

Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann.
Tsamardinos, I., Brown, L. E., & Aliferis, C. F. (2003). Algorithms for large scale Markov blanket discovery. Proceedings of FLAIRS-16, 376–380.