40. PC-MB — PC Markov Blanket Search

Type: Constraint-based (local)
Output: CPDAG (local to a target variable)

PcMb is a local variant of PC/CPC designed to recover the Markov blanket of a single target variable rather than the entire graph.
Given a target T, PcMb uses conditional independence tests to construct a local CPDAG whose structure encodes all Markov blankets of T that are compatible with the CI information.

---

40.1. Key Idea

PcMb applies PC-style conditional independence reasoning but restricted to the target T.

A Markov blanket of T is any set of variables that renders T independent of all others when conditioned on.
PcMb identifies:

• which variables remain adjacent to T after CI pruning, and

• which orientations around T are required or possible.

The resulting local CPDAG describes the entire family of Markov blankets consistent with the CI information.

---

40.2. When to Use

Use PcMb when:

• You care about a single target variable (e.g., an outcome or label).

• You want a constraint-based Markov blanket without learning the full graph.

• The number of variables is large and a full CPDAG or PAG is too expensive.

• You want all possible Markov blankets, not one heuristic choice.

PcMb is especially helpful for:

• Feature selection or classification

• High-dimensional settings

• Comparing constraint-based and score-based MB learners (PcMb vs FgesMb)

---

40.3. Prior Knowledge Support

PcMb respects all Tetrad background knowledge:

• Required edges

• Forbidden edges

• Tier / temporal constraints

All knowledge is enforced during adjacency pruning and orientation.

---

40.4. Strengths

• Efficient when focusing on one target

• Returns all possible Markov blankets (via the local CPDAG)

• Conservative collider handling (CPC-style) reduces false positive orientations

• Fully compatible with Tetrad knowledge constraints

• Uses standard CI tests (Fisher Z, G-test, KCI, RCIT, BF tests, etc.)

---

40.5. Limitations

• Can still be CI-test intensive when T has many neighbors

• Sensitive to finite-sample CI errors (as with PC/CPC)

• Must be run separately for each target if many MBs are desired

• Does not produce a full-graph CPDAG or MAG/PAG

---

40.6. Key Parameters in Tetrad

All parameters appear in the GUI (camelCase form) and scripting interfaces.

Parameter (camelCase)	Description
target	The distinguished variable T.
independenceTest	CI test used (Fisher Z, G-test, KCI, RCIT, basis-function tests, etc.).
significanceLevel	Alpha level for CI tests.
depth	Maximum conditioning-set size.
colliderOrientationStyle	PC, PC-Max, or CPC-style collider logic.
stableFas	Use order-independent adjacency search.
verbose	Print detailed logs.

---

40.7. Reference

Bai, X., Padman, R., Ramsey, J., & Spirtes, P. (2008).
Tabu search-enhanced graphical models for classification in high dimensions.
INFORMS Journal on Computing, 20(3), 423–437.

---

40.8. Summary

PcMb is a local PC/CPC-style Markov blanket learner for a single target T.
It builds a local CPDAG encoding all Markov blankets of T consistent with the observed CI structure and any supplied background knowledge.