Probabilistic Graphical Models - Learning Module

Loading content...

0/245

Markov Blanket

The Minimal Sufficient Information Set

Imagine you're building a machine learning model to predict whether a patient has heart disease. You have access to hundreds of patient attributes: demographics, symptoms, lab results, lifestyle factors, and genetic markers. Which features should you include?

Including irrelevant features wastes computation and can hurt generalization. But how do you know which features are truly relevant?

Here's a profound insight from probabilistic graphical models: there exists a minimal sufficient set—a smallest collection of variables that captures ALL the predictive information about your target. Given this set, all other variables become irrelevant. This set is called the Markov blanket.

The Markov blanket of a variable $X$ is:

The minimal set of other variables $MB(X)$ such that $X$ is conditionally independent of all remaining variables given $MB(X)$.

In other words, if you know the Markov blanket, you know everything there is to know about $X$—the rest of the universe provides no additional information.

This concept has far-reaching implications for:

Feature selection: Only Markov blanket variables matter for prediction
Local computation: Inference about $X$ only requires $MB(X)$
Causal discovery: The blanket reveals direct causal connections
Distributed systems: Each node only needs to communicate with its blanket

What You Will Learn

By the end of this page, you will understand the Markov blanket: its definition for both directed and undirected graphs, algorithms to compute it, theoretical properties, and practical applications. You will see why the Markov blanket is a fundamental concept that appears throughout machine learning, from feature selection to distributed inference.

Formal Definition of the Markov Blanket

Let's establish the formal definition with mathematical precision.

Definition (Markov Blanket):

For a random variable $X$ in a joint distribution $P(X_1, \ldots, X_n)$, the Markov blanket of $X$, denoted $MB(X)$, is the minimal set of variables such that:

$$X \perp!!!\perp {X_1, \ldots, X_n} \setminus ({X} \cup MB(X)) \mid MB(X)$$

In words: Given the Markov blanket, $X$ is conditionally independent of all other variables.

Key properties:

Sufficiency: $MB(X)$ contains all information about $X$ that the other variables provide
Minimality: No proper subset of $MB(X)$ satisfies the independence property
Uniqueness: For positive distributions, the Markov blanket is unique

Graphical characterization:

The beauty of graphical models is that the Markov blanket can be read directly from the graph structure—no probability calculations required.

Markov Blanket in Undirected Graphs (MRFs):

For Markov Random Fields, the Markov blanket is simply the neighbors:

$$MB(X) = \text{Neighbors}(X)$$

This follows directly from the local Markov property: a node is independent of non-neighbors given its neighbors.

Markov Blanket in Directed Graphs (Bayesian Networks):

For Bayesian networks, the Markov blanket consists of three types of nodes:

$$MB(X) = \text{Parents}(X) \cup \text{Children}(X) \cup \text{Parents of Children}(X)$$

Or equivalently:

$$MB(X) = \text{Parents}(X) \cup \text{Children}(X) \cup \text{Spouses}(X)$$

Where "spouses" are other parents of X's children (co-parents).

Why include spouses?

Recall the explaining away effect: if $Y$ is a child of both $X$ and $Z$, then conditioning on $Y$ makes $X$ and $Z$ dependent. So to shield $X$ from the rest of the graph, we need to include not just $Y$, but also $Z$ (the spouse)—otherwise, information could flow through the activated collider.

Markov Blanket in Different Graph Types
Graph Type	Markov Blanket Components	Intuition
Undirected (MRF)	Neighbors	Direct connections are all that matter
Directed (BN)	Parents + Children + Spouses	Include co-parents to block collider paths

Visualizing the Blanket

Think of the Markov blanket as a 'shield' around a node. For undirected graphs, it's the immediate neighbors. For directed graphs, it's more complex because of colliders: you need parents (direct causes), children (direct effects), AND spouses (to block explaining away paths). The blanket completely shields the node from the rest of the network.

Computing the Markov Blanket

Let's implement algorithms to compute Markov blankets from graph structure.

Algorithm for Undirected Graphs (O(degree)):

Simply return the neighbors—trivial!

Algorithm for Directed Graphs (O(n + e)):

Find all parents of the target node
Find all children of the target node
For each child, find its parents (the spouses)
Return the union of these sets, excluding the target itself

markov_blanket.py
Python
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
"""
Computing Markov Blankets in Graphical Models
"""
 
from collections import defaultdict
 
class BayesianNetwork:
    def __init__(self):
        self.parents = defaultdict(set)   # child -> parents
        self.children = defaultdict(set)  # parent -> children
        self.nodes = set()
    
    def add_edge(self, parent, child):
        """Add directed edge parent -> child"""
        self.parents[child].add(parent)
        self.children[parent].add(child)
        self.nodes.update([parent, child])
    
    def markov_blanket(self, node):
        """
        Compute the Markov blanket of a node in a Bayesian network.
        MB(X) = Parents(X) ∪ Children(X) ∪ Spouses(X)
        
        Returns: Set of nodes in the Markov blanket
        """
        blanket = set()
        
        # 1. Add parents
        blanket.update(self.parents[node])
        
        # 2. Add children
        blanket.update(self.children[node])
        
        # 3. Add spouses (other parents of children)
        for child in self.children[node]:
            for spouse in self.parents[child]:
                if spouse != node:
                    blanket.add(spouse)
        
        return blanket
 
 
class MarkovRandomField:
    def __init__(self):
        self.neighbors = defaultdict(set)
        self.nodes = set()
    
    def add_edge(self, u, v):
        """Add undirected edge u -- v"""
        self.neighbors[u].add(v)
        self.neighbors[v].add(u)
        self.nodes.update([u, v])
    
    def markov_blanket(self, node):
        """
        Compute the Markov blanket of a node in an MRF.
        MB(X) = Neighbors(X)
        """
        return self.neighbors[node].copy()
 
 
# Example: Medical Diagnosis Bayesian Network
print("=" * 60)
print("Markov Blankets in a Medical Diagnosis Network")
print("=" * 60)
 
bn = BayesianNetwork()
# Structure:
#   Genetics    Lifestyle
#       \        /
#        \      /
#         Disease
#        /   |   \
#       /    |    \
#   Symptom1 Symptom2 TestResult
 
bn.add_edge('Genetics', 'Disease')
bn.add_edge('Lifestyle', 'Disease')
bn.add_edge('Disease', 'Symptom1')
bn.add_edge('Disease', 'Symptom2')
bn.add_edge('Disease', 'TestResult')
# Add another cause of Symptom1 (makes Disease have a spouse via Symptom1)
bn.add_edge('Environment', 'Symptom1')
 
print("\nNetwork structure:")
print("  Genetics → Disease ← Lifestyle")
print("  Disease → Symptom1 ← Environment")
print("  Disease → Symptom2")
print("  Disease → TestResult")
print()
 
for node in sorted(bn.nodes):
    mb = bn.markov_blanket(node)
    print(f"MB({node}): {mb}")
 
# Verify using d-separation
print("\n" + "=" * 60)
print("Verification: Is Disease ⟂ Environment | MB(Disease)?")
print("=" * 60)
 
mb_disease = bn.markov_blanket('Disease')
print(f"MB(Disease) = {mb_disease}")
print()
 
# Disease should be independent of all non-blanket nodes given blanket
non_blanket = bn.nodes - mb_disease - {'Disease'}
print(f"Non-blanket nodes: {non_blanket}")
print()
 
# Environment is in the blanket (it's a spouse!)
# What about if we had another node?
bn.add_edge('Weather', 'Lifestyle')  # Weather affects Lifestyle
 
print("After adding Weather → Lifestyle:")
for node in sorted(bn.nodes):
    mb = bn.markov_blanket(node)
    print(f"  MB({node}): {mb}")
 
mb_disease = bn.markov_blanket('Disease')
print(f"\nIs Weather outside MB(Disease)? {'Weather' not in mb_disease}")
print("Disease ⟂ Weather | MB(Disease) should hold")

Why Spouses Matter

The spouse inclusion is crucial. Without it, if we conditioned only on parents and children, the child would be an unblocked collider connecting the target to its spouses. By including spouses in the blanket, we ensure that even conditioning on children (which activates them as colliders) cannot let information flow from outside the blanket.

Proving the Markov Blanket Property

Let's prove that the Markov blanket as defined does indeed shield a node from all others.

Theorem: In a Bayesian network, for any node $X$:

$$X \perp!!!\perp (V \setminus {X} \setminus MB(X)) \mid MB(X)$$

where $MB(X) = \text{Parents}(X) \cup \text{Children}(X) \cup \text{Spouses}(X)$.

Proof using d-separation:

We need to show that conditioning on $MB(X)$ blocks all paths from $X$ to any node $Y \notin MB(X) \cup {X}$.

Consider any path from $X$ to $Y$. The path must start at $X$ and end at $Y$.

Case 1: Path goes through a parent of X

Since all parents of $X$ are in $MB(X)$, the path is blocked at the parent (it's a chain or fork, and we're conditioning on it).

Case 2: Path goes through a child of X

Let the path be $X \to C \to \cdots \to Y$ where $C$ is a child of $X$.

Subcase 2a: Next step is to another parent of $C$ (a spouse). The spouse is in $MB(X)$, so the path is blocked.

Subcase 2b: Next step is to a child of $C$ (a grandchild of $X$). $C$ is in $MB(X)$, and $C$ is a chain node on this path (both edges go "downward"). Since we condition on $C$, the path is blocked.

Case 3: Path goes "up" from X to a child, then "up" to a spouse

This is: $X \to C \leftarrow S$ where $S$ is a spouse. $S$ is in $MB(X)$, so the path is blocked at $S$.

Continuing the proof:

Case 4: Path stays entirely above X (ancestors)

If we go to a parent $P$ of $X$ and continue upward, $P$ is in $MB(X)$, so the chain is blocked at $P$.

Case 5: Path goes to a child, then the child is a collider

If the path is $X \to C \leftarrow Y'$ going toward $Y$:

If $C$ has no other children along the path, it's a collider and blocked unless $C$ or its descendants are conditioned
But $C$ is in $MB(X)$, so $C$ IS conditioned
This activates the collider, which is why we need to include $S$ (the spouses)
Since all spouses are in $MB(X)$, the path is blocked at the spouse

Summary:

Every path from $X$ to a node outside $MB(X)$ must pass through either:

A parent (conditioned → blocked)
A child, then a spouse (both conditioned → blocked)
A child, then a grandchild (child conditioned as chain → blocked)

Thus, all paths are blocked, proving $X \perp!!!\perp V \setminus {X} \setminus MB(X) \mid MB(X)$. ∎

The Shield Metaphor

Think of the Markov blanket as a perimeter defense. Parents block incoming causal influence. Children (with their spouses) block outgoing paths. The spouses specifically guard against the 'explaining away' attack—without them, evidence about a child could flow to other causes.

Markov Blanket for Feature Selection

One of the most important applications of Markov blankets is optimal feature selection.

Theorem (Optimal Feature Set):

For predicting a target variable $Y$ from features $X_1, \ldots, X_n$, the Markov blanket $MB(Y)$ is the optimal feature set in the following sense:

Sufficiency: $MB(Y)$ contains all predictive information: $$P(Y | X_1, \ldots, X_n) = P(Y | MB(Y))$$
Minimality: No proper subset of $MB(Y)$ is sufficient: $$\forall S \subsetneq MB(Y): P(Y | X_1, \ldots, X_n) \neq P(Y | S)$$

Implications:

Features outside $MB(Y)$ are provably irrelevant for predicting $Y$
You cannot improve prediction by adding non-blanket features
You cannot remove any blanket feature without losing predictive power

This makes Markov blanket discovery the gold standard for feature selection—it finds the exact set of features you need, no more and no less.

mb_feature_selection.py
Python
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
"""
Markov Blanket-Based Feature Selection
 
We demonstrate that the Markov blanket is sufficient for prediction
by comparing prediction accuracy with different feature sets.
"""
 
import numpy as np
from sklearn.linear_model import LogisticRegression
from sklearn.model_selection import cross_val_score
from collections import defaultdict
 
def generate_data_from_bn(n_samples=2000, seed=42):
    """
    Generate data from a known Bayesian network structure:
    
        X1   X2   X3 (irrelevant, no connection to Y)
         \   |
          v  v
           Y  
          / \
         v   v
        X4   X5
        |
        v
        X6 (grandchild, not in MB(Y))
    
    MB(Y) = {X1, X2, X4, X5}
    Irrelevant: {X3, X6}
    """
    np.random.seed(seed)
    
    # Generate features
    X1 = np.random.randn(n_samples)
    X2 = np.random.randn(n_samples)
    X3 = np.random.randn(n_samples)  # Completely independent of Y
    
    # Y depends on X1 and X2
    logit_Y = 0.8 * X1 + 0.6 * X2
    prob_Y = 1 / (1 + np.exp(-logit_Y))
    Y = (np.random.rand(n_samples) < prob_Y).astype(int)
    
    # X4 and X5 depend on Y (children)
    X4 = Y + 0.5 * np.random.randn(n_samples)
    X5 = 0.7 * Y + 0.5 * np.random.randn(n_samples)
    
    # X6 depends on X4 (grandchild of Y)
    X6 = 0.6 * X4 + 0.4 * np.random.randn(n_samples)
    
    features = np.column_stack([X1, X2, X3, X4, X5, X6])
    feature_names = ['X1', 'X2', 'X3', 'X4', 'X5', 'X6']
    
    return features, Y, feature_names
 
# Generate data
X, y, feature_names = generate_data_from_bn()
 
# Define feature sets
all_features = [0, 1, 2, 3, 4, 5]
markov_blanket = [0, 1, 3, 4]  # X1, X2, X4, X5
parents_only = [0, 1]  # X1, X2
children_only = [3, 4]  # X4, X5
irrelevant_included = [0, 1, 2, 3, 4, 5]  # All including irrelevant
without_children = [0, 1, 2]  # Parents + irrelevant, no children
 
feature_sets = [
    ('All features', all_features),
    ('Markov Blanket', markov_blanket),
    ('Parents only', parents_only),
    ('Children only', children_only),
    ('Without children', without_children),
]
 
print("Feature Selection via Markov Blanket")
print("=" * 60)
print("True DAG structure:")
print("  X1 → Y ← X2")
print("  Y → X4 → X6")
print("  Y → X5")
print("  X3 (independent)")
print()
print(f"True MB(Y) = {{X1, X2, X4, X5}}")
print()
 
print("Cross-validation accuracy comparison:")
print("-" * 60)
 
for name, indices in feature_sets:
    X_subset = X[:, indices]
    scores = cross_val_score(LogisticRegression(), X_subset, y, cv=5)
    selected = [feature_names[i] for i in indices]
    print(f"{name:<25} Features: {selected}")
    print(f"                          Accuracy: {scores.mean():.4f} (+/- {scores.std()*2:.4f})")
    print()
 
print("-" * 60)
print("Observations:")
print("1. Markov blanket achieves same accuracy as all features")
print("2. Parents alone are nearly as good (Y directly depends on them)")
print("3. Children alone have lower accuracy (indirect information)")
print("4. Adding irrelevant features doesn't help")

Learning MB from Data

In practice, we often don't know the true graph structure. Algorithms like IAMB (Incremental Association Markov Blanket) and MMMB (Max-Min Markov Blanket) learn the Markov blanket directly from data using conditional independence tests. These are more statistically efficient than learning the full graph structure.

Local Computation and Distributed Inference

The Markov blanket has profound implications for computation—it enables locality.

Principle of Local Computation:

For any computation involving node $X$, you only need to consider $X$ and its Markov blanket. Everything else is conditionally irrelevant.

Application 1: Gibbs Sampling

In Gibbs sampling, we iteratively sample each variable from its conditional distribution:

$$X_i^{(t+1)} \sim P(X_i | X_{-i}^{(t)})$$

The Markov blanket property means:

$$P(X_i | X_{-i}) = P(X_i | MB(X_i))$$

We only need to look at the Markov blanket to sample $X_i$—not the entire state. For large networks, this is a massive computational saving.

Application 2: Distributed Inference

In a distributed system, each node can:

Maintain its own state
Communicate only with Markov blanket neighbors
Perform local inference updates

No global coordination needed! This enables scalable probabilistic inference on large networks.

Application 3: Modular Testing

To test whether a single CPD is correctly specified, you only need test data for the node and its Markov blanket—not the entire network.

gibbs_sampling_local.py
Python
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
"""
Gibbs Sampling with Markov Blanket Locality
 
Demonstrates how Gibbs sampling only needs the Markov blanket
for each conditional update.
"""
 
import numpy as np
from collections import defaultdict
 
class DiscreteBayesNet:
    def __init__(self):
        self.parents = defaultdict(list)
        self.children = defaultdict(list)
        self.cpds = {}  # node -> CPD function
        self.nodes = []
    
    def add_node(self, name, parents, cpd):
        """
        Add a node with its parents and CPD.
        cpd: function(parent_values) -> probability distribution
        """
        self.nodes.append(name)
        self.parents[name] = parents
        for p in parents:
            self.children[p].append(name)
        self.cpds[name] = cpd
    
    def markov_blanket(self, node):
        """Get the Markov blanket of a node."""
        mb = set(self.parents[node])  # Parents
        mb.update(self.children[node])  # Children
        for child in self.children[node]:
            mb.update(self.parents[child])  # Spouses
        mb.discard(node)
        return mb
    
    def gibbs_sample_node(self, node, current_state):
        """
        Sample a new value for 'node' given the current state.
        
        Key insight: We only need values from the Markov blanket!
        
        P(X | MB(X)) ∝ P(X | Parents(X)) * Π_{C ∈ Children(X)} P(C | Parents(C))
        """
        mb = self.markov_blanket(node)
        
        # Get parent values
        parent_vals = tuple(current_state[p] for p in self.parents[node])
        
        # For each possible value of node, compute unnormalized probability
        probs = []
        for val in [0, 1]:  # Assuming binary
            # P(node = val | parents)
            p = self.cpds[node](parent_vals)[val]
            
            # Multiply by P(child | child's parents) for each child
            for child in self.children[node]:
                child_parent_vals = []
                for cp in self.parents[child]:
                    if cp == node:
                        child_parent_vals.append(val)
                    else:
                        child_parent_vals.append(current_state[cp])
                child_val = current_state[child]
                p *= self.cpds[child](tuple(child_parent_vals))[child_val]
            
            probs.append(p)
        
        # Normalize and sample
        probs = np.array(probs)
        probs = probs / probs.sum()
        return np.random.choice([0, 1], p=probs)
 
# Example: Simple network A → B → C
bn = DiscreteBayesNet()
bn.add_node('A', [], lambda _: [0.3, 0.7])  # P(A=0)=0.3, P(A=1)=0.7
bn.add_node('B', ['A'], lambda pa: [0.9, 0.1] if pa[0]==0 else [0.2, 0.8])
bn.add_node('C', ['B'], lambda pb: [0.8, 0.2] if pb[0]==0 else [0.3, 0.7])
 
print("Gibbs Sampling Demonstration")
print("=" * 50)
print("Network: A → B → C")
print()
 
for node in bn.nodes:
    mb = bn.markov_blanket(node)
    print(f"MB({node}) = {mb}")
 
print()
 
# Run Gibbs sampling
np.random.seed(42)
state = {'A': 0, 'B': 0, 'C': 0}
samples = []
 
print("Gibbs sampling (100 iterations):")
for i in range(100):
    # Sample each node in turn
    for node in bn.nodes:
        mb = bn.markov_blanket(node)
        state[node] = bn.gibbs_sample_node(node, state)
    
    if i >= 50:  # Burn-in
        samples.append(state.copy())
 
# Estimate marginals
print("\nEstimated marginals (from samples):")
for node in bn.nodes:
    counts = sum(1 for s in samples if s[node] == 1)
    print(f"  P({node}=1) ≈ {counts/len(samples):.3f}")
 
print("\nKey insight: Each Gibbs update only looked at the Markov blanket!")
print("For B, we only needed A and C, not global state.")

Scalability Through Locality

The Markov blanket enables "think locally, act locally" computation. In a network with millions of nodes, each update operation only touches a small neighborhood. This is why probabilistic graphical models can scale to massive networks—the blanket creates boundaries that limit computational dependencies.

Markov Blanket Discovery Algorithms

When we don't know the graph structure, we need to discover the Markov blanket from data. Several algorithms have been developed for this purpose.

Algorithm 1: IAMB (Incremental Association Markov Blanket)

Two-phase approach:

Growing phase:

Start with empty blanket $MB = \emptyset$
Repeat: Add the variable most dependent on target given current $MB$
Stop when no variable shows significant dependence

Shrinking phase:

For each variable in $MB$:
Test if removing it maintains conditional independence
Remove if so (it was a false positive)

Algorithm 2: MMMB (Max-Min Markov Blanket)

Uses a "max-min" heuristic:

For each candidate variable, compute minimum dependence over all subsets
Add the variable with maximum of these minima
Repeat until no variable shows significant dependence

Algorithm 3: PCMB (derived from PC algorithm)

Based on constraint-based structure learning:

Run PC algorithm on local neighborhood
Extract Markov blanket from learned structure

Sample complexity considerations:

Algorithm	Complexity	Sample Requirement	Strengths
IAMB	$O(n^2)$ CI tests	Moderate	Simple, fast
MMMB	$O(n \cdot 2^{	MB	})$
PCMB	$O(n^k)$	High	Sound and complete

Practical recommendations:

Start with IAMB for quick results
Use MMMB when IAMB gives inconsistent results
Consider ensemble methods that combine multiple algorithms
Validate with domain knowledge when possible

Key insight: All these algorithms ultimately rely on conditional independence tests. The quality of the discovered Markov blanket depends on:

Sample size (more data → better CI tests)
Test accuracy (right statistical test for data type)
Faithfulness assumption (no accidental independencies)

Markov Blanket vs Full Structure Learning

Learning just the Markov blanket is often easier and more data-efficient than learning the full network structure. If your goal is prediction of a specific target, MB discovery is all you need. It avoids learning irrelevant parts of the network, reducing computational cost and statistical errors.

Advanced Topics and Extensions

Let's explore some advanced aspects of Markov blankets.

Markov Blanket in Factor Graphs:

In factor graphs, the Markov blanket of variable $X$ includes all variables that share a factor with $X$:

$$MB(X) = \bigcup_{f: X \in \text{scope}(f)} \text{scope}(f) \setminus {X}$$

This generalizes both the BN and MRF definitions.

Dynamic Markov Blankets:

In temporal models (e.g., Dynamic Bayesian Networks), the Markov blanket may change over time:

$$MB(X_t) = \text{temporal parents} \cup \text{contemporaneous blanket}$$

This is crucial for filtering and prediction in time-series models.

Approximate Markov Blankets:

When the true blanket is large, we may use approximations:

k-nearest in dependency: Top k most dependent variables
Dependency thresholding: Variables with dependence above threshold
Regularized selection: LASSO-style penalized selection

Markov Blanket in Continuous and Mixed Models:

For Gaussian graphical models, the Markov blanket corresponds to non-zero entries in the precision matrix (inverse covariance):

$$MB(X_i) = {X_j : (\Sigma^{-1})_{ij} \neq 0}$$

This connects graphical model theory to sparse precision matrix estimation (graphical LASSO).

Causal Markov Blanket:

In causal graphs, the Markov blanket has causal interpretation:

Parents: Direct causes of $X$
Children: Direct effects of $X$
Spouses: Variables that share effects with $X$

This makes MB discovery relevant for causal feature selection—features in $MB(Y)$ may be causally relevant for interventions on $Y$.

Blanket Stability:

An important property is stability—does the estimated blanket change with small data perturbations? Stable blankets are more reliable for feature selection. Methods like stability selection can identify robust blanket members.

The Blanket as a Local Neighborhood

The Markov blanket defines a 'local universe' for each variable. This locality principle appears throughout probabilistic AI: in message passing algorithms, in gradient computation, in variational approximations. Understanding the blanket concept deeply enriches your understanding of all these methods.

Summary and Module Conclusion

Let's consolidate the essential knowledge about Markov blankets:

Key Takeaways

•Definition: The Markov blanket $MB(X)$ is the minimal set such that $X \perp!!!\perp V \setminus {X} \setminus MB(X) | MB(X)$.
•In MRFs: $MB(X) = \text{Neighbors}(X)$ — just the direct connections.
•In BNs: $MB(X) = \text{Parents} \cup \text{Children} \cup \text{Spouses}$ — must include co-parents to block colliders.
•Feature selection: The Markov blanket is the optimal, minimal sufficient feature set for prediction.
•Local computation: Gibbs sampling, message passing, and distributed inference only need MB interactions.
•Discovery: Algorithms like IAMB and MMMB can learn the blanket from data without full structure learning.
•Causality: In causal graphs, the blanket reveals direct causal connections.

Module 1 Complete: Probabilistic Graphical Models

Congratulations! You've completed the foundational module on Probabilistic Graphical Models. Let's review what you've mastered:

PGM Overview: Why graphical models matter—combining probability and graph theory to represent complex distributions compactly.
Directed vs Undirected: Bayesian networks for causal/generative models, Markov Random Fields for symmetric correlations, and their different factorization properties.
Conditional Independence: The mathematical foundation—formal definitions, graphoid axioms, and the factorization-independence equivalence.
D-Separation: The algorithmic tool for reading independence from directed graphs—chains, forks, colliders, and the complete criterion.
Markov Blanket: The minimal sufficient set—how to compute it, why it matters, and applications from feature selection to distributed inference.

You now have the foundational understanding needed to work with graphical models. The subsequent modules in this chapter will build on these foundations, covering Bayesian Networks, Markov Random Fields, inference algorithms, and specific models like HMMs and CRFs.

Module Complete

You've mastered the foundational concepts of Probabilistic Graphical Models. You understand how graphs encode probability distributions, how conditional independence enables tractable computation, how to read independence from directed graphs via d-separation, and how Markov blankets define minimal sufficient information sets. These concepts form the bedrock for all advanced topics in graphical models, from exact inference algorithms to approximate methods to learning. Proceed to Module 2 to dive deep into Bayesian Networks.