cognitive/knowledge_base/mathematics/conditional_independence.md

title	type	status	created	complexity	processing_priority	tags	semantic_relations
Conditional Independence	concept	stable	2024-03-15	intermediate	1	mathematics probability statistics graphical_models	type links foundation_for markov_blanket probabilistic_graphical_models type links implements probability_theory information_theory type links relates bayesian_networks markov_random_fields causal_inference

Conditional Independence

Overview

Conditional Independence is a fundamental concept in probability theory and statistics that describes when the occurrence of one event provides no information about another event, given knowledge of a third event. This concept is crucial for understanding probabilistic graphical models, Markov blankets, and efficient inference algorithms.

Mathematical Foundation

Definition

Two random variables X and Y are conditionally independent given Z if:

P(X,Y|Z) = P(X|Z)P(Y|Z)

Equivalently:

P(X|Y,Z) = P(X|Z)

Properties

Chain Rule Decomposition

P(X_1,...,X_n) = \prod_{i=1}^n P(X_i|X_{1:i-1})

D-separation

For nodes X, Y, Z in a Bayesian network:

X \perp\!\!\!\perp Y | Z \iff P(X|Y,Z) = P(X|Z)

Implementation

Testing Conditional Independence

class ConditionalIndependenceTester:
    def __init__(self,
                 data: np.ndarray,
                 alpha: float = 0.05):
        """Initialize CI tester.
        
        Args:
            data: Data matrix
            alpha: Significance level
        """
        self.data = data
        self.alpha = alpha
    
    def partial_correlation_test(self,
                               x: int,
                               y: int,
                               z: List[int]) -> bool:
        """Test CI using partial correlation.
        
        Args:
            x: First variable index
            y: Second variable index
            z: Conditioning set indices
            
        Returns:
            is_independent: Whether variables are CI
        """
        # Compute partial correlation
        corr = self.compute_partial_correlation(x, y, z)
        
        # Fisher z-transform
        z_score = self.fisher_z_transform(corr, len(self.data))
        
        # Test significance
        return abs(z_score) < stats.norm.ppf(1 - self.alpha/2)
    
    def mutual_information_test(self,
                              x: int,
                              y: int,
                              z: List[int]) -> bool:
        """Test CI using conditional mutual information.
        
        Args:
            x: First variable index
            y: Second variable index
            z: Conditioning set indices
            
        Returns:
            is_independent: Whether variables are CI
        """
        # Estimate conditional mutual information
        cmi = self.estimate_cmi(x, y, z)
        
        # Apply threshold test
        return cmi < self.compute_threshold()

Graphical Model Implementation

class ConditionalIndependenceGraph:
    def __init__(self,
                 n_nodes: int):
        """Initialize CI graph.
        
        Args:
            n_nodes: Number of nodes
        """
        self.n_nodes = n_nodes
        self.adjacency = np.zeros((n_nodes, n_nodes))
        self.separating_sets = {}
    
    def add_edge(self,
                i: int,
                j: int):
        """Add edge between nodes.
        
        Args:
            i: First node
            j: Second node
        """
        self.adjacency[i,j] = 1
        self.adjacency[j,i] = 1
    
    def find_separating_set(self,
                          i: int,
                          j: int) -> Set[int]:
        """Find separating set between nodes.
        
        Args:
            i: First node
            j: Second node
            
        Returns:
            sep_set: Separating set
        """
        # Implement separation set search
        pass
    
    def is_conditionally_independent(self,
                                  i: int,
                                  j: int,
                                  z: Set[int]) -> bool:
        """Check if nodes are conditionally independent.
        
        Args:
            i: First node
            j: Second node
            z: Conditioning set
            
        Returns:
            is_ci: Whether nodes are CI
        """
        return self._check_separation(i, j, z)

Applications

Structure Learning

PC Algorithm

• Start with complete graph
• Remove edges based on CI tests
• Orient remaining edges
• Infer causal structure

FCI Algorithm

• Handle latent confounders
• Test ancestral relationships
• Build PAG representation

Probabilistic Inference

Belief Propagation

• Message passing
• Factor graph operations
• Marginal computation

Variational Inference

• Mean field approximation
• Factorized distributions
• Evidence lower bound

Best Practices

Testing

1. Choose appropriate test
2. Consider sample size
3. Handle multiple testing
4. Validate assumptions

Implementation

1. Efficient data structures
2. Numerical stability
3. Sparse representations
4. Caching results

Validation

1. Cross-validation
2. Robustness checks
3. Sensitivity analysis
4. Benchmark comparison

Common Issues

Technical Challenges

1. Finite sample effects
2. Curse of dimensionality
3. Computational complexity
4. Numerical precision

Solutions

1. Regularization
2. Efficient algorithms
3. Approximation methods
4. Robust statistics

Related Documentation

• probability_theory
• markov_blanket
• graphical_models
• causal_inference