cognitive/knowledge_base/mathematics/probabilistic_graphical_models.md

title	type	status	created	complexity	processing_priority	tags	semantic_relations
Probabilistic Graphical Models	concept	stable	2024-03-15	advanced	1	mathematics probability graphical_models machine_learning	type links foundation_for markov_blanket bayesian_networks markov_random_fields type links implements probability_theory graph_theory conditional_independence type links relates variational_inference belief_propagation causal_inference information_theory

Probabilistic Graphical Models

Overview

Probabilistic Graphical Models (PGMs) provide a framework for representing and reasoning about complex probability distributions using graph structures. They combine probability theory with graph theory to create powerful tools for modeling uncertainty, causality, and dependencies in complex systems.

Mathematical Foundation

Graph Representation

Directed Graphs (Bayesian Networks)

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

where:

• X_i are random variables
• Pa(X_i) are parents of X_i

Undirected Graphs (Markov Random Fields)

P(X_1,...,X_n) = \frac{1}{Z}\prod_{C \in \mathcal{C}} \phi_C(X_C)

where:

• \mathcal{C} are maximal cliques
• \phi_C are potential functions
• Z is partition function

Implementation

Graph Structure

class ProbabilisticGraph:
    def __init__(self,
                 nodes: List[str],
                 edges: List[Tuple[str, str]],
                 directed: bool = True):
        """Initialize probabilistic graph.
        
        Args:
            nodes: List of node names
            edges: List of edges
            directed: Whether graph is directed
        """
        self.nodes = nodes
        self.edges = edges
        self.directed = directed
        
        # Initialize graph structure
        self.adjacency = self._build_adjacency()
        self.factors = {}
        
    def _build_adjacency(self) -> Dict[str, Set[str]]:
        """Build adjacency structure.
        
        Returns:
            adjacency: Adjacency dictionary
        """
        adj = {node: set() for node in self.nodes}
        
        for i, j in self.edges:
            adj[i].add(j)
            if not self.directed:
                adj[j].add(i)
        
        return adj
    
    def add_factor(self,
                  variables: List[str],
                  factor: torch.Tensor):
        """Add factor to graph.
        
        Args:
            variables: Variables in factor
            factor: Factor tensor
        """
        key = tuple(sorted(variables))
        self.factors[key] = factor
    
    def get_markov_blanket(self,
                          node: str) -> Set[str]:
        """Get Markov blanket of node.
        
        Args:
            node: Target node
            
        Returns:
            blanket: Markov blanket
        """
        if self.directed:
            # For Bayesian networks
            parents = self.get_parents(node)
            children = self.get_children(node)
            spouses = set().union(*[
                self.get_parents(child)
                for child in children
            ])
            return parents.union(children).union(spouses)
        else:
            # For Markov random fields
            return self.adjacency[node]

Inference Algorithms

class InferenceEngine:
    def __init__(self,
                 graph: ProbabilisticGraph):
        """Initialize inference engine.
        
        Args:
            graph: Probabilistic graph
        """
        self.graph = graph
        
    def variable_elimination(self,
                           query_var: str,
                           evidence: Dict[str, Any]) -> torch.Tensor:
        """Perform variable elimination.
        
        Args:
            query_var: Query variable
            evidence: Evidence dictionary
            
        Returns:
            distribution: Resulting distribution
        """
        # Get elimination ordering
        ordering = self.get_elimination_ordering(query_var, evidence)
        
        # Initialize factor list
        factors = self.graph.factors.copy()
        
        # Eliminate variables
        for var in ordering:
            # Collect relevant factors
            relevant_factors = self.get_relevant_factors(var, factors)
            
            # Multiply factors
            product = self.multiply_factors(relevant_factors)
            
            # Marginalize
            new_factor = self.marginalize(product, var)
            
            # Update factor list
            factors[self.get_factor_key(new_factor)] = new_factor
        
        return self.normalize(factors[query_var])
    
    def belief_propagation(self,
                          max_iterations: int = 100,
                          tolerance: float = 1e-6) -> Dict[str, torch.Tensor]:
        """Perform belief propagation.
        
        Args:
            max_iterations: Maximum iterations
            tolerance: Convergence tolerance
            
        Returns:
            beliefs: Node beliefs
        """
        # Initialize messages
        messages = self.initialize_messages()
        
        # Message passing
        for _ in range(max_iterations):
            old_messages = messages.copy()
            
            # Update messages
            for i, j in self.graph.edges:
                messages[(i,j)] = self.compute_message(i, j, messages)
            
            # Check convergence
            if self.check_convergence(messages, old_messages, tolerance):
                break
        
        # Compute final beliefs
        return self.compute_beliefs(messages)

Learning Algorithms

class StructureLearning:
    def __init__(self,
                 data: torch.Tensor,
                 nodes: List[str]):
        """Initialize structure learning.
        
        Args:
            data: Training data
            nodes: Node names
        """
        self.data = data
        self.nodes = nodes
        
    def score_based_learning(self,
                           score_fn: str = 'bic') -> ProbabilisticGraph:
        """Learn structure using score-based method.
        
        Args:
            score_fn: Scoring function
            
        Returns:
            graph: Learned graph
        """
        # Initialize empty graph
        graph = ProbabilisticGraph(self.nodes, [])
        
        # Hill climbing
        while True:
            best_score = float('-inf')
            best_operation = None
            
            # Try all possible operations
            for op in self.get_possible_operations(graph):
                # Apply operation
                new_graph = self.apply_operation(graph, op)
                
                # Compute score
                score = self.compute_score(new_graph, score_fn)
                
                if score > best_score:
                    best_score = score
                    best_operation = op
            
            if best_operation is None:
                break
                
            # Apply best operation
            graph = self.apply_operation(graph, best_operation)
        
        return graph
    
    def constraint_based_learning(self) -> ProbabilisticGraph:
        """Learn structure using constraint-based method.
        
        Returns:
            graph: Learned graph
        """
        # Start with complete graph
        edges = [
            (i, j) for i in self.nodes
            for j in self.nodes if i != j
        ]
        graph = ProbabilisticGraph(self.nodes, edges)
        
        # Remove edges based on CI tests
        for i in self.nodes:
            for j in self.nodes:
                if i == j:
                    continue
                    
                # Find separating set
                sep_set = self.find_separating_set(i, j, graph)
                
                if sep_set is not None:
                    graph.remove_edge(i, j)
        
        # Orient edges
        return self.orient_edges(graph)

Applications

Bayesian Networks

Causal Modeling

• Causal relationships
• Intervention analysis
• Counterfactual reasoning

Expert Systems

• Medical diagnosis
• Fault diagnosis
• Decision support

Markov Random Fields

Computer Vision

• Image segmentation
• Object recognition
• Scene understanding

Natural Language

• Part-of-speech tagging
• Named entity recognition
• Semantic parsing

Best Practices

Model Design

1. Choose appropriate graph type
2. Define clear semantics
3. Handle missing data
4. Consider scalability

Implementation

1. Efficient data structures
2. Optimize inference
3. Manage memory
4. Profile performance

Validation

1. Cross-validation
2. Structure validation
3. Parameter validation
4. Inference validation

Common Issues

Technical Challenges

1. Inference intractability
2. Structure learning complexity
3. Parameter estimation
4. Model selection

Solutions

1. Approximate inference
2. Sparse structures
3. Parameter tying
4. Regularization

Related Documentation

• probability_theory
• graph_theory
• markov_blanket
• conditional_independence
• variational_inference
• belief_propagation