Algorithms of Bayesian Networks

Bayesian networks are graphical models that represent probabilistic dependencies among variables, combining graph theory and probability theory. The core algorithms of Bayesian networks revolve around three key functions: structure learning, parameter learning, and inference.

Structure Learning The objective of structure learning is to determine dependency relationships among nodes (variables) in the network, i.e., constructing a Directed Acyclic Graph (DAG). Commonly used methods include: Constraint-based methods: Utilize statistical tests (such as chi-square tests) to assess conditional independence between variables and infer the structure accordingly. Score-based methods: Evaluate the fitness of different network structures using scoring functions (e.g., BIC, AIC), and employ search algorithms (such as greedy search, Monte Carlo methods) to find the optimal structure. Hybrid methods: Combine the above two strategies by first narrowing the search space with constraint-based approaches, then optimizing with score-based methods.

Parameter Learning After determining the network structure, parameter learning estimates the Conditional Probability Distribution (CPD) for each node. Key approaches include: Maximum Likelihood Estimation (MLE): Directly computes probabilities based on data frequencies, suitable for scenarios with sufficient data. Bayesian Estimation: Introduces prior distributions (e.g., Dirichlet distribution) to avoid overfitting through posterior probability calculations, ideal for small datasets.

Probabilistic Inference Inference is the core application of Bayesian networks, used to answer conditional probability queries (e.g., "Given certain evidence, what is the probability of the target variable?"). Common algorithms include: Exact Inference: Methods like Variable Elimination and Junction Tree Algorithm efficiently compute probabilities through decomposition and pruning techniques. Approximate Inference: Approaches such as Markov Chain Monte Carlo (MCMC) and Variational Inference are suitable for large-scale or complex networks where exact methods are computationally expensive.

Extending the Concept Bayesian networks demonstrate excellent performance in fields like medical diagnosis and financial risk assessment, but their effectiveness highly depends on structural accuracy. Recently, hybrid methods integrating deep learning (e.g., Variational Autoencoders) have emerged as research hotspots to enhance modeling capabilities for high-dimensional data.