Bayesian Networks: Learning and Inference

Bayesian Networks are graphical models used to represent probabilistic dependencies among variables, combining knowledge from graph theory and probability theory. They are widely applied in machine learning, data mining, and artificial intelligence. Implementing Bayesian network learning and inference in MATLAB can leverage built-in toolboxes or third-party libraries to streamline development.

### Fundamental Concepts of Bayesian Networks A Bayesian Network consists of a Directed Acyclic Graph (DAG) and Conditional Probability Tables (CPTs). Nodes represent random variables, edges indicate dependency relationships between variables, and CPTs quantify the probability distributions of these dependencies.

### Learning Bayesian Networks In MATLAB, learning Bayesian networks typically involves two components: structure learning and parameter learning. Structure Learning: Infers dependency relationships between variables from data. Common methods include constraint-based approaches (e.g., PC algorithm) and score-based methods (e.g., K2 algorithm). Parameter Learning: Estimates CPTs using data given a fixed network structure. Methods include Maximum Likelihood Estimation (MLE) or Bayesian estimation. MATLAB's Statistics and Machine Learning Toolbox provides partial functionality, while third-party libraries like Bayes Net Toolbox (BNT) offer more flexible implementations. For parameter learning, MATLAB's `mle` function can be adapted or custom implementations using expectation-maximization algorithms may be developed.

### Inference in Bayesian Networks Inference refers to using observed variables to deduce probability distributions of unknown variables. MATLAB supports various inference methods: Exact Inference: Such as variable elimination and junction tree algorithms, suitable for small-scale networks. Approximate Inference: Including Markov Chain Monte Carlo (MCMC) or belief propagation, ideal for large-scale or complex networks. Using MATLAB functions and scripts, tasks like conditional probability queries and Maximum a Posteriori (MAP) estimation can be efficiently implemented. The `bayesnet` class or BNT functions like `jtree_inf_engine` facilitate these operations.

### Practical Applications Bayesian Networks find extensive applications in medical diagnosis, financial risk assessment, and natural language processing. MATLAB's matrix computation capabilities make it particularly suitable for probability calculations, while graphical tools like the Bayesian Network Toolbox enable visualization of network structures and inference results. Code implementation often involves data preprocessing with `table2array`, probability updates using Bayesian rules, and visualization with `graph` objects.

In summary, MATLAB provides robust computational and visualization support for Bayesian network learning and inference. Combined with its rich toolboxes, developers can efficiently build and optimize probabilistic models through structured scripting and algorithm integration.