Belief Propagation in Graphical Models

Belief Propagation (BP) is an algorithm used for inference in probabilistic graphical models, such as Bayesian Networks and Markov Random Fields. It efficiently computes the marginal probabilities of variables given observed evidence by passing messages between nodes in a graph.

Message Passing in Belief Propagation

Belief Propagation works by exchanging messages between nodes based on local computations. It operates in two main ways:

  1. Sum-Product Algorithm (For Marginal Probabilities): Used in probabilistic inference, it computes the probability of a variable by summing over all possible values of other variables.

  2. Max-Product Algorithm (For MAP Estimation): Finds the most probable state configuration by maximizing probabilities instead of summing.

Steps of Message Passing:

  1. Initialization: Nodes initialize messages based on prior distributions.

  2. Message Passing: Each node sends messages to its neighbors, summarizing its belief about the variable.

  3. Updating Beliefs: Nodes update their beliefs based on incoming messages.

  4. Convergence: The process continues until beliefs stabilize or reach a stopping criterion.

Inference in Probabilistic Networks

  • Tree-Structured Graphs: BP guarantees exact inference.

  • Loopy Graphs (Approximate Inference): Loopy Belief Propagation (LBP) is used, though it may not always converge.

Applications of Belief Propagation

  • Error correction in coding theory (e.g., Turbo codes).

  • Image denoising and segmentation in computer vision.

  • Natural language processing for dependency parsing.

Belief Propagation is a fundamental technique for efficiently inferring probabilities in complex probabilistic networks.