BNS

Bayesian networks, often abbreviated as BNs, are probabilistic graphical models that represent a set of variables and their conditional dependencies via a directed acyclic graph. Each node corresponds to a random variable and each edge denotes a direct probabilistic influence. The joint distribution over all variables factorizes as the product of the conditional distributions of each variable given its parents: P(X1,…,Xn) = ∏i P(Xi | Pa(Xi)).

The graph structure encodes conditional independence assumptions, which allows complex multivariate distributions to be represented compactly. Nodes can represent discrete or continuous variables, and networks can be single or mixed type (for example, Gaussian Bayesian networks handle continuous variables with linear Gaussian conditional distributions).

Inference in Bayesian networks seeks to compute posterior distributions of interest given observed evidence. Exact inference uses algorithms such as variable elimination or junction tree, while approximate methods include Monte Carlo sampling and particle filtering. The computational complexity varies with network size and topology; exact inference is tractable for certain structures but NP-hard in general, making approximate methods common for large or complex networks.

Learning in Bayesian networks occurs in two main areas: parameter learning and structure learning. Parameter learning estimates the conditional distributions P(Xi | Pa(Xi)) from data (with or without prior knowledge). Structure learning aims to identify the graph that best explains the data, often using score-based approaches (e.g., BIC, BDe) or constraint-based approaches, and can be computationally challenging.

BNs are used in diverse fields such as medicine, finance, engineering, and bioinformatics for tasks including diagnosis, risk assessment, decision support, and causal inference. They provide a principled framework for reasoning under uncertainty and integrating prior knowledge with observed data.