Bayesiannätverk

Bayesian networks, also known as Bayesian belief networks or probabilistic directed acyclic graphical models, are a type of probabilistic graphical model that represents a set of random variables and their conditional dependencies via a directed acyclic graph (DAG). Each node in the graph represents a random variable, which can be a propositional logic variable, a discovery from a data mining process, a function of random variables, a hypothesis, or any other type of computational expression. The edges in the DAG represent conditional probabilities between variables. Specifically, an edge from node A to node B indicates that variable B is directly dependent on variable A.

The structure of a Bayesian network is a DAG, meaning there are no directed cycles. This property is crucial for defining a joint probability distribution over all variables in the network. The network encodes the joint probability distribution of a set of variables X1, X2, ..., Xn as the product of the conditional probabilities of each variable given its parents in the graph: P(X1, ..., Xn) = product of P(Xi | Parents(Xi)) for i = 1 to n. This factorization allows for a more compact and interpretable representation of complex probabilistic relationships compared to explicitly defining the entire joint probability distribution.

Bayesian networks are widely used in artificial intelligence and machine learning for tasks such as probabilistic inference, causal reasoning, and decision making under uncertainty. They can be used to model expert knowledge or learned from data. Applications include medical diagnosis, spam filtering, and risk assessment. Inference in Bayesian networks involves computing the probability of certain events given evidence, which can be done using exact or approximate algorithms.