# Bayesian updating in causal probabilistic networks by local computations

Using a Bayesian network can save considerable amounts of memory, if the dependencies in the joint distribution are sparse.For example, a naive way of storing the conditional probabilities of 10 two-valued variables as a table requires storage space for values.The minimal set of nodes which d-separates node X from all other nodes is given by Xs Markov blanket.A path p (allowing paths that are not directed) is said to be d-separated (or blocked) by a set of nodes Z if and only if one of the following holds: A set Z is said to d-separate x from y in a directed acyclic graph G if all paths from x to y in G are d-separated by Z.Once fully specified, a Bayesian network compactly represents the joint probability distribution (JPD) and, thus, can be used for computing the posterior probabilities of any subset of variables given evidence about any other subset.

Two nodes are (unconditionally) independent if the two nodes have no common ancestors (since this is equivalent to saying all paths between these nodes contain at least one collider, which is equivalent to saying that the two nodes are d-separated by the empty set).

Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

The term "Bayesian networks" was coined by Pearl (1985) to emphasize three aspects: Formally, Bayesian networks are directed acyclic graphs whose nodes represent variables, and whose arcs encode conditional independencies between the variables.

Probabilistic models based on directed acyclic graphs (DAG) have a long and rich tradition, beginning with the work of geneticist Sewall Wright in the 1920s. Within statistics, such models are known as directed graphical models; within cognitive science and artificial intelligence, such models are known as Bayesian networks. Thomas Bayes (1702-1761), whose rule for updating probabilities in the light of new evidence is the foundation of the approach. Bayes addressed both the case of discrete probability distributions of data and the more complicated case of continuous probability distributions.

In the discrete case, Bayes’ theorem relates the conditional and marginal probabilities of events.

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If, on the other hand, we wish to answer an interventional question: "What is the likelihood that it would rain, given that we wet the grass?

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