This article forex statistics and probabilities a list of references, but its sources remain unclear because it has insufficient inline citations. Rain influences whether the sprinkler is activated, and both rain and the sprinkler influence whether the grass is wet.

Formally, Bayesian networks are DAGs whose nodes represent variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Efficient algorithms exist that perform inference and learning in Bayesian networks. Suppose that there are two events which could cause grass to be wet: either the sprinkler is on or it’s raining. The model can answer questions like “What is the probability that it is raining, given the grass is wet?

If, on the other hand, we wish to answer an interventional question: “What is the probability that it would rain, given that we wet the grass? These predictions may not be feasible when some of the variables are unobserved, as in most policy evaluation problems. A back-door path is one that ends with an arrow into X. Sets that satisfy the back-door criterion are called “sufficient” or “admissible. Using a Bayesian network can save considerable amounts of memory, if the dependencies in the joint distribution are sparse. There are three main inference tasks for Bayesian networks.

Because a Bayesian network is a complete model for the variables and their relationships, it can be used to answer probabilistic queries about them. This process of computing the posterior distribution of variables given evidence is called probabilistic inference. In order to fully specify the Bayesian network and thus fully represent the joint probability distribution, it is necessary to specify for each node X the probability distribution for X conditional upon X’s parents. The distribution of X conditional upon its parents may have any form.

Often these conditional distributions include parameters which are unknown and must be estimated from data, sometimes using the maximum likelihood approach. A more fully Bayesian approach to parameters is to treat parameters as additional unobserved variables and to compute a full posterior distribution over all nodes conditional upon observed data, then to integrate out the parameters. This approach can be expensive and lead to large dimension models, so in practice classical parameter-setting approaches are more common. In the simplest case, a Bayesian network is specified by an expert and is then used to perform inference.