This article includes 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 hierarchical models bayesian statistics forex 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. In other applications the task of defining the network is too complex for humans. In this case the network structure and the parameters of the local distributions must be learned from data. Algorithms have been developed to systematically determine the skeleton of the underlying graph and, then, orient all arrows whose directionality is dictated by the conditional independencies observed.
An alternative method of structural learning uses optimization based search. It requires a scoring function and a search strategy. A common scoring function is posterior probability of the structure given the training data, like the BIC or the BDeu. A particularly fast method for exact BN learning is to cast the problem as an optimization problem, and solve it using integer programming. In order to deal with problems with thousands of variables, it is necessary to use a different approach. One is to first sample one ordering, and then find the optimal BN structure with respect to that ordering. This implies working on the search space of the possible orderings, which is convenient as it is smaller than the space of network structures.
Multiple orderings are then sampled and evaluated. Another method consists of focusing on the sub-class of decomposable models, for which the MLE have a closed form. It is then possible to discover a consistent structure for hundreds of variables. Yet, being a global property of the graph, it considerably increases the difficulty of the learning process. In this context it is possible to use the concept of K-tree for effective learning. This is the simplest example of a hierarchical Bayes model. Eventually the process must terminate, with priors that do not depend on any other unmentioned parameters.
You can help by adding to it. This shrinkage is a typical behavior in hierarchical Bayes models. There are several equivalent definitions of a Bayesian network. V be a set of random variables indexed by V. The difference between the two expressions is the conditional independence of the variables from any of their non-descendants, given the values of their parent variables. Note that the set of parents is a subset of the set of non-descendants because the graph is acyclic.
To develop a Bayesian network, we often first develop a DAG G such that we believe X satisfies the local Markov property with respect to G. Sometimes this is done by creating a causal DAG. We then ascertain the conditional probability distributions of each variable given its parents in G. The Markov blanket of a node is the set of nodes consisting of its parents, its children, and any other parents of its children.