Bayesian network

A Bayesian network or Bayesian belief network or just belief network is a form of probabilistic graphical model. Bayesian network represents joint probability distribution of a set of variables with explicit independency assumptions.

Definition

A Bayesian network is a directed acyclic graph of nodes representing variables and arcs representing probabilistic dependency relations among the variables. If there is an arc from node A to another node B, then variable B depends directly on variable A and A is called a parent of B. If the variable represented by a node has a known value then the node is said to be an evidence node. A node can represent any kind of variable, be it an measured parameter, a latent variable or an hypothesis. Nodes are not restricted to representing random variables; this is what is "Bayesian" about a Bayesian network. Let the variables be X₁, ..., X_n. Let parents(A) be the parents of the node A. Then the joint distribution for X₁ through X_n is represented as the product of the probability distributions $\Pr[X_{i}\mid \operatorname {parents} (X_{i}))]$ for i = 1 to n. If X_i has no parents, its probability distribution is said to be unconditional, otherwise it is conditional.

Questions about incongruent dependence among variables can be answered by studying the graph alone. It can be shown that conditional independence is represented in the graph by the graphical property of d-separation: nodes X and Y are d-separated in the graph, given specified evidence nodes, if and only if variables X and Y are independent given the corresponding evidence variables. The set of all other nodes on which node X can directly depend is given by X's Markov blanket.

Causal Bayesian networks

A causal Bayesian network is a Bayesian network where the directed arcs of the graph are interpreted as representing causal relations in some real domain. The directed arcs do not have to be interpreted as representing causal relations; however in practice knowledge about causal relations is very often used as a guide in drawing Bayesian network graphs, thus resulting in causal Bayesian networks.

Structure learning

In the simplest case, a Bayesian network is specified by an expert and is then used to perform inference after some of the nodes are fixed to observed values. In some applications, such as finding gene regulatory networks, a more complex problem of finding dependencies between variables arises. This can be solved by learning a Bayesian network that fits to the data.

Learning the structure of a Bayesian network (i.e., the graph) is a very important part of machine learning. Given the information that the data is being generated by a Bayesian network and that all the variables are visible in every iteration, the following methods are used to learn the structure of the acyclic graph and the conditional probability table associated with it. The elements of a structure finding algorithm are a scoring function and a search strategy. The time requirement of an exhaustive search returning back a structure that maximizes the score is superexponential in the number of variables. A local search algorithm makes incremental changes aimed at improving the score of the structure. A global search algorithm like Markov chain Monte Carlo can avoid getting trapped in local minima. Friedman et. al. talk about using mutual information between variables and finding a structure that maximizes this. They do this by restricting the parent candidate set to k nodes and exhaustively searching therein.

Parameter learning

In order to fully specify the Bayesian network and thus fully represent the joing probability distribution, it is necessary to further 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. It is common to work with discrete or Gaussian distributions since that simplifies calculations. Sometimes only constraints on a distribution are known; one can then use the principle of maximum entropy to determine a single distribution, the one with the greatest entropy given the constraints. (Analogously, in the specific context of a dynamic Bayesian network, one commonly specifies the conditional distribution for the hidden state's temporal evolution to maximize the entropy rate of the implied stochastic process.)

Often these conditional distributions include parameters which are unknown and must be estimated from data, sometimes using the maximum likelihood approach. Direct maximization of the likelihood (or of the posterior probability) is often complex when there are unobserved variables. A classical approach to this problem is the expectation-maximization algorithm which alternates computing expected values of the unobserved variables conditional on observed data, with maximizing the complete likelihood (or posterior) assuming that previously computed expected values are correct. Under mild regularity conditions this process converges on maximum likelihood (or maximum posterior) values for parameters. 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 practise classical parameter-setting approaches are more common.

Inference

The goal of inference is typically to find the distribution of a subset of the variables, conditional upon some other subset of variables with known values (the evidence), with any remaining variables integrated out. This is known as the posterior distribution of the subset of the variables given the evidence. The posterior gives a universal sufficient statistic for detection applications, when one wants to choose values for the variable subset which minimize some expected loss function, for instance the probability of decision error. A Bayesian network can thus be considered a mechanism for automatically constructing extensions of Bayes' theorem to more complex problems. The most common exact inference methods are variable elimination which eliminates (by integration or summation) the non-observed non-query variables one by one by distributing the sum over the product, clique tree propagation which caches the computation so that the many variables can be queried at one time, and new evidence can be propagated quickly, recursive conditioning which allows for a space-time tradeoff but still allowing for the efficiency of variable elimination when enough space is used. All of these methods have complexity that is exponential in tree width. The most common approximate inference algorithms are stochastic MCMC simulation, mini-bucket elimination which generalizes loopy belief propagation, and variational methods.

Applications

Bayesian networks are used for modelling knowledge in gene regulatory networks, medicine, engineering, text analysis, image processing, data fusion, and decision support systems.

See also

• Bayesian inference
• Bayes' theorem
• Bayesian statistics
• Graphical model
• Machine learning

Links and software

• Association for Uncertainty in Artificial Intelligence: http://www.auai.org/
• Intro to Bayesian networks : http://www.niedermayer.ca/papers/bayesian/bayes.html
• Bayesia: http://www.bayesia.com
• GeNIe & SMILE: http://genie.sis.pitt.edu (offered gratis)
• Hugin: http://www.hugin.com
• Netica: http://www.norsys.com
• BNet: http://www.cra.com/bnet

References

• Finn V. Jensen. Bayesian Networks and Decision Graphs. Springer, 2001.
• Judea Pearl, Stuart Russell. Bayesian Networks. UCLA Cognitive Systems Laboratory, Technical Report (R-277), November 2000.
• Judea Pearl, Stuart Russell. Bayesian Networks, in M. A. Arbib (Ed.), Handbook of Brain Theory and Neural Networks, pp. 157–160, Cambridge, MA: MIT Press, 2003, ISBN 0262011972.
• Enrique Castillo, José Manuel Gutiérrez, and Ali S. Hadi. Expert Systems and Probabilistic Network Models. New York: Springer-Verlag, 1997. ISBN 0-387-94858-9
• Judea Pearl. Fusion, propagation, and structuring in belief networks. Artificial Intelligence 29(3):241–288, 1986.
• J.W. Comley and D.L. Dowe, "Minimum Message Length, MDL and Generalised Bayesian Networks with Asymmetric Languages", chapter 11 (pp265–294) in P. Grunwald, M.A. Pitt and I.J. Myung (eds)., Advances in Minimum Description Length: Theory and Applications, Cambridge, MA: MIT Press, April 2005, ISBN 0262072629. (This paper puts decision trees in internal nodes of Bayes networks using Minimum Message Length (MML). An earlier version is Comley and Dowe (2003), .pdf.)
• Christian Borgelt and Rudolf Kruse. Graphical Models – Methods for Data Analysis and Mining, Chichester, UK: Wiley, 2002, ISBN 0-470-84337-3
• Nevin Lianwen Zhang and David Poole, A simple approach to Bayesian network computations, Proceedings of the Tenth Biennial Canadian Artificial Intelligence Conference (AI-94), Banff, May 1994, 171-178. This paper presents variable elimination for belief networks.