Bayesian network

A Bayesian network, Bayes network, belief network, Bayes(ian) model or probabilistic directed acyclic graphical model is a probabilistic graphical model (a type of statistical model) that represents a set of random variables and their conditional dependencies via a directed acyclic graph (DAG). For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.
Formally, Bayesian networks are DAGs whose nodes represent random variables in the Bayesian sense: they may be observable quantities, latent variables, unknown parameters or hypotheses. Edges represent conditional dependencies; nodes that are not connected represent variables that are conditionally independent of each other. Each node is associated with a probability function that takes as input a particular set of values for the node's parent variables and gives the probability of the variable represented by the node. For example, if the parents are Boolean variables then the probability function could be represented by a table of entries, one entry for each of the possible combinations of its parents being true or false. Similar ideas may be applied to undirected, and possibly cyclic, graphs; such are called Markov networks.
Efficient algorithms exist that perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (e.g. speech signals or protein sequences) are called dynamic Bayesian networks. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called influence diagrams.
Contents
Example
Suppose that there are two events which could cause grass to be wet: either the sprinkler is on or it's raining. Also, suppose that the rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler is usually not turned on). Then the situation can be modeled with a Bayesian network (shown). All three variables have two possible values, T (for true) and F (for false).
The joint probability function is:
where the names of the variables have been abbreviated to G = Grass wet (yes/no), S = Sprinkler turned on (yes/no), and R = Raining (yes/no).
The model can answer questions like "What is the probability that it is raining, given the grass is wet?" by using the conditional probability formula and summing over all nuisance variables:
Using the expansion for the joint probability function and the conditional probabilities stated in the diagram, one can evaluate each term in the sums in the numerator and denominator. For example,
Then the numerical results (subscripted by the associated variable values) are
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?" the answer would be governed by the postintervention joint distribution function obtained by removing the factor from the preintervention distribution. As expected, the likelihood of rain is unaffected by the action: .
If, moreover, we wish to predict the impact of turning the sprinkler on, we have
with the term removed, showing that the action has an effect on the grass but not on the rain.
These predictions may not be feasible when some of the variables are unobserved, as in most policy evaluation problems. The effect of the action can still be predicted, however, whenever a criterion called "backdoor" is satisfied.^{1} It states that, if a set Z of nodes can be observed that dseparates (or blocks) all backdoor paths from X to Y then . A backdoor path is one that ends with an arrow into X. Sets that satisfy the backdoor criterion are called "sufficient" or "admissible." For example, the set Z = R is admissible for predicting the effect of S = T on G, because R dseparate the (only) backdoor path S ← R → G. However, if S is not observed, there is no other set that dseparates this path and the effect of turning the sprinkler on (S = T) on the grass (G) cannot be predicted from passive observations. We then say that P(Gdo(S = T)) is not "identified." This reflects the fact that, lacking interventional data, we cannot determine if the observed dependence between S and G is due to a causal connection or is spurious (apparent dependence arising from a common cause, R). (see Simpson's paradox)
To determine whether a causal relation is identified from an arbitrary Bayesian network with unobserved variables, one can use the three rules of "docalculus"^{1}^{2} and test whether all do terms can be removed from the expression of that relation, thus confirming that the desired quantity is estimable from frequency data.^{3}
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 twovalued variables as a table requires storage space for values. If the local distributions of no variable depends on more than 3 parent variables, the Bayesian network representation only needs to store at most values.
One advantage of Bayesian networks is that it is intuitively easier for a human to understand (a sparse set of) direct dependencies and local distributions than complete joint distribution.
Inference and learning
There are three main inference tasks for Bayesian networks.
Inferring unobserved variables
Because a Bayesian network is a complete model for the variables and their relationships, it can be used to answer probabilistic queries about them. For example, the network can be used to find out updated knowledge of the state of a subset of variables when other variables (the evidence variables) are observed. This process of computing the posterior distribution of variables given evidence is called probabilistic inference. 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 applying Bayes' theorem to complex problems.
The most common exact inference methods are: variable elimination, which eliminates (by integration or summation) the nonobserved nonquery variables one by one by distributing the sum over the product; clique tree propagation, which caches the computation so that many variables can be queried at one time and new evidence can be propagated quickly; and recursive conditioning and AND/OR search, which allow for a spacetime tradeoff and match the efficiency of variable elimination when enough space is used. All of these methods have complexity that is exponential in the network's treewidth. The most common approximate inference algorithms are importance sampling, stochastic MCMC simulation, minibucket elimination, loopy belief propagation, generalized belief propagation, and variational methods.
Parameter learning
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. 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 expectationmaximization 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 practice classical parametersetting approaches are more common.
Structure learning
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.
Automatically learning the graph structure of a Bayesian network is a challenge pursued within machine learning. The basic idea goes back to a recovery algorithm developed by Rebane and Pearl (1987)^{4} and rests on the distinction between the three possible types of adjacent triplets allowed in a directed acyclic graph (DAG):
Type 1 and type 2 represent the same dependencies ( and are independent given ) and are, therefore, indistinguishable. Type 3, however, can be uniquely identified, since and are marginally independent and all other pairs are dependent. Thus, while the skeletons (the graphs stripped of arrows) of these three triplets are identical, the directionality of the arrows is partially identifiable. The same distinction applies when and have common parents, except that one must first condition on those parents. 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.^{1}^{5}^{6}^{7}
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. The time requirement of an exhaustive search returning a structure that maximizes the score is superexponential in the number of variables. A local search strategy 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.^{8}^{9} 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.
Another method consists of focusing on the subclass of decomposable models, for which the MLE have a closed form. It is then possible to discover a consistent structure for hundreds of variables.^{10}
Statistical introduction
Given data and parameter , a simple Bayesian analysis starts with a prior probability (prior) and likelihood to compute a posterior probability .
Often the prior on depends in turn on other parameters that are not mentioned in the likelihood. So, the prior must be replaced by a likelihood , and a prior on the newly introduced parameters is required, resulting in a posterior probability
This is the simplest example of a hierarchical Bayes model.^{clarification needed]]}
The process may be repeated; for example, the parameters may depend in turn on additional parameters , which will require their own prior. Eventually the process must terminate, with priors that do not depend on any other unmentioned parameters.
Introductory examples
This section requires expansion. (March 2009) 
Suppose we have measured the quantities each with normally distributed errors of known standard deviation ,
Suppose we are interested in estimating the . An approach would be to estimate the using a maximum likelihood approach; since the observations are independent, the likelihood factorizes and the maximum likelihood estimate is simply
However, if the quantities are related, so that for example we may think that the individual have themselves been drawn from an underlying distribution, then this relationship destroys the independence and suggests a more complex model, e.g.,
with improper priors flat, flat. When , this is an identified model (i.e. there exists a unique solution for the model's parameters), and the posterior distributions of the individual will tend to move, or shrink away from the maximum likelihood estimates towards their common mean. This shrinkage is a typical behavior in hierarchical Bayes models.
Restrictions on priors
Some care is needed when choosing priors in a hierarchical model, particularly on scale variables at higher levels of the hierarchy such as the variable in the example. The usual priors such as the Jeffreys prior often do not work, because the posterior distribution will be improper (not normalizable), and estimates made by minimizing the expected loss will be inadmissible.
Definitions and concepts
There are several equivalent definitions of a Bayesian network. For all the following, let G = (V,E) be a directed acyclic graph (or DAG), and let X = (X_{v})_{v ∈ V} be a set of random variables indexed by V.
Factorization definition
X is a Bayesian network with respect to G if its joint probability density function (with respect to a product measure) can be written as a product of the individual density functions, conditional on their parent variables:^{11}
where pa(v) is the set of parents of v (i.e. those vertices pointing directly to v via a single edge).
For any set of random variables, the probability of any member of a joint distribution can be calculated from conditional probabilities using the chain rule (given a topological ordering of X) as follows:^{11}
Compare this with the definition above, which can be written as:
for each which is a parent of
The difference between the two expressions is the conditional independence of the variables from any of their nondescendants, given the values of their parent variables.
Local Markov property
X is a Bayesian network with respect to G if it satisfies the local Markov property: each variable is conditionally independent of its nondescendants given its parent variables:^{12}
where de(v) is the set of descendants of v.
This can also be expressed in terms similar to the first definition, as
 for each which is not a descendant of for each which is a parent of
Note that the set of parents is a subset of the set of nondescendants because the graph is acyclic.
Developing Bayesian networks
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. In many cases, in particular in the case where the variables are discrete, if we define the joint distribution of X to be the product of these conditional distributions, then X is a Bayesian network with respect to G.^{13}
Markov blanket
The Markov blanket of a node is the set of nodes consisting of its parents, its children, and any other parents of its children. This set renders it independent of the rest of the network; the joint distribution of the variables in the Markov blanket of a node is sufficient knowledge for calculating the distribution of the node. X is a Bayesian network with respect to G if every node is conditionally independent of all other nodes in the network, given its Markov blanket.^{12}
dseparation
This definition can be made more general by defining the "d"separation of two nodes, where d stands for directional.^{14}^{15} Let P be a trail (that is, a collection of edges which is like a path, but each of whose edges may have any direction) from node u to v. Then P is said to be dseparated by a set of nodes Z if and only if (at least) one of the following holds:
 P contains a chain, u ← m ← v, such that the middle node m is in Z,
 P contains a fork, u ← m → v, such that the middle node m is in Z, or
 P contains an inverted fork (or collider), u → m ← v, such that the middle node m is not in Z and no descendant of m is in Z.
Thus u and v are said to be dseparated by Z if all trails between them are dseparated. If u and v are not dseparated, they are called dconnected.
X is a Bayesian network with respect to G if, for any two nodes u, v:
where Z is a set which dseparates u and v. (The Markov blanket is the minimal set of nodes which dseparates node v from all other nodes.)
Hierarchical models
The term hierarchical model is sometimes considered a particular type of Bayesian network, but has no formal definition. Sometimes the term is reserved for models with three or more levels of random variables; other times, it is reserved for models with latent variables. In general, however, any moderately complex Bayesian network is usually termed "hierarchical".
Causal networks
Although Bayesian networks are often used to represent causal relationships, this need not be the case: a directed edge from u to v does not require that X_{v} is causally dependent on X_{u}. This is demonstrated by the fact that Bayesian networks on the graphs:
are equivalent: that is they impose exactly the same conditional independence requirements.
A causal network is a Bayesian network with an explicit requirement that the relationships be causal. The additional semantics of the causal networks specify that if a node X is actively caused to be in a given state x (an action written as do(X=x)), then the probability density function changes to the one of the network obtained by cutting the links from X's parents to X, and setting X to the caused value x.^{1} Using these semantics, one can predict the impact of external interventions from data obtained prior to intervention.
Applications
Bayesian networks are used for modelling knowledge in computational biology and bioinformatics (gene regulatory networks, protein structure, gene expression analysis,^{16} sports betting,^{17}^{18} learning epistasis from GWAS data sets ^{19}) medicine,^{20} biomonitoring,^{21} document classification, information retrieval,^{22} semantic search,^{23} image processing, data fusion, decision support systems,^{24} engineering, gaming and law.^{25}^{26}^{27} There are texts applying Bayesian networks to bioinformatics ^{28} and financial and marketing informatics.^{29}
Software
 WinBUGS
 OpenBUGS (website), further (open source) development of WinBUGS.
 OpenMarkov, open source software and API implemented in Java
 Graphical Models Toolkit (GMTK) — GMTK is an open source, publicly available toolkit for rapidly prototyping statistical models using dynamic graphical models (DGMs) and dynamic Bayesian networks (DBNs). GMTK can be used for applications and research in speech and language processing, bioinformatics, activity recognition, and any time series application.
 Just another Gibbs sampler (JAGS) (website)
 Stan (software) (website) — Stan is an opensource package for obtaining Bayesian inference using the NoUTurn sampler, a variant of Hamiltonian Monte Carlo. It’s somewhat like BUGS, but with a different language for expressing models and a different sampler for sampling from their posteriors. RStan is the R interface to Stan.
 PyMC — PyMC is a python module that implements Bayesian statistical models and fitting algorithms, including Markov chain Monte Carlo. Its flexibility and extensibility make it applicable to a large suite of problems. Along with core sampling functionality, PyMC includes methods for summarizing output, plotting, goodnessoffit and convergence diagnostics.
 GeNIe&Smile (website) — SMILE is a C++ library for BN and ID, and GeNIe is a GUI for it
 SamIam (website), a Javabased system with GUI and Java API
 Bayes Server  User Interface and API for Bayesian networks, includes support for time series and sequences
 Belief and Decision Networks on AIspace
 BayesiaLab by Bayesia
 Hugin
 Netica by Norsys
 dVelox by Apara Software
 System Modeler by Inatas AB
History
The term "Bayesian networks" was coined by Judea Pearl in 1985 to emphasize three aspects:^{30}
 The often subjective nature of the input information.
 The reliance on Bayes' conditioning as the basis for updating information.
 The distinction between causal and evidential modes of reasoning, which underscores Thomas Bayes' posthumously published paper of 1763.^{31}
In the late 1980s Judea Pearl's text Probabilistic Reasoning in Intelligent Systems^{32} and Richard E. Neapolitan's text Probabilistic Reasoning in Expert Systems^{33} summarized the properties of Bayesian networks and established Bayesian networks as a field of study.
Informal variants of such networks were first used by legal scholar John Henry Wigmore, in the form of Wigmore charts, to analyse trial evidence in 1913.^{26}^{:66–76} Another variant, called path diagrams, was developed by the geneticist Sewall Wright^{34} and used in social and behavioral sciences (mostly with linear parametric models).
See also
 Artificial intelligence
 Bayes' theorem
 Bayesian inference
 Bayesian probability
 Bayesian programming
 Belief propagation
 Chow–Liu tree
 Computational intelligence
 Computational phylogenetics
 Deep belief network
 Dempster–Shafer theory – a Generalization of Bayes' theorem
 Dynamic Bayesian network
 Expectation–maximization algorithm
 Factor graph
 Graphical model
 Hierarchical temporal memory
 Influence diagram
 Judea Pearl
 Kalman filter
 Machine learning
 Memoryprediction framework
 Mixture distribution
 Mixture model
 Naive Bayes classifier
 Path analysis
 Polytree
 Sensor fusion
 Sequence alignment
 Speech recognition
 Structural equation modeling
 Subjective logic
 Variableorder Bayesian network
 Wigmore chart
 World view
Notes
 ^ ^{a} ^{b} ^{c} ^{d} Pearl, Judea (2000). Causality: Models, Reasoning, and Inference. Cambridge University Press. ISBN 0521773628. OCLC 42291253.
 ^ J., Pearl (1994). "A Probabilistic Calculus of Actions". In Lopez de Mantaras, R.; Poole, D. UAI'94 Proceedings of the Tenth international conference on Uncertainty in artificial intelligence. San Mateo CA: Morgan Kaufman. pp. 454–462. ISBN 1558603328.
 ^ I. Shpitser, J. Pearl, "Identification of Conditional Interventional Distributions" In R. Dechter and T.S. Richardson (Eds.), Proceedings of the TwentySecond Conference on Uncertainty in Artificial Intelligence, 437–444, Corvallis, OR: AUAI Press, 2006.
 ^ Rebane, G. and Pearl, J., "The Recovery of Causal Polytrees from Statistical Data," Proceedings, 3rd Workshop on Uncertainty in AI, (Seattle, WA) pages 222–228, 1987
 ^ Spirtes, P.; Glymour, C. (1991). "An algorithm for fast recovery of sparse causal graphs" (PDF). Social Science Computer Review 9 (1): 62–72. doi:10.1177/089443939100900106.
 ^ Spirtes, Peter; Glymour, Clark N.; Scheines, Richard (1993). Causation, Prediction, and Search (1st ed.). SpringerVerlag. ISBN 9780387979793.
 ^ Verma, Thomas; Pearl, Judea (1991). "Equivalence and synthesis of causal models". In Bonissone, P.; Henrion, M.; Kanal, L.N. et al. UAI '90 Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence. Elsevier. pp. 255–270. ISBN 0444892648.
 ^ Friedman, N.; Geiger, D.; Goldszmidt, M. (1997). Machine Learning 29 (2/3): 131. doi:10.1023/A:1007465528199.
 ^ Friedman, N.; Linial, M.; Nachman, I.; Pe'er, D. (2000). "Using Bayesian Networks to Analyze Expression Data". Journal of Computational Biology 7 (3–4): 601–620. doi:10.1089/106652700750050961. PMID 11108481.
 ^ Petitjean, F.; Webb, G.I.; Nicholson, A.E. (2013). "Scaling loglinear analysis to highdimensional data". International Conference on Data Mining. Dallas, TX, USA: IEEE.
 ^ ^{a} ^{b} Russell & Norvig 2003, p. 496.
 ^ ^{a} ^{b} Russell & Norvig 2003, p. 499.
 ^ Neapolitan, Richard E. (2004). Learning Bayesian networks. Prentice Hall. ISBN 9780130125347.
 ^ Geiger, Dan; Verma, Thomas; Pearl, Judea (1990). "Identifying independence in Bayesian Networks" (PDF). Networks 20: 507–534. doi:10.1177/089443939100900106.
 ^ Richard Scheines, Dseparation
 ^ N. Friedman, M. Linial, I. Nachman, D. Pe'er (August 2000). "Using Bayesian Networks to Analyze Expression Data". Journal of Computational Biology (Larchmont, New York: Mary Ann Liebert, Inc.) 7 (3/4): 601–620. doi:10.1089/106652700750050961. ISSN 10665277. PMID 11108481.
 ^ Constantinou, Anthony; Fenton, N.; Neil, M. (2012). "pifootball: A Bayesian network model for forecasting Association Football match outcomes". KnowledgeBased Systems 36: 322–339. Retrieved 25 March 2014.
 ^ Constantinou, Anthony; Fenton, N.; Neil, M. (2013). "Profiting from an inefficient Association Football gambling market: Prediction, Risk and Uncertainty using Bayesian networks.". KnowledgeBased Systems 50: 60–86. Retrieved 25 March 2014.
 ^ Jiang, X.; Neapolitan, R.E.; Barmada, M.M.; Visweswaran, S. (2011). "Learning Genetic Epistasis using Bayesian Network Scoring Criteria". BMC Bioinformatics 12: 89. doi:10.1186/147121051289. PMC 3080825. PMID 21453508.
 ^ J. Uebersax (2004). Genetic Counseling and Cancer Risk Modeling: An Application of Bayes Nets. Marbella, Spain: Ravenpack International.
 ^ Jiang X, Cooper GF. (July–August 2010). "A Bayesian spatiotemporal method for disease outbreak detection". J Am Med Inform Assoc 17 (4): 462–71. doi:10.1136/jamia.2009.000356. PMC 2995651. PMID 20595315.
 ^ Luis M. de Campos, Juan M. FernándezLuna and Juan F. Huete (2004). "Bayesian networks and information retrieval: an introduction to the special issue". Information Processing & Management (Elsevier) 40 (5): 727–733. doi:10.1016/j.ipm.2004.03.001. ISBN 0471141828.
 ^ Christos L. Koumenides and Nigel R. Shadbolt. 2012. Combining link and contentbased information in a Bayesian inference model for entity search. In Proceedings of the 1st Joint International Workshop on EntityOriented and Semantic Search (JIWES '12). ACM, New York, NY, USA, , Article 3 , 6 pages. DOI=10.1145/2379307.2379310
 ^ F.J. Díez, J. Mira, E. Iturralde and S. Zubillaga (1997). "DIAVAL, a Bayesian expert system for echocardiography". Artificial Intelligence in Medicine (Elsevier) 10 (1): 59–73. PMID 9177816.
 ^ G. A. Davis (2003). "Bayesian reconstruction of traffic accidents". Law, Probability and Risk 2 (2): 69–89. doi:10.1093/lpr/2.2.69.
 ^ ^{a} ^{b} J. B. Kadane and D. A. Schum (1996). A Probabilistic Analysis of the Sacco and Vanzetti Evidence. New York: Wiley. ISBN 0471141828.
 ^ O. Pourret, P. Naim and B. Marcot (2008). Bayesian Networks: A Practical Guide to Applications. Chichester, UK: Wiley. ISBN 9780470060308.
 ^ Neapolitan, Richard (2009). Probabilistic Methods for Bioinformatics. Burlington, MA: Morgan Kaufmann. p. 406. ISBN 9780123704764.
 ^ Neapolitan, Richard, and Xia Jiang (2007). Probabilistic Methods for Financial and Marketing Informatics. Burlingon, MA: Morgan Kaufmann. p. 432. ISBN 0123704774.
 ^ Pearl, J. (1985). "Bayesian Networks: A Model of SelfActivated Memory for Evidential Reasoning" (UCLA Technical Report CSD850017). Proceedings of the 7th Conference of the Cognitive Science Society, University of California, Irvine, CA. pp. 329–334. Retrieved 20090501.
 ^ Bayes, T.; Price, Mr. (1763). "An Essay towards solving a Problem in the Doctrine of Chances". Philosophical Transactions of the Royal Society 53: 370–418. doi:10.1098/rstl.1763.0053.
 ^ Pearl, J. Probabilistic Reasoning in Intelligent Systems. San Francisco CA: Morgan Kaufmann. p. 1988. ISBN 1558604790.
 ^ Neapolitan, Richard E. (1989). Probabilistic reasoning in expert systems: theory and algorithms. Wiley. ISBN 9780471618409.
 ^ Wright, S. (1921). "Correlation and Causation" (PDF). Journal of Agricultural Research 20 (7): 557–585.
General references
 BenGal, Irad (2007). Bayesian Networks (PDF). In Ruggeri, Fabrizio; Kennett, Ron S.; Faltin, Frederick W. "Encyclopedia of Statistics in Quality and Reliability". Encyclopedia of Statistics in Quality and Reliability. John Wiley & Sons. doi:10.1002/9780470061572.eqr089. ISBN 9780470018613.
 Bertsch McGrayne, Sharon. The Theory That Would not Die. Yale.
 Borgelt, Christian; Kruse, Rudolf (March 2002). Graphical Models: Methods for Data Analysis and Mining. Chichester, UK: Wiley. ISBN 0470843373.
 Borsuk, Mark Edward (2008). "Ecological informatics: Bayesian networks". In Jørgensen , Sven Erik, Fath, Brian. Encyclopedia of Ecology. Elsevier. ISBN 9780444520333.
 Castillo, Enrique; Gutiérrez, José Manuel; Hadi, Ali S. (1997). "Learning Bayesian Networks". Expert Systems and Probabilistic Network Models. Monographs in computer science. New York: SpringerVerlag. pp. 481–528. ISBN 0387948589.
 Comley, Joshua W.; Dowe, David L. (October 2003). "Minimum Message Length and Generalized Bayesian Nets with Asymmetric Languages". written at Victoria, Australia. In Grünwald, Peter D.; Myung, In Jae; Pitt, Mark A. Advances in Minimum Description Length: Theory and Applications. Neural information processing series. Cambridge, Massachusetts: Bradford Books (MIT Press) (published April 2005). pp. 265–294. 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.)
 Darwiche, Adnan (2009). Modeling and Reasoning with Bayesian Networks. Cambridge University Press. ISBN 9780521884389.
 Dowe, David L. (2010). MML, hybrid Bayesian network graphical models, statistical consistency, invariance and uniqueness, in Handbook of Philosophy of Science (Volume 7: Handbook of Philosophy of Statistics), Elsevier, ISBN 9780444518620, pp 901–982.
 Fenton, Norman; Neil, Martin E. (November 2007). Managing Risk in the Modern World: Applications of Bayesian Networks – A Knowledge Transfer Report from the London Mathematical Society and the Knowledge Transfer Network for Industrial Mathematics. London (England): London Mathematical Society.
 Fenton, Norman; Neil, Martin E. (July 23, 2004). "Combining evidence in risk analysis using Bayesian Networks" (PDF). Safety Critical Systems Club Newsletter 13 (4) (Newcastle upon Tyne, England). pp. 8–13.
 Andrew Gelman; John B Carlin; Hal S Stern; Donald B Rubin (2003). "Part II: Fundamentals of Bayesian Data Analysis: Ch.5 Hierarchical models". Bayesian Data Analysis. CRC Press. pp. 120–. ISBN 9781584883883.
 Heckerman, David (March 1, 1995). "Tutorial on Learning with Bayesian Networks". In Jordan, Michael Irwin. Learning in Graphical Models. Adaptive Computation and Machine Learning. Cambridge, Massachusetts: MIT Press (published 1998). pp. 301–354. ISBN 0262600323..
 Also appears as Heckerman, David (March 1997). "Bayesian Networks for Data Mining". Data Mining and Knowledge Discovery (Netherlands: Springer Netherlands) 1 (1): 79–119. doi:10.1023/A:1009730122752. ISSN 13845810.
 An earlier version appears as Technical Report MSRTR9506, Microsoft Research, March 1, 1995. The paper is about both parameter and structure learning in Bayesian networks.
 Jensen, Finn V; Nielsen, Thomas D. (June 6, 2007). Bayesian Networks and Decision Graphs. Information Science and Statistics series (2nd ed.). New York: SpringerVerlag. ISBN 9780387682815.
 Korb, Kevin B.; Nicholson, Ann E. (December 2010). Bayesian Artificial Intelligence. CRC Computer Science & Data Analysis (2nd ed.). Chapman & Hall (CRC Press). doi:10.1007/s1004400402145. ISBN 1584883871.
 Lunn, D.; et al., D; Thomas, A; Best, N (2009). "The BUGS project: Evolution, critique and future directions". Statistics in Medicine 28 (25): 3049–3067. doi:10.1002/sim.3680. PMID 19630097.
 Neil, Martin; Fenton, Norman E.; Tailor, Manesh (August 2005). "Using Bayesian Networks to Model Expected and Unexpected Operational Losses" (pdf). In Greenberg, Michael R. Risk Analysis: an International Journal (John Wiley & Sons) 25 (4): 963–972. doi:10.1111/j.15396924.2005.00641.x. PMID 16268944.
 Pearl, Judea (September 1986). "Fusion, propagation, and structuring in belief networks". Artificial Intelligence (Elsevier) 29 (3): 241–288. doi:10.1016/00043702(86)90072X. ISSN 00043702.
 Pearl, Judea (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Representation and Reasoning Series (2nd printing ed.). San Francisco, California: Morgan Kaufmann. ISBN 0934613737.
 Pearl, Judea; Russell, Stuart (November 2002). "Bayesian Networks". In Arbib, Michael A.. Handbook of Brain Theory and Neural Networks. Cambridge, Massachusetts: Bradford Books (MIT Press). pp. 157–160. ISBN 0262011972.
 Russell, Stuart J.; Norvig, Peter (2003), Artificial Intelligence: A Modern Approach (2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, ISBN 0137903952.
 Zhang, Nevin Lianwen; Poole, David (May 1994). "A simple approach to Bayesian network computations". Proceedings of the Tenth Biennial Canadian Artificial Intelligence Conference (AI94). (Banff, Alberta): 171–178. This paper presents variable elimination for belief networks.
 Computational Intelligence: A Methodological Introduction by Kruse, Borgelt, Klawonn, Moewes, Steinbrecher, Held, 2013, Springer, ISBN 9781447150121
 Graphical Models  Representations for Learning, Reasoning and Data Mining, 2nd Edition, by Borgelt, Steinbrecher, Kruse, 2009, J. Wiley & Sons, ISBN 9780470749562
External links
 A tutorial on learning with Bayesian Networks
 An Introduction to Bayesian Networks and their Contemporary Applications
 Online Tutorial on Bayesian nets and probability
 WebApp to create Bayesian nets and run it with a Monte Carlo method
 Continuous Time Bayesian Networks
 Bayesian Networks: Explanation and Analogy
 A live tutorial on learning Bayesian networks
 A hierarchical Bayes Model for handling sample heterogeneity in classification problems, provides a classification model taking into consideration the uncertainty associated with measuring replicate samples.
 Hierarchical Naive Bayes Model for handling sample uncertainty, shows how to perform classification and learning with continuous and discrete variables with replicated measurements.
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