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A '''Bayesian network''' is a form of probabilistic [[graphical model]], also known as '''Bayesian belief network''' or just '''belief network'''.
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A '''Bayesian network''' (or a '''belief network''') is a [[probabilistic graphical model]] that represents a set of [[variable]]s and their probabilistic independencies. 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. The term "Bayesian networks" was coined by Pearl (1985) to emphasize three aspects:
   
A Bayesian network can be represented by a [[graph (mathematics)|graph]] (as in [[graph theory]]) with [[probabilities]] attached. Thus, a Bayesian network represents a set of [[variable]]s together with a [[joint probability distribution]]
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#The often subjective nature of the input information.
with explicit [[statistical independence|independence]] assumptions.
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#The reliance on Bayes's conditioning as the basis for updating information.
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#The distinction between causal and evidential modes of reasoning, which underscores [[Thomas Bayes]]'s posthumous paper of 1763.<ref>{{Cite journal
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|author = Thomas Bayes
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|year = 1763
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|title = An Essay towards solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, F.R.S., communicated by Mr. Price, in a letter to John Canton, A.M., F.R.S.
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|journal = Philosophical Transactions of the Royal Society of London
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|volume = 53
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|pages = 370–418
  +
}}</ref>
   
== Definition ==
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Formally, Bayesian networks are [[directed acyclic graph]]s whose nodes represent variables, and whose arcs encode conditional independencies between the variables. Nodes can represent any kind of variable, be it a measured parameter, a [[latent variable]] or a hypothesis. They are not restricted to representing [[random variable]]s, which represents another "[[Bayesian]]" aspect of a Bayesian network. Efficient algorithms exist that perform [[inference]] and learning in Bayesian networks. Bayesian networks that model sequences of variables (such as for example [[speech recognition|speech signals]] or [[peptide sequence|protein sequences]]) are called [[dynamic Bayesian network]]s. Generalizations of Bayesian networks that can represent and solve decision problems under uncertainty are called [[influence diagrams]].
   
A Bayesian network is a [[directed acyclic graph]] whose
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==Definitions and concepts==
* nodes represent variables,
 
* arcs represent statistical dependence relations among the variables and local probability distributions for each variable given values of its parents.
 
   
Nodes can represent any kind of variable, be it a measured parameter, a [[latent variable]] or a hypothesis. They are not restricted to representing [[random variable]]s; this is what is "[[Bayesian]]" about a Bayesian network.
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If there is an [[Graph (mathematics)|arc]] from node ''A'' to another node ''B'', ''A'' is called a ''parent'' of ''B'', and ''B'' is a ''child'' of ''A''. The set of parent nodes of a node ''X''<sub>i</sub> is denoted by parents(''X''<sub>i</sub>). A [[directed acyclic graph]] is a Bayesian Network relative to a set of variables if the [[joint distribution]] of the node values can be written as the product of the local distributions of each node and its parents:
   
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 for each variable ''X''<sub>i</sub>, ''i''= 1 to ''n'', the set of parent variables is denoted by parents(''X''<sub>i</sub>) then the [[joint distribution]] of the variables is product of the local distributions
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:<math>\mathrm P(X_1, \ldots, X_n) = \prod_{i=1}^n \mathrm P(X_i \mid \operatorname{parents}(X_i)).\,</math>
:<math>Pr(X_1, \ldots, X_n) = \prod_{i=1}^n Pr(X_i \mid \operatorname{parents}(X_i))</math>
 
   
If ''X''<sub>i</sub> has no parents, its local probability distribution is said to be ''unconditional'', otherwise it is ''conditional''. If the variable represented by a node is ''observed'', then the node is said to be an ''evidence'' node.
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If node <math>X_i</math> has no parents, its local probability distribution is said to be ''unconditional'', otherwise it is ''conditional''. If the value of a node is ''observed'', then the node is said to be an ''evidence'' node.
   
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|''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]].
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===Independencies and ''d''-separation===
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The graph encodes independencies between variables. [[Conditional independence]] can be determined by the graphical property of ''' ''d''-separation'''. If two sets of nodes ''X'' and ''Y'' are ''d''-separated in the graph by a third set ''Z'', then the corresponding variable sets ''X'' and ''Y'' are independent given the variables in ''Z''. The minimal set of nodes which ''d''-separates node ''X'' from all other nodes is given by ''X''s [[Markov blanket]].
   
One advantage of Bayesian networks is that it is intuitively easier for a human to understand direct dependencies and local distributions than complete joint distribution.
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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:
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# p contains a ''chain'' i -> m -> j such that the middle node m is in Z,
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# p contains a ''fork'' i <- m -> j such that the middle node m is in Z,
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# p contains an ''inverted fork'' (or ''collider'') i -> m <- j such that the middle node m is not in Z and no descendant of m is in Z.
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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. The 'd' in d-separation stands for 'directional', since the behavior of a three node link on a path depends on the direction of the arrows in the link.
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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).
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===Causal Bayesian networks===
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A Bayesian network is a carrier of the conditional independencies of a set of variables, not of their causal connections. However, causal relations can be modelled by the closely related [[causal Bayesian network]]. The additional semantics of the causal Bayesian networks specify that if a node ''X'' is actively caused to be in a given state ''x'' (an operation written as ''do(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'' (Pearl, 2000). Using this semantics, one can predict the impact of external interventions from data obtained prior to intervention.
   
 
== Example ==
 
== Example ==
   
[[Image:SimpleBayesNet.svg|thumb|right|A simple Bayesian network.]]
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[[Image:SimpleBayesNet.svg|400px|thumb|right|A simple Bayesian network.]]
   
If there are two reasons which could cause grass to be wet, either the sprinkler is on or it's raining, then the situation can be modelled with adjacent Bayesian network. Here, all variables have two possible states T (for true) and F (for false).
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Suppose that there are two reasons 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 modelled with the adjacent Bayesian network. All three variables have two possible values T (for true) and F (for false).
   
The joint probability function is
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The joint probability function is:
:Pr(GRASSWET, SPRINKLER, RAIN) = Pr(GRASSWET | SPRINKLER, RAIN) Pr(SPRINKLER | RAIN) Pr(RAIN)
 
   
The model can answer questions like "What is the likelihood that it is raining, given the grass is wet?" by using the [[conditional probability]] formula and summing over all nuisance variables:
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<math>\mathrm P(G,S,R)=\mathrm P(G|S,R)\mathrm P(S|R)\mathrm P(R)</math>
:<math>Pr(\mathit{RAIN}=T \mid \mathit{GRASSWET}=T) = \frac{Pr(\mathit{GRASSWET}=T, \mathit{RAIN}=T)}{Pr(\mathit{GRASSWET}=T)}</math>
 
:<math>= \frac{\sum_{\mathit{SPRINKLER} \in \{T, F\}} Pr(\mathit{GRASSWET}=T, \mathit{SPRINKLER}, \mathit{RAIN}=T)}{\sum_{\mathit{SPRINKLER}, \mathit{RAIN} \in \{T, F\}} Pr(\mathit{GRASSWET}=T, \mathit{SPRINKLER}, \mathit{RAIN})}</math>
 
   
Substituting numerical values yields Pr(RAIN=T | GRASSWET=T) = 891/2491 35.77%.
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where the names of the variables have been abbreviated to ''G = Grass wet'', ''S = Sprinkler'', and ''R = Rain''.
   
== Causal Bayesian networks ==
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The model can answer questions like "What is the likelihood that it is raining, given the grass is wet?" by using the [[conditional probability]] formula and summing over all [[nuisance variable]]s:
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:
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<math> \mathrm P(\mathit{R}=T \mid \mathit{G}=T)
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=\frac{\mathrm P(\mathit{G}=T,\mathit{R}=T)}{\mathrm P(\mathit{G}=T)}
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=\frac{\sum_{\mathit{S} \in \{T, F\}}\mathrm P(\mathit{G}=T,\mathit{S},\mathit{R}=T)}{\sum_{\mathit{S}, \mathit{R} \in \{T, F\}} \mathrm P(\mathit{G}=T,\mathit{S},\mathit{R})}
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</math>
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:<math> = \frac{(0.99 * 0.01 * 0.2 = 0.00198_{TTT}) + (0.8 * 0.99 * 0.2 = 0.1584_{TFT})}{0.00198_{TTT} + 0.288_{TTF} + 0.1584_{TFT} + 0_{TFF}} \approx 35.77 %.</math>
   
A causal Bayesian network is a Bayesian network where the directed arcs of the graph are interpreted as representing [[causality|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.
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As in the example numerator is pointed out explicitly, the joint probability function is used to calculate each iteration of the summation function. In the [[numerator]] marginalizing over <math>\mathit{S}</math> and in the [[denominator]] marginalizing over <math>\mathit{S}</math> and <math>\mathit{R}</math>.
   
== Structure learning ==
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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?" the answer would be governed by the post-intervention joint distribution function <math>\mathrm P(S,R|do(G=T)) = P(S|R) P(R)</math> obtained by removing the factor <math>\mathrm P(G|S,R)</math> from the pre-intervention distribution. As expected, the likelihood of rain is unaffected by the action: <math>\mathrm P(R|do(G=T)) = P(R)</math>.
   
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.
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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 <math>2^{10} = 1024</math> 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 <math>10*2^3 = 80</math> values.
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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.
  +
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== Inference ==
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  +
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.
   
Learning the structure of a Bayesian network (i.e., the graph) is a very important part of [[machine learning]]. Assuming that the data is generated from a Bayesian network and that all the variables are visible in every iteration, optimization based search method can be used to find the structure of the network. 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 back 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 [[maxima and minima|local minima]]. Friedman et al.{{fact}} 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.
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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 many variables can be queried at one time and new evidence can be propagated quickly; and [[recursive conditioning]], which allows for a space-time tradeoff and matches 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 stochastic [[Markov chain Monte Carlo|MCMC]] simulation, [[mini-bucket elimination]] which generalizes [[loopy belief propagation]], and [[variational Bayes|variational methods]].
   
 
== Parameter learning ==
 
== Parameter learning ==
   
In order to fully specify the Bayesian network and thus fully represent the joint 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 [[information entropy|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.)
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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 [[normal distribution|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 [[information entropy|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.
 
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 ==
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== 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.
   
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 constructing extensions of [[Bayes' theorem]] to more complex problems.
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Learning the structure of a Bayesian network (i.e., the graph) is a challenge pursued within [[machine learning]]. The basic idea goes back to a recovery algorithm
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developed by Rebane and Pearl (1987)<ref>Rebane, G. and Pearl, J., "The Recovery of Causal Poly-trees from Statistical Data," ''Proceedings, 3rd Workshop on Uncertainty in AI,'' (Seattle, WA) pp. 222-228,1987</ref> and rests
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on the distinction between the three possible types of
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adjacent triplets allowed in a directed acyclic graph (DAG):
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<ol>
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<li> <math>X \rightarrow Y \rightarrow Z</math>
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<li> <math>X \leftarrow Y \rightarrow Z</math>
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<li> <math>X \rightarrow Y \leftarrow Z</math>
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</ol>
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Type 1 and type 2 represent the same dependencies (i.e., <math>X</math> and <math>Z</math> are independent given <math>Y</math>) and are, therefore, indistinguishable. Type 3, however, can be uniquely identified, since <math>X</math> and <math>Z</math> 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 <math>X</math> and <math>Z</math> 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.<ref>{{Cite book
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| first = Judea
  +
| last = Pearl
  +
| authorlink = Judea Pearl
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| title = Causality: Models, Reasoning, and Inference
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| publisher = [[Cambridge University Press]]
  +
| year = 2000
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| isbn = 0-521-77362-8
  +
}}</ref><ref>P. Spirtes and C. Glymour, "An algorithm for fast recovery of sparse causal graphs", ''Social Science Computer Review,'' Vol. 9, pp. 62-72, 1991.
  +
</ref><ref>P. Spirtes, C. Glymour, and R. Scheines, ''Causation, Prediction, and Search,'' New York: Springer-Verlag, 1993
  +
</ref><ref>T. Verma and J. Pearl, "Equivalence and Synthesis of Causal Models," ''Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence,'' (July, Cambridge, MA), pp. 220-227, 1990. Reprinted in P. Bonissone, M. Henrion, L. N. Kanal and J. F. Lemmer (editors), ''Uncertainty in Artificial Intelligence 6,'' Amsterdam: Elsevier Science Publishers, B.V., pp. 225-268, 1991</ref>
   
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 many variables can be queried at one time and new evidence can be propagated quickly; and [[recursive conditioning]], which allows for a space-time tradeoff and matches 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 stochastic [[Markov_chain_Monte_Carlo|MCMC]] simulation, [[mini-bucket elimination]] which generalizes [[loopy belief propagation]], and [[variational Bayes|variational methods]].
+
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 back 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 [[maxima and minima|local minima]]. Friedman et al.{{Fact|date=February 2007}} 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.
   
 
== Applications ==
 
== Applications ==
  +
Bayesian networks are used for [[mathematical model|modelling]] knowledge in [[bioinformatics]] ([[gene regulatory network]]s, [[protein structure]]), [[medicine]], [[document classification]], [[image processing]], [[data fusion]], [[decision support system]]s,{{Fact|date=October 2007}} [[engineering]]<ref name="davis">Davis (2003)</ref> and [[law]]<ref>Kadane & Schum (1996)</ref><ref name="davis"/>.
   
Bayesian networks are used for [[mathematical model|modelling]] knowledge in [[gene regulatory network]]s, [[medicine]], [[engineering]], [[text analysis]], [[image processing]], [[data fusion]], and [[decision support system]]s.
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== History ==
  +
Informal variants of such networks were first used by [[legal scholar]] [[John Henry Wigmore]], in the form of [[Wigmore chart]]s, to analyse [[trial (law)|trial]] [[evidence (law)|evidence]] in [[1913]].<ref>Kadane & Schum (1996) 66-76</ref> Another variant, called [[path analysis (statistics)|path diagrams]] was developed by the geneticist [[Sewall Wright]]<ref>Wright, S. (1921) "Correlation and Causation," ''Journal of Agricultural Research'', 20:557-585.</ref> and used in social and
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behavioral sciences (mostly with linear parametric models).
   
 
==See also==
 
==See also==
  +
<div style="-moz-column-count:2; column-count:2;">
 
*[[Bayesian inference]]
 
*[[Bayesian inference]]
 
*[[Bayes' theorem]]
 
*[[Bayes' theorem]]
 
*[[Bayesian statistics]]
 
*[[Bayesian statistics]]
  +
*[[Chow-Liu tree]]
 
*[[Graphical model]]
 
*[[Graphical model]]
  +
*[[Influence diagram]]
 
*[[Machine learning]]
 
*[[Machine learning]]
 
*[[Polytree]]
 
*[[Polytree]]
  +
*[[Variable-order Bayesian network]]
  +
*[[Structural equation modeling]]
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*[[World view]]
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</div>
   
== Links and software ==
+
== External links ==
  +
*Tutorial on Learning with Bayesian Networks: http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06
 
*Association for Uncertainty in Artificial Intelligence: http://www.auai.org/
 
*Association for Uncertainty in Artificial Intelligence: http://www.auai.org/
*Intro to Bayesian networks : http://www.niedermayer.ca/papers/bayesian/bayes.html
+
*Intro to Bayesian networks: http://www.niedermayer.ca/papers/bayesian/bayes.html
 
*On-line Tutorial on Bayesian nets and probability: http://www.dcs.qmw.ac.uk/%7Enorman/BBNs/BBNs.htm
 
*On-line Tutorial on Bayesian nets and probability: http://www.dcs.qmw.ac.uk/%7Enorman/BBNs/BBNs.htm
  +
  +
'''Software'''
  +
:'''Free and open source software'''
  +
*dlib C++ Library: http://dclib.sourceforge.net/
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*Kevin Murphy's Bayesian Network Toolbox for MatLab: http://bnt.sourceforge.net/
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*GeNIe & SMILE: http://genie.sis.pitt.edu
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*OpenBayes: http://www.openbayes.org
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*RISO: http://sourceforge.net/projects/riso/ (distributed belief networks)
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*[http://www.dynamics.unam.edu/DinamicaNoLineal3/bansy3.htm BANSY3] - Freeware. From the Non Linear Dynamics Laboratory. Mathematics Department, Science School, UNAM.
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*SamIam: http://reasoning.cs.ucla.edu/samiam
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*Ace, the Bayesian network compiler: http://reasoning.cs.ucla.edu/ace
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*BN4R: http://bn4r.rubyforge.org/
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*JavaBayes, Bayesian Networks in Java: http://www.pmr.poli.usp.br/ltd/Software/javabayes/
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  +
:'''Commercial software'''
 
*AgenaRisk Bayesian network tool: http://www.agenarisk.com
 
*AgenaRisk Bayesian network tool: http://www.agenarisk.com
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*BayesBuilder: http://www.snn.ru.nl/nijmegen/index.php3?page=31
 
*Bayesian network application library: http://www.norsys.com/netlibrary/index.htm
 
*Bayesian network application library: http://www.norsys.com/netlibrary/index.htm
 
*Bayesia: http://www.bayesia.com
 
*Bayesia: http://www.bayesia.com
*GeNIe & SMILE: http://genie.sis.pitt.edu
 
 
*Hugin: http://www.hugin.com
 
*Hugin: http://www.hugin.com
 
*Netica: http://www.norsys.com
 
*Netica: http://www.norsys.com
 
*BNet: http://www.cra.com/bnet
 
*BNet: http://www.cra.com/bnet
 
*Dezide: http://www.dezide.com
 
*Dezide: http://www.dezide.com
*SamIam: http://reasoning.cs.ucla.edu/samiam
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*Promedas (Bayesian medical decision support): http://www.promedas.nl
*OpenBayes: http://www.openbayes.org
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*ProBayes: http://www.probayes.com
 
* MSBNx: a component-centric toolkit for modeling and inference with Bayesian Network (from [[Microsoft]] Research): http://research.microsoft.com/adapt/MSBNx/
 
* MSBNx: a component-centric toolkit for modeling and inference with Bayesian Network (from [[Microsoft]] Research): http://research.microsoft.com/adapt/MSBNx/
  +
* dVelox: http://www.aparasw.com/en/index.php?option=com_content&task=view&id=65&Itemid=102
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* Causeway: http://www.inet.saic.com/
   
 
== References ==
 
== References ==
<div style="font-size:90%">
+
{{reflist}}
* Finn V. Jensen. Bayesian Networks and Decision Graphs. Springer, 2001.
+
  +
  +
<div class="references-small">
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* I. Ben-Gal (2007), [http://www.eng.tau.ac.il/~bengal/BN.pdf Bayesian Networks], in F. Ruggeri, R. Kenett, and F. Faltin (editors), Encyclopedia of Statistics in Quality and Reliability, John Wiley & Sons.
  +
* Enrique Castillo, José Manuel Gutiérrez, and Ali S. Hadi (1997). ''Expert Systems and Probabilistic Network Models''. New York: [[Springer Science+Business Media|Springer-Verlag]]. ISBN 0-387-94858-9
  +
*{{cite journal | author=G. A. Davis | title=Bayesian reconstruction of traffic accidents | journal=Law, Probability and Risk | year=2003 | volume=2 | pages=69-89 }}
  +
* N. E. Fenton, and M. Neil, "Combining evidence in risk analysis using Bayesian Networks". https://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdf
  +
* {{Cite book
  +
| author = Finn V. Jensen
  +
| title = Bayesian Networks and Decision Graphs
  +
| publisher = [[Springer Science+Business Media|Springer]]
  +
| year = 2001
  +
}}
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*{{ cite book | author=J. B. Kadane and D. A. Schum | title=A Probabilistic Analysis of the Sacco and Vanzetti Evidence | location=New York | publisher=Wiley | id=ISBN 0-471-14182-8 | year=1996 }}
 
<!-- These links don't work! * Eugene Charniak. [http://www.cs.ubc.ca/~murphyk/Bayes/Charniak_91.pdf Bayesian Networks Without Tears]. This is a nice intro for non-specialists.
 
<!-- These links don't work! * Eugene Charniak. [http://www.cs.ubc.ca/~murphyk/Bayes/Charniak_91.pdf Bayesian Networks Without Tears]. This is a nice intro for non-specialists.
* Kevin Murphy. An introduction to graphical models. 2001. http://www.ai.mit.edu/~murphyk/Papers/intro_gm.pdf -->
+
* Kevin Murphy (2001). An introduction to graphical models. http://www.ai.mit.edu/~murphyk/Papers/intro_gm.pdf -->
* Judea Pearl, Stuart Russell. Bayesian Networks. [[UCLA]] Cognitive Systems Laboratory, Technical Report (R-277), November 2000.
+
* M. Neil, N. E. Fenton, and M. Tailor (2005), "Using Bayesian Networks to model Expected and Unexpected Operational Losses", Risk Analysis: An International Journal, Vol. 25(4), pp. 963-972. http://www.dcs.qmul.ac.uk/~norman/papers/oprisk.pdf
* Judea Pearl, Stuart Russell. Bayesian Networks, in M. A. Arbib (Ed.), ''Handbook of Brain Theory and Neural Networks'', pp. 157&ndash;160, Cambridge, MA: [[MIT Press]], 2003, ISBN 0-262-01197-2.
+
* Judea Pearl (1985). "Bayesian Networks: A Model of Self-Activated Memory for Evidential Reasoning". In Proceedings of the 7<sup>th</sup> Conference of the Cognitive Science Society, University of California, Irvine, CA, pp. 329-334, August 15-17.
* Neil M, Fenton N, Tailor M, "Using Bayesian Networks to model Expected and Unexpected Operational Losses", Risk Analysis: An International Journal, Vol 25(4), 963-972, 2005. http://www.dcs.qmul.ac.uk/~norman/papers/oprisk.pdf
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* Judea Pearl (1986), "Fusion, propagation, and structuring in belief networks". ''Artificial Intelligence'' '''29'''(3):241&ndash;288.
* Enrique Castillo, José Manuel Gutiérrez, and Ali S. Hadi. ''Expert Systems and Probabilistic Network Models''. New York: [[Springer Science+Business Media|Springer-Verlag]], 1997. ISBN 0-387-94858-9
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* {{Cite book
* Fenton NE and Neil M, "Combining evidence in risk analysis using Bayesian Networks". https://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdf
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| author = Judea Pearl
* Judea Pearl. Fusion, propagation, and structuring in belief networks. ''Artificial Intelligence'' '''29'''(3):241&ndash;288, 1986.
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| authorlink = Judea Pearl
* Judea Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, 1988, ISBN 0-934613-73-7
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| title = Probabilistic Reasoning in Intelligent Systems
* J.W. Comley and [http://www.csse.monash.edu.au/~dld D.L. Dowe], "[http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2005 Minimum Message Length, MDL and Generalised Bayesian Networks with Asymmetric Languages]", [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10478&mode=toc chapter 11] (pp[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p265.jpg 265]&ndash;[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p294.jpg 294]) in P. Grunwald, M.A. Pitt and I.J. Myung (eds)., [http://mitpress.mit.edu/catalog/item/default.asp?sid=4C100C6F-2255-40FF-A2ED-02FC49FEBE7C&ttype=2&tid=10478 Advances in Minimum Description Length: Theory and Applications], Cambridge, MA: [[MIT Press]], April 2005, ISBN 0-262-07262-9. (This paper puts [[decision tree]]s in internal nodes of Bayes networks using [http://www.csse.monash.edu.au/~dld/MML.html Minimum Message Length] ([[Minimum message length|MML]]). An earlier version is [http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2003 Comley and Dowe (2003)], [http://www.csse.monash.edu.au/~dld/Publications/2003/Comley+Dowe03_HICS2003_GeneralBayesianNetworksAsymmetricLanguages.pdf .pdf].)
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| publisher = [[Morgan Kaufmann]]
  +
| year = 1988
  +
| isbn = 0-934613-73-7
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}}
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* {{Cite book
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| author = Judea Pearl
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| authorlink = Judea Pearl
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| title = Causality: Models, Reasoning, and Inference
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| publisher = [[Cambridge University Press]]
  +
| year = 2000
  +
| isbn = 0-521-77362-8
  +
}}
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* Judea Pearl and Stuart Russell. Bayesian Networks, in M. A. Arbib (editor), ''Handbook of Brain Theory and Neural Networks'', pp. 157&ndash;160, Cambridge, MA: [[MIT Press]], 2003, ISBN 0-262-01197-2.
  +
* J. W. Comley and [http://www.csse.monash.edu.au/~dld D. L. Dowe], "[http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2005 Minimum Message Length, MDL and Generalised Bayesian Networks with Asymmetric Languages]", [http://mitpress.mit.edu/catalog/item/default.asp?ttype=2&tid=10478&mode=toc chapter 11] (pp[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p265.jpg 265]&ndash;[http://www.csse.monash.edu.au/~dld/Publications/2005/ComleyDowe2005MMLGeneralizedBayesianNetsAsymmetricLanguages_p294.jpg 294]) in P. Grunwald, M. A. Pitt and I. J. Myung (editors), [http://mitpress.mit.edu/catalog/item/default.asp?sid=4C100C6F-2255-40FF-A2ED-02FC49FEBE7C&ttype=2&tid=10478 Advances in Minimum Description Length: Theory and Applications], Cambridge, MA: [[MIT Press]], April 2005, ISBN 0-262-07262-9. (This paper puts [[decision tree]]s in internal nodes of Bayes networks using [http://www.csse.monash.edu.au/~dld/MML.html Minimum Message Length] ([[Minimum message length|MML]]). An earlier version is [http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2003 Comley and Dowe (2003)], [http://www.csse.monash.edu.au/~dld/Publications/2003/Comley+Dowe03_HICS2003_GeneralBayesianNetworksAsymmetricLanguages.pdf .pdf].)
 
* Christian Borgelt and Rudolf Kruse. [http://fuzzy.cs.uni-magdeburg.de/books/gm/ Graphical Models &ndash; Methods for Data Analysis and Mining], Chichester, UK: [[Wiley]], 2002, ISBN 0-470-84337-3
 
* Christian Borgelt and Rudolf Kruse. [http://fuzzy.cs.uni-magdeburg.de/books/gm/ Graphical Models &ndash; Methods for Data Analysis and Mining], Chichester, UK: [[Wiley]], 2002, ISBN 0-470-84337-3
* [http://www.cs.ust.hk/faculty/lzhang/bio.html Nevin Lianwen Zhang] and [http://www.cs.ubc.ca/spider/poole/ David Poole], [http://www.cs.ubc.ca/spider/poole/papers/canai94.pdf 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.
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* {{Cite book
* [http://research.microsoft.com/~heckerman/ David Heckerman], [http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06 A Tutorial on Learning with Bayesian Networks]. In Learning in Graphical Models, M. Jordan, ed.. MIT Press, Cambridge, MA, 1999. Also appears as Technical Report MSR-TR-95-06, Microsoft Research, March, 1995. An earlier version appears as Bayesian Networks for Data Mining, Data Mining and Knowledge Discovery, 1:79-119, 1997. This paper is all about parameter and structure learning in Bayesian networks.
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| author = Kevin B. Korb
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| coauthors = Ann E. Nicholson
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| url = http://www.csse.monash.edu.au/bai
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| title = Bayesian Artificial Intelligence
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| publisher = [[CRC Press]]
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| year = 2004
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* [http://www.cs.ust.hk/faculty/lzhang/bio.html Nevin Lianwen Zhang] and [http://www.cs.ubc.ca/spider/poole/ David Poole], [http://www.cs.ubc.ca/spider/poole/papers/canai94.pdf A simple approach to Bayesian network computations], Proceedings of the Tenth Biennial Canadian Artificial Intelligence Conference (AI-94), Banff, AB, May 1994, 171-178. This paper presents variable elimination for belief networks.
  +
* [http://research.microsoft.com/~heckerman/ David Heckerman], [http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06 A Tutorial on Learning with Bayesian Networks]. In Learning in Graphical Models, M. Jordan, ed. MIT Press, Cambridge, MA, 1999. Also appears as Technical Report MSR-TR-95-06, Microsoft Research, March, 1995. An earlier version appears as Bayesian Networks for Data Mining, Data Mining and Knowledge Discovery, 1:79-119, 1997. The paper is about both parameter and structure learning in Bayesian networks.
 
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{{enWP| Bayesian network}}
 
{{enWP| Bayesian network}}

Latest revision as of 10:33, February 11, 2008

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A Bayesian network (or a belief network) is a probabilistic graphical model that represents a set of variables and their probabilistic independencies. 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. The term "Bayesian networks" was coined by Pearl (1985) to emphasize three aspects:

  1. The often subjective nature of the input information.
  2. The reliance on Bayes's conditioning as the basis for updating information.
  3. The distinction between causal and evidential modes of reasoning, which underscores Thomas Bayes's posthumous paper of 1763.[1]

Formally, Bayesian networks are directed acyclic graphs whose nodes represent variables, and whose arcs encode conditional independencies between the variables. Nodes can represent any kind of variable, be it a measured parameter, a latent variable or a hypothesis. They are not restricted to representing random variables, which represents another "Bayesian" aspect of a Bayesian network. Efficient algorithms exist that perform inference and learning in Bayesian networks. Bayesian networks that model sequences of variables (such as for example 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.

Definitions and conceptsEdit

If there is an arc from node A to another node B, A is called a parent of B, and B is a child of A. The set of parent nodes of a node Xi is denoted by parents(Xi). A directed acyclic graph is a Bayesian Network relative to a set of variables if the joint distribution of the node values can be written as the product of the local distributions of each node and its parents:

\mathrm P(X_1, \ldots, X_n) = \prod_{i=1}^n \mathrm P(X_i \mid \operatorname{parents}(X_i)).\,

If node X_i has no parents, its local probability distribution is said to be unconditional, otherwise it is conditional. If the value of a node is observed, then the node is said to be an evidence node.

Independencies and d-separationEdit

The graph encodes independencies between variables. Conditional independence can be determined by the graphical property of d-separation. If two sets of nodes X and Y are d-separated in the graph by a third set Z, then the corresponding variable sets X and Y are independent given the variables in Z. 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:

  1. p contains a chain i -> m -> j such that the middle node m is in Z,
  2. p contains a fork i <- m -> j such that the middle node m is in Z,
  3. p contains an inverted fork (or collider) i -> m <- j such that the middle node m is not in Z and no descendant of m is in Z.

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. The 'd' in d-separation stands for 'directional', since the behavior of a three node link on a path depends on the direction of the arrows in the link.

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).

Causal Bayesian networksEdit

A Bayesian network is a carrier of the conditional independencies of a set of variables, not of their causal connections. However, causal relations can be modelled by the closely related causal Bayesian network. The additional semantics of the causal Bayesian networks specify that if a node X is actively caused to be in a given state x (an operation written as do(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 (Pearl, 2000). Using this semantics, one can predict the impact of external interventions from data obtained prior to intervention.

Example Edit

SimpleBayesNet

A simple Bayesian network.

Suppose that there are two reasons 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 modelled with the adjacent Bayesian network. All three variables have two possible values T (for true) and F (for false).

The joint probability function is:

\mathrm P(G,S,R)=\mathrm P(G|S,R)\mathrm P(S|R)\mathrm P(R)

where the names of the variables have been abbreviated to G = Grass wet, S = Sprinkler, and R = Rain.

The model can answer questions like "What is the likelihood that it is raining, given the grass is wet?" by using the conditional probability formula and summing over all nuisance variables:

 \mathrm P(\mathit{R}=T \mid \mathit{G}=T) 
=\frac{\mathrm P(\mathit{G}=T,\mathit{R}=T)}{\mathrm P(\mathit{G}=T)} 
=\frac{\sum_{\mathit{S} \in \{T, F\}}\mathrm P(\mathit{G}=T,\mathit{S},\mathit{R}=T)}{\sum_{\mathit{S}, \mathit{R} \in \{T, F\}} \mathrm P(\mathit{G}=T,\mathit{S},\mathit{R})}

 = \frac{(0.99 * 0.01 * 0.2 = 0.00198_{TTT}) + (0.8 * 0.99 * 0.2 = 0.1584_{TFT})}{0.00198_{TTT} + 0.288_{TTF} + 0.1584_{TFT} + 0_{TFF}} \approx 35.77 %.

As in the example numerator is pointed out explicitly, the joint probability function is used to calculate each iteration of the summation function. In the numerator marginalizing over \mathit{S} and in the denominator marginalizing over \mathit{S} and \mathit{R}.

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 post-intervention joint distribution function \mathrm P(S,R|do(G=T)) = P(S|R) P(R) obtained by removing the factor \mathrm P(G|S,R) from the pre-intervention distribution. As expected, the likelihood of rain is unaffected by the action: \mathrm P(R|do(G=T)) = P(R).

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 2^{10} = 1024 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 10*2^3 = 80 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 Edit

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 non-observed non-query 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, which allows for a space-time tradeoff and matches 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 stochastic MCMC simulation, mini-bucket elimination which generalizes loopy belief propagation, and variational methods.

Parameter learning Edit

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 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.

Structure learning Edit

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.

Learning the structure of a Bayesian network (i.e., the graph) is a challenge pursued within machine learning. The basic idea goes back to a recovery algorithm developed by Rebane and Pearl (1987)[2] and rests on the distinction between the three possible types of adjacent triplets allowed in a directed acyclic graph (DAG):

  1. X \rightarrow Y \rightarrow Z
  2. X \leftarrow Y \rightarrow Z
  3. X \rightarrow Y \leftarrow Z

Type 1 and type 2 represent the same dependencies (i.e., X and Z are independent given Y) and are, therefore, indistinguishable. Type 3, however, can be uniquely identified, since X and Z 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 X and Z 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.[3][4][5][6]

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 back 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.[How to reference and link to summary or text] 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.

Applications Edit

Bayesian networks are used for modelling knowledge in bioinformatics (gene regulatory networks, protein structure), medicine, document classification, image processing, data fusion, decision support systems,[How to reference and link to summary or text] engineering[7] and law[8][7].

History Edit

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.[9] Another variant, called path diagrams was developed by the geneticist Sewall Wright[10] and used in social and behavioral sciences (mostly with linear parametric models).

See alsoEdit

External links Edit

Software

Free and open source software
Commercial software

References Edit

  1. Thomas Bayes (1763). An Essay towards solving a Problem in the Doctrine of Chances. By the late Rev. Mr. Bayes, F.R.S., communicated by Mr. Price, in a letter to John Canton, A.M., F.R.S.. Philosophical Transactions of the Royal Society of London 53: 370–418.
  2. Rebane, G. and Pearl, J., "The Recovery of Causal Poly-trees from Statistical Data," Proceedings, 3rd Workshop on Uncertainty in AI, (Seattle, WA) pp. 222-228,1987
  3. Pearl, Judea (2000). Causality: Models, Reasoning, and Inference, Cambridge University Press.
  4. P. Spirtes and C. Glymour, "An algorithm for fast recovery of sparse causal graphs", Social Science Computer Review, Vol. 9, pp. 62-72, 1991.
  5. P. Spirtes, C. Glymour, and R. Scheines, Causation, Prediction, and Search, New York: Springer-Verlag, 1993
  6. T. Verma and J. Pearl, "Equivalence and Synthesis of Causal Models," Proceedings of the Sixth Conference on Uncertainty in Artificial Intelligence, (July, Cambridge, MA), pp. 220-227, 1990. Reprinted in P. Bonissone, M. Henrion, L. N. Kanal and J. F. Lemmer (editors), Uncertainty in Artificial Intelligence 6, Amsterdam: Elsevier Science Publishers, B.V., pp. 225-268, 1991
  7. 7.0 7.1 Davis (2003)
  8. Kadane & Schum (1996)
  9. Kadane & Schum (1996) 66-76
  10. Wright, S. (1921) "Correlation and Causation," Journal of Agricultural Research, 20:557-585.


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