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{{Short description|Statistical model}} {{more footnotes needed|date=February 2011}} {{Bayesian statistics}} <!-- Note: to keep the citation format consistent, please use the "cite" family of templates. --> A '''Bayesian network''' (also known as a '''Bayes network''', '''Bayes net''', '''belief network''', or '''decision network''') is a [[probabilistic graphical model]] that represents a set of variables and their [[Conditional dependence|conditional dependencies]] via a [[directed acyclic graph]] (DAG).<ref>{{Cite book |url=https://onlinelibrary.wiley.com/doi/book/10.1002/9780470061572 |title=Encyclopedia of Statistics in Quality and Reliability |date=2007-12-14 |publisher=Wiley |isbn=978-0-470-01861-3 |editor-last=Ruggeri |editor-first=Fabrizio |edition=1 |pages=1 |language=en |doi=10.1002/9780470061572.eqr089 |editor-last2=Kenett |editor-first2=Ron S. |editor-last3=Faltin |editor-first3=Frederick W.}}</ref> While it is one of several forms of [[causal notation]], causal networks are special cases of Bayesian networks. Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. 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. Efficient algorithms can perform [[inference]] and [[machine learning|learning]] in Bayesian networks. Bayesian networks that model sequences of variables (''e.g.'' [[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 diagram]]s. {{Toclimit|3}} ==Graphical model== Formally, Bayesian networks are [[directed acyclic graph]]s (DAGs) whose nodes represent variables in the [[Bayesian probability|Bayesian]] sense: they may be observable quantities, [[latent variable]]s, unknown parameters or hypotheses. Each edge represents a direct conditional dependency. Any pair of nodes that are not connected (i.e. no path connects one node to the other) represent variables that are [[conditional independence|conditionally independent]] of each other. Each node is associated with a [[Probability distribution|probability function]] that takes, as input, a particular set of values for the node's [[Glossary of graph theory#Directed acyclic graphs|parent]] variables, and gives (as output) the probability (or probability distribution, if applicable) of the variable represented by the node. For example, if <math>m</math> parent nodes represent <math>m</math> [[Boolean data type|Boolean variables]], then the probability function could be represented by a table of <small><math>2^m</math></small> entries, one entry for each of the <small><math>2^m</math></small> possible parent combinations. Similar ideas may be applied to undirected, and possibly cyclic, graphs such as [[Markov network]]s. ==Example== [[Image:SimpleBayesNet.svg|400px|thumb|right|A simple Bayesian network with [[conditional probability table]]s ]] Suppose we want to model the dependencies between three variables: the sprinkler (or more appropriately, its state - whether it is on or not), the presence or absence of rain and whether the grass is wet or not. Observe that two events can cause the grass to become wet: an active sprinkler or rain. Rain has a direct effect on the use of the sprinkler (namely that when it rains, the sprinkler usually is not active). This situation can be modeled with a Bayesian network (shown to the right). Each variable has two possible values, T (for true) and F (for false). The [[Joint probability distribution|joint probability function]] is, by the [[chain rule of probability]], : <math>\Pr(G,S,R)=\Pr(G\mid S,R) \Pr(S\mid R)\Pr(R)</math> where ''G'' = "Grass wet (true/false)", ''S'' = "Sprinkler turned on (true/false)", and ''R'' = "Raining (true/false)". The model can answer questions about the presence of a cause given the presence of an effect (so-called inverse probability) 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 variable]]s: : <math>\Pr(R=T\mid G=T) =\frac{\Pr(G=T,R=T)}{\Pr(G=T)} = \frac{\sum_{x \in \{T, F\}}\Pr(G=T, S=x,R=T)}{\sum_{x, y \in \{T, F\}} \Pr(G=T,S=x,R=y)}</math> Using the expansion for the joint probability function <math>\Pr(G,S,R)</math> and the conditional probabilities from the [[Conditional probability table|conditional probability tables (CPTs)]] stated in the diagram, one can evaluate each term in the sums in the numerator and denominator. For example, : <math>\begin{align} \Pr(G=T, S=T,R=T) & = \Pr(G=T\mid S=T,R=T)\Pr(S=T\mid R=T)\Pr(R=T) \\ & = 0.99 \times 0.01 \times 0.2 \\ & = 0.00198. \end{align} </math> Then the numerical results (subscripted by the associated variable values) are : <math>\Pr(R=T\mid G=T) = \frac{ 0.00198_{TTT} + 0.1584_{TFT} }{ 0.00198_{TTT} + 0.288_{TTF} + 0.1584_{TFT} + 0.0_{TFF} } = \frac{891}{2491}\approx 35.77 \%.</math> To answer an interventional question, such as "What is the probability that it would rain, given that we wet the grass?" the answer is governed by the post-intervention joint distribution function : <math>\Pr(S,R\mid\text{do}(G=T)) = \Pr(S\mid R) \Pr(R)</math> obtained by removing the factor <math>\Pr(G\mid S,R)</math> from the pre-intervention distribution. The do operator forces the value of G to be true. The probability of rain is unaffected by the action: : <math>\Pr(R\mid\text{do}(G=T)) = \Pr(R).</math> To predict the impact of turning the sprinkler on: : <math>\Pr(R,G\mid\text{do}(S=T)) = \Pr(R)\Pr(G\mid R,S=T)</math> with the term <math>\Pr(S=T\mid R)</math> removed, showing that the action affects the grass but not the rain. These predictions may not be feasible given unobserved variables, as in most policy evaluation problems. The effect of the action <math>\text{do}(x)</math> can still be predicted, however, whenever the back-door criterion is satisfied.<ref name=pearl2000/><ref>{{cite web|url=http://bayes.cs.ucla.edu/BOOK-2K/ch3-3.pdf |title=The Back-Door Criterion |access-date=2014-09-18}}</ref> It states that, if a set ''Z'' of nodes can be observed that [[#d-separation|''d''-separates]]<ref>{{cite web|url=http://bayes.cs.ucla.edu/BOOK-09/ch11-1-2-final.pdf |title=d-Separation without Tears |access-date=2014-09-18}}</ref> (or blocks) all back-door paths from ''X'' to ''Y'' then : <math>\Pr(Y,Z\mid\text{do}(x)) = \frac{\Pr(Y,Z,X=x)}{\Pr(X=x\mid Z)}.</math> A back-door path is one that ends with an arrow into ''X''. Sets that satisfy the back-door criterion are called "sufficient" or "admissible." For example, the set ''Z'' = ''R'' is admissible for predicting the effect of ''S'' = ''T'' on ''G'', because ''R'' ''d''-separates the (only) back-door path ''S'' ← ''R'' → ''G''. However, if ''S'' is not observed, no other set ''d''-separates this path and the effect of turning the sprinkler on (''S'' = ''T'') on the grass (''G'') cannot be predicted from passive observations. In that case ''P''(''G'' | do(''S'' = ''T'')) is not "identified". This reflects the fact that, lacking interventional data, 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 "''do''-calculus"<ref name="pearl2000"/><ref name="pearl-r212">{{cite conference |url=http://dl.acm.org/ft_gateway.cfm?id=2074452&ftid=1062250&dwn=1&CFID=161588115&CFTOKEN=10243006 |title=A Probabilistic Calculus of Actions | vauthors = Pearl J |year=1994 | veditors = Lopez de Mantaras R, Poole D | book-title = UAI'94 Proceedings of the Tenth international conference on Uncertainty in artificial intelligence |publisher=[[Morgan Kaufmann]] |location=San Mateo CA |pages=454–462 |isbn=1-55860-332-8 |arxiv=1302.6835 |bibcode=2013arXiv1302.6835P }}</ref> 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.<ref>{{cite book | vauthors = Shpitser I, Pearl J | chapter = Identification of Conditional Interventional Distributions | veditors = Dechter R, Richardson TS | title = Proceedings of the Twenty-Second Conference on Uncertainty in Artificial Intelligence | pages = 437–444 | location = Corvallis, OR | publisher = AUAI Press | year = 2006 | arxiv = 1206.6876 }}</ref> Using a Bayesian network can save considerable amounts of memory over exhaustive probability tables, 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 no variable's local distribution depends on more than three parent variables, the Bayesian network representation stores at most <math>10\cdot2^3 = 80</math> 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 distributions. ==Inference and learning== Bayesian networks perform three main inference tasks: ===Inferring unobserved variables=== Because a Bayesian network is a complete model for its variables and their relationships, it can be used to answer probabilistic queries about them. For example, the network can be used to update 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 choosing values for the variable subset that 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; [[Junction tree algorithm|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 [[space–time 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 [[Markov chain Monte Carlo|MCMC]] simulation, mini-bucket elimination, [[loopy belief propagation]], [[generalized belief propagation]] and [[variational Bayes|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 [[normal distribution|Gaussian distributions]] since that simplifies calculations. Sometimes only constraints on 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]], the conditional distribution for the hidden state's temporal evolution is commonly specified to maximize the [[entropy rate]] of the implied stochastic process.) Often these conditional distributions include parameters that are unknown and must be estimated from data, e.g., via the [[maximum likelihood]] approach. Direct maximization of the likelihood (or of the [[posterior probability]]) is often complex given 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 them 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, making classical parameter-setting approaches more tractable. ===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 (BN) is a challenge pursued within [[machine learning]]. The basic idea goes back to a recovery algorithm developed by Rebane and [[Judea Pearl|Pearl]]<ref>{{cite book | vauthors = Rebane G, Pearl J | chapter = The Recovery of Causal Poly-trees from Statistical Data| title = Proceedings, 3rd Workshop on Uncertainty in AI | location = Seattle, WA | pages = 222–228 | year = 1987 | arxiv = 1304.2736}}</ref> and rests on the distinction between the three possible patterns allowed in a 3-node DAG: {| class="wikitable" |+Junction patterns !Pattern !Model |- |Chain !<math>X \rightarrow Y \rightarrow Z</math> |- |Fork |<math>X \leftarrow Y \rightarrow Z</math> |- |Collider |<math>X \rightarrow Y \leftarrow Z</math> |} The first 2 represent the same dependencies (<math>X</math> and <math>Z</math> are independent given <math>Y</math>) and are, therefore, indistinguishable. The collider, 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 independences observed.<ref name="pearl2000">{{Cite book | first = Judea | last = Pearl | author-link = Judea Pearl | title = Causality: Models, Reasoning, and Inference |url={{google books |plainurl=y |id=LLkhAwAAQBAJ}}| publisher = [[Cambridge University Press]] | year = 2000 | isbn = 978-0-521-77362-1 | oclc = 42291253 }}</ref><ref>{{cite journal | vauthors = Spirtes P, Glymour C |title=An algorithm for fast recovery of sparse causal graphs |journal=Social Science Computer Review |volume=9 |issue=1 |pages=62–72 |year=1991 |doi=10.1177/089443939100900106 |s2cid=38398322 |url=http://repository.cmu.edu/cgi/viewcontent.cgi?article=1316&context=philosophy |format=PDF|citeseerx=10.1.1.650.2922 }}</ref><ref>{{cite book |first1=Peter |last1=Spirtes |first2=Clark N. |last2=Glymour |first3=Richard |last3=Scheines | name-list-style = vanc |title=Causation, Prediction, and Search |url={{google books |plainurl=y |id=VkawQgAACAAJ}} |year=1993 |publisher=Springer-Verlag |isbn=978-0-387-97979-3 |edition=1st}}</ref><ref>{{cite conference |title=Equivalence and synthesis of causal models |url={{google books |plainurl=y |id=ikuuHAAACAAJ}}|first1=Thomas |last1=Verma |first2=Judea |last2=Pearl |year=1991 | veditors = Bonissone P, Henrion M, Kanal LN, Lemmer JF | book-title = UAI '90 Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence |publisher=Elsevier |pages=255–270 |isbn=0-444-89264-8 }}</ref> An alternative method of structural learning uses optimization-based search. It requires a [[scoring function]] and a search strategy. A common scoring function is [[posterior probability]] of the structure given the training data, like the [[Bayesian information criterion|BIC]] or the BDeu. The time requirement of an [[exhaustive search]] returning a structure that maximizes the score is [[Tetration|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.<ref>{{cite journal |last1=Friedman |first1=Nir |last2=Geiger |first2=Dan |last3=Goldszmidt |first3=Moises | name-list-style = vanc |date=November 1997 |title=Bayesian Network Classifiers |journal=Machine Learning |volume=29 |issue=2–3 |pages=131–163 |doi=10.1023/A:1007465528199|doi-access=free }}</ref><ref>{{cite journal | vauthors = Friedman N, Linial M, Nachman I, Pe'er D | title = Using Bayesian networks to analyze expression data | journal = Journal of Computational Biology | volume = 7 | issue = 3–4 | pages = 601–20 | date = August 2000 | pmid = 11108481 | doi = 10.1089/106652700750050961 | citeseerx = 10.1.1.191.139 }}</ref> discuss 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. A particularly fast method for exact BN learning is to cast the problem as an optimization problem, and solve it using [[integer programming]]. Acyclicity constraints are added to the integer program (IP) during solving in the form of [[Cutting-plane method|cutting planes]].<ref>{{Cite journal|last=Cussens|first=James | name-list-style = vanc |year=2011|title=Bayesian network learning with cutting planes|url=https://dslpitt.org/papers/11/p153-cussens.pdf|archive-url=https://web.archive.org/web/20220327163338/https://dslpitt.org/papers/11/p153-cussens.pdf|url-status=usurped|archive-date=March 27, 2022|journal=Proceedings of the 27th Conference Annual Conference on Uncertainty in Artificial Intelligence|pages=153–160|bibcode=2012arXiv1202.3713C |arxiv=1202.3713 }}</ref> Such method can handle problems with up to 100 variables. In order to deal with problems with thousands of variables, a different approach is necessary. One is to first sample one ordering, and then find the optimal BN structure with respect to that ordering. This implies working on the search space of the possible orderings, which is convenient as it is smaller than the space of network structures. Multiple orderings are then sampled and evaluated. This method has been proven to be the best available in literature when the number of variables is huge.<ref>{{cite book | vauthors = Scanagatta M, de Campos CP, Corani G, Zaffalon M | chapter-url = https://papers.nips.cc/paper/5803-learning-bayesian-networks-with-thousands-of-variables | chapter = Learning Bayesian Networks with Thousands of Variables | title = NIPS-15: Advances in Neural Information Processing Systems | volume = 28 | pages = 1855–1863 | year = 2015 | publisher = Curran Associates }}</ref> Another method consists of focusing on the sub-class of decomposable models, for which the [[Maximum likelihood estimate|MLE]] have a closed form. It is then possible to discover a consistent structure for hundreds of variables.<ref name="Petitjean">{{cite conference |url=http://www.tiny-clues.eu/Research/Petitjean2013-ICDM.pdf |title= Scaling log-linear analysis to high-dimensional data | vauthors = Petitjean F, Webb GI, Nicholson AE |year=2013 |publisher=IEEE |conference=International Conference on Data Mining |location=Dallas, TX, USA }}</ref> Learning Bayesian networks with bounded treewidth is necessary to allow exact, tractable inference, since the worst-case inference complexity is exponential in the treewidth k (under the exponential time hypothesis). Yet, as a global property of the graph, it considerably increases the difficulty of the learning process. In this context it is possible to use [[K-tree]] for effective learning.<ref>M. Scanagatta, G. Corani, C. P. de Campos, and M. Zaffalon. [http://papers.nips.cc/paper/6232-learning-treewidth-bounded-bayesian-networks-with-thousands-of-variables Learning Treewidth-Bounded Bayesian Networks with Thousands of Variables.] In NIPS-16: Advances in Neural Information Processing Systems 29, 2016.</ref> ==Statistical introduction== {{Main|Bayesian statistics|Multilevel model}} Given data <math>x\,\!</math> and parameter <math>\theta</math>, a simple [[Bayesian statistics|Bayesian analysis]] starts with a [[prior probability]] (''prior'') <math>p(\theta)</math> and [[likelihood function|likelihood]] <math>p(x\mid\theta)</math> to compute a [[posterior probability]] <math>p(\theta\mid x) \propto p(x\mid\theta)p(\theta)</math>. Often the prior on <math>\theta</math> depends in turn on other parameters <math>\varphi</math> that are not mentioned in the likelihood. So, the prior <math>p(\theta)</math> must be replaced by a likelihood <math>p(\theta\mid \varphi)</math>, and a prior <math>p(\varphi)</math> on the newly introduced parameters <math>\varphi</math> is required, resulting in a posterior probability : <math>p(\theta,\varphi\mid x) \propto p(x\mid\theta)p(\theta\mid\varphi)p(\varphi).</math> This is the simplest example of a [[Bayesian hierarchical modeling|''hierarchical Bayes model'']]. The process may be repeated; for example, the parameters <math>\varphi</math> may depend in turn on additional parameters <math>\psi\,\!</math>, which require their own prior. Eventually the process must terminate, with priors that do not depend on unmentioned parameters. ===Introductory examples=== {{Expand section|date=March 2009|reason=More examples needed}} Given the measured quantities <math>x_1,\dots,x_n\,\!</math>each with [[Normal distribution|normally distributed]] errors of known [[standard deviation]] <math>\sigma\,\!</math>, : <math> x_i \sim N(\theta_i, \sigma^2) </math> Suppose we are interested in estimating the <math>\theta_i</math>. An approach would be to estimate the <math>\theta_i</math> using a [[maximum likelihood]] approach; since the observations are independent, the likelihood factorizes and the maximum likelihood estimate is simply : <math> \theta_i = x_i. </math> However, if the quantities are related, so that for example the individual <math>\theta_i</math>have themselves been drawn from an underlying distribution, then this relationship destroys the independence and suggests a more complex model, e.g., : <math> x_i \sim N(\theta_i,\sigma^2), </math> : <math> \theta_i\sim N(\varphi, \tau^2), </math> with [[improper prior]]s <math>\varphi\sim\text{flat}</math>, <math>\tau\sim\text{flat} \in (0,\infty)</math>. When <math>n\ge 3</math>, this is an ''identified model'' (i.e. there exists a unique solution for the model's parameters), and the posterior distributions of the individual <math>\theta_i</math> will tend to move, or ''[[Shrinkage estimator|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 <math>\tau\,\!</math> in the example. The usual priors such as the [[Jeffreys prior]] often do not work, because the posterior distribution will not be normalizable and estimates made by minimizing the [[Loss function#Expected loss|expected loss]] will be [[admissible decision rule|inadmissible]]. ==Definitions and concepts== {{See also|Glossary of graph theory#Directed acyclic graphs}} Several equivalent definitions of a Bayesian network have been offered. For the following, let ''G'' = (''V'',''E'') be a [[directed acyclic graph]] (DAG) and let ''X'' = (''X''<sub>''v''</sub>), ''v'' ∈ ''V'' be a set of [[random variable]]s 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:{{sfn|Russell|Norvig|2003|p=496}} : <math> p (x) = \prod_{v \in V} p \left(x_v \,\big|\, x_{\operatorname{pa}(v)} \right) </math> 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 (probability)|chain rule]] (given a [[topological ordering]] of ''X'') as follows:{{sfn|Russell|Norvig|2003|p=496}} : <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P \left(X_v=x_v \mid X_{v+1}=x_{v+1}, \ldots, X_n=x_n \right)</math> Using the definition above, this can be written as: : <math>\operatorname P(X_1=x_1, \ldots, X_n=x_n) = \prod_{v=1}^n \operatorname P (X_v=x_v \mid X_j=x_j \text{ for each } X_j\, \text{ that is a parent of } X_v\, )</math> The difference between the two expressions is the [[conditional independence]] of the variables from any of their non-descendants, given the values of their parent variables. ===Local Markov property=== ''X'' is a Bayesian network with respect to ''G'' if it satisfies the ''local Markov property'': each variable is [[Conditional independence|conditionally independent]] of its non-descendants given its parent variables:{{sfn|Russell|Norvig|2003|p=499}} :<math> X_v \perp\!\!\!\perp X_{V \,\smallsetminus\, \operatorname{de}(v)} \mid X_{\operatorname{pa}(v)} \quad\text{for all }v \in V</math> where de(''v'') is the set of descendants and ''V'' \ de(''v'') is the set of non-descendants of ''v''. This can be expressed in terms similar to the first definition, as :<math> \begin{align} & \operatorname P(X_v=x_v \mid X_i=x_i \text{ for each } X_i \text{ that is not a descendant of } X_v\, ) \\[6pt] = {} & P(X_v=x_v \mid X_j=x_j \text{ for each } X_j \text{ that is a parent of } X_v\, ) \end{align} </math> The set of parents is a subset of the set of non-descendants because the graph is [[Cycle (graph theory)|acyclic]]. ===Marginal independence structure=== In general, learning a Bayesian network from data is known to be [[NP-hardness|NP-hard]].<ref>{{cite journal |last1=Chickering |first1=David M. |last2=Heckerman |first2=David |last3=Meek |first3=Christopher |title=Large-sample learning of Bayesian networks is NP-hard |journal=Journal of Machine Learning Research |date=2004 |volume=5 |pages=1287–1330 |url=https://www.jmlr.org/papers/volume5/chickering04a/chickering04a.pdf}}</ref> This is due in part to the [[combinatorial explosion]] of [[Directed acyclic graph#Combinatorial enumeration|enumerating DAGs]] as the number of variables increases. Nevertheless, insights about an underlying Bayesian network can be learned from data in polynomial time by focusing on its marginal independence structure:<ref>{{cite journal |last1=Deligeorgaki |first1=Danai |last2=Markham |first2=Alex |last3=Misra |first3=Pratik |last4=Solus |first4=Liam |title=Combinatorial and algebraic perspectives on the marginal independence structure of Bayesian networks |journal=Algebraic Statistics |date=2023 |volume=14 |issue=2 |pages=233–286 |doi=10.2140/astat.2023.14.233|arxiv=2210.00822 }}</ref> while the conditional independence statements of a distribution modeled by a Bayesian network are encoded by a DAG (according to the factorization and Markov properties above), its marginal independence statements—the conditional independence statements in which the conditioning set is empty—are encoded by a [[Graph (discrete mathematics)|simple undirected graph]] with special properties such as equal [[Intersection number (graph theory)|intersection]] and [[Independent set (graph theory)|independence numbers]]. ===Developing Bayesian networks=== Developing a Bayesian network often begins with creating a DAG ''G'' such that ''X'' satisfies the local Markov property with respect to ''G''. Sometimes this is a [[Causal graph|causal]] DAG. The conditional probability distributions of each variable given its parents in ''G'' are assessed. In many cases, in particular in the case where the variables are discrete, if the joint distribution of ''X'' is the product of these conditional distributions, then ''X'' is a Bayesian network with respect to ''G''.<ref>{{cite book |first=Richard E. |last=Neapolitan | name-list-style = vanc |title=Learning Bayesian networks |url={{google books |plainurl=y |id=OlMZAQAAIAAJ}} |year=2004 |publisher=Prentice Hall |isbn=978-0-13-012534-7 }}</ref> ===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. The Markov blanket renders the node 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]].{{sfn|Russell|Norvig|2003|p=499}} ===={{anchor|d-separation}}''d''-separation==== This definition can be made more general by defining the "d"-separation of two nodes, where d stands for directional.<ref name=pearl2000/> We first define the "d"-separation of a trail and then we will define the "d"-separation of two nodes in terms of that. Let ''P'' be a trail from node ''u'' to ''v''. A trail is a loop-free, undirected (i.e. all edge directions are ignored) path between two nodes. Then ''P'' is said to be ''d''-separated by a set of nodes ''Z'' if any of the following conditions holds: *''P'' contains (but does not need to be entirely) a directed chain, <math> u \cdots \leftarrow m \leftarrow \cdots v</math> or <math> u \cdots \rightarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'', *''P'' contains a fork, <math> u \cdots \leftarrow m \rightarrow \cdots v</math>, such that the middle node ''m'' is in ''Z'', or *''P'' contains an inverted fork (or collider), <math> u \cdots \rightarrow m \leftarrow \cdots v</math>, such that the middle node ''m'' is not in ''Z'' and no descendant of ''m'' is in ''Z''. The nodes ''u'' and ''v'' are ''d''-separated by ''Z'' if all trails between them are ''d''-separated. If ''u'' and ''v'' are not d-separated, they are d-connected. ''X'' is a Bayesian network with respect to ''G'' if, for any two nodes ''u'', ''v'': : <math>X_u \perp\!\!\!\perp X_v \mid X_Z</math> where ''Z'' is a set which ''d''-separates ''u'' and ''v''. (The [[Markov blanket]] is the minimal set of nodes which ''d''-separates node ''v'' from all other nodes.) ===Causal networks=== Although Bayesian networks are often used to represent [[causality|causal]] relationships, this need not be the case: a directed edge from ''u'' to ''v'' does not require that ''X<sub>v</sub>'' be causally dependent on ''X<sub>u</sub>''. This is demonstrated by the fact that Bayesian networks on the graphs: :<math> a \rightarrow b \rightarrow c \qquad \text{and} \qquad a \leftarrow b \leftarrow c </math> are equivalent: that is they impose exactly the same conditional independence requirements. A causal network is a Bayesian network with the requirement that the relationships be causal. The additional semantics of 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 that of the network obtained by cutting the links from the parents of ''X'' to ''X'', and setting ''X'' to the caused value ''x''.<ref name=pearl2000/> Using these semantics, the impact of external interventions from data obtained prior to intervention can be predicted. ==Inference complexity and approximation algorithms== In 1990, while working at Stanford University on large bioinformatic applications, Cooper proved that exact inference in Bayesian networks is [[NP-hard]].<ref> {{cite journal | first = Gregory F. | last = Cooper | name-list-style = vanc | title = The Computational Complexity of Probabilistic Inference Using Bayesian Belief Networks | url = https://stat.duke.edu/~sayan/npcomplete.pdf | journal = Artificial Intelligence | volume = 42 | issue = 2–3 | date = 1990 | pages = 393–405 | doi = 10.1016/0004-3702(90)90060-d | s2cid = 43363498 }} </ref> This result prompted research on approximation algorithms with the aim of developing a tractable approximation to probabilistic inference. In 1993, Paul Dagum and [[Michael Luby]] proved two surprising results on the complexity of approximation of probabilistic inference in Bayesian networks.<ref> {{cite journal | vauthors = Dagum P, Luby M | author-link1 = Paul Dagum | author-link2 = Michael Luby | title = Approximating probabilistic inference in Bayesian belief networks is NP-hard | journal = Artificial Intelligence | volume = 60 | issue = 1 | date = 1993 | pages = 141–153 | doi = 10.1016/0004-3702(93)90036-b | citeseerx = 10.1.1.333.1586 }} </ref> First, they proved that no tractable [[deterministic algorithm]] can approximate probabilistic inference to within an [[absolute error]] ''ɛ'' < 1/2. Second, they proved that no tractable [[randomized algorithm]] can approximate probabilistic inference to within an absolute error ''ɛ'' < 1/2 with confidence probability greater than 1/2. At about the same time, [[Dan Roth|Roth]] proved that exact inference in Bayesian networks is in fact [[Sharp-P-complete|#P-complete]] (and thus as hard as counting the number of satisfying assignments of a [[conjunctive normal form]] formula (CNF)) and that approximate inference within a factor 2<sup>''n''<sup>1−''ɛ''</sup></sup> for every ''ɛ'' > 0, even for Bayesian networks with restricted architecture, is NP-hard.<ref>D. Roth, [http://cogcomp.cs.illinois.edu/page/publication_view/5 On the hardness of approximate reasoning], IJCAI (1993)</ref><ref>D. Roth, [http://cogcomp.cs.illinois.edu/papers/hardJ.pdf On the hardness of approximate reasoning], Artificial Intelligence (1996)</ref> In practical terms, these complexity results suggested that while Bayesian networks were rich representations for AI and machine learning applications, their use in large real-world applications would need to be tempered by either topological structural constraints, such as naïve Bayes networks, or by restrictions on the conditional probabilities. The bounded variance algorithm<ref>{{cite journal | vauthors = Dagum P, Luby M | author-link1 = Paul Dagum | author-link2 = Michael Luby | title = An optimal approximation algorithm for Bayesian inference | url = http://icsi.berkeley.edu/~luby/PAPERS/bayesian.ps | journal = Artificial Intelligence | volume = 93 | issue = 1–2 | date = 1997 | pages = 1–27 | doi = 10.1016/s0004-3702(97)00013-1 | citeseerx = 10.1.1.36.7946 | access-date = 2015-12-19 | archive-url = https://web.archive.org/web/20170706064354/http://www1.icsi.berkeley.edu/~luby/PAPERS/bayesian.ps | archive-date = 2017-07-06 }}</ref> developed by Dagum and Luby was the first provable fast approximation algorithm to efficiently approximate probabilistic inference in Bayesian networks with guarantees on the error approximation. This powerful algorithm required the minor restriction on the conditional probabilities of the Bayesian network to be bounded away from zero and one by <math>1/p(n)</math> where <math>p(n)</math> was any polynomial of the number of nodes in the network, <math>n</math>. ==Software== <!--Entries in this list should be "notable" with a sourced Wikipedia article. See WP:GNG and WP:WTAF. --> Notable software for Bayesian networks include: * [[Just another Gibbs sampler]] (JAGS) – Open-source alternative to WinBUGS. Uses Gibbs sampling. * [[OpenBUGS]] – Open-source development of WinBUGS. * [[SPSS Modeler]] – Commercial software that includes an implementation for Bayesian networks. * [[Stan (software)]] – Stan is an open-source package for obtaining Bayesian inference using the No-U-Turn sampler (NUTS),<ref>{{Cite arXiv |eprint = 1111.4246 |last1 = Hoffman|first1 = Matthew D.|last2 = Gelman|first2 = Andrew|title = The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo|year = 2011| class=stat.CO }}</ref> a variant of Hamiltonian Monte Carlo. * [[PyMC]] – A Python library implementing an embedded domain specific language to represent bayesian networks, and a variety of samplers (including NUTS) * [[WinBUGS]] – One of the first computational implementations of MCMC samplers. No longer maintained. ==History== The term Bayesian network was coined by [[Judea Pearl]] in 1985 to emphasize:<ref>{{cite conference |last=Pearl |first=J. | name-list-style = vanc |author-link=Judea Pearl |year=1985 |title=Bayesian Networks: A Model of Self-Activated Memory for Evidential Reasoning |conference=Proceedings of the 7th Conference of the Cognitive Science Society, University of California, Irvine, CA |pages=329–334 |url=http://ftp.cs.ucla.edu/tech-report/198_-reports/850017.pdf|access-date=2009-05-01 |format=UCLA Technical Report CSD-850017}}</ref> *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<ref>{{Cite journal | last1 = Bayes | first1 = T. | name-list-style = vanc | author-link = Thomas Bayes | year = 1763 | title = An Essay Towards Solving a Problem in the Doctrine of Chances | journal = [[Philosophical Transactions of the Royal Society]] | volume = 53 | pages = 370–418 | doi = 10.1098/rstl.1763.0053 | last2 = Price | title-link = An Essay Towards Solving a Problem in the Doctrine of Chances | doi-access = free }}</ref> In the late 1980s Pearl's ''Probabilistic Reasoning in Intelligent Systems''<ref>{{cite book | vauthors = Pearl J |title=Probabilistic Reasoning in Intelligent Systems |publisher=[[Morgan Kaufmann]] |location=San Francisco CA |isbn=978-1-55860-479-7 |page=1988 |url={{google books |plainurl=y |id=AvNID7LyMusC}}|date=1988-09-15 }}</ref> and [[Richard E. Neapolitan|Neapolitan]]'s ''Probabilistic Reasoning in Expert Systems''<ref>{{cite book |first=Richard E. |last=Neapolitan | name-list-style = vanc |title=Probabilistic reasoning in expert systems: theory and algorithms |url={{google books |plainurl=y |id=7X5KLwEACAAJ}} |year=1989 |publisher=Wiley |isbn=978-0-471-61840-9}}</ref> summarized their properties and established them as a field of study. == See also == {{Portal|Mathematics}} {{columns-list|colwidth=30em| *[[Bayesian epistemology]] *[[Bayesian programming]] *[[Causal inference]] *[[Causal loop diagram]] *[[Chow–Liu tree]] *[[Computational intelligence]] *[[Computational phylogenetics]] *[[Deep belief network]] *[[Dempster–Shafer theory]] – a generalization of Bayes' theorem *[[Expectation–maximization algorithm]] *[[Factor graph]] *[[Hierarchical temporal memory]] *[[Kalman filter]] *[[Memory-prediction framework]] *[[Mixture distribution]] *[[Mixture model]] *[[Naive Bayes classifier]] *[[Plate notation]] *[[Polytree]] *[[Sensor fusion]] *[[Sequence alignment]] *[[Structural equation modeling]] *[[Subjective logic]] *[[Variable-order Bayesian network]] }} ==Notes== {{Reflist}} == References == {{Refbegin}} * {{cite encyclopedia | last = Ben Gal | first = Irad | title = Support-Page | editor-last1 = Ruggeri | editor-first1 = Fabrizio | editor-last2 = Kennett | editor-first2 = Ron S. | editor-last3 = Faltin | editor-first3 = Frederick W | name-list-style = vanc | encyclopedia = Encyclopedia of Statistics in Quality and Reliability | chapter-url = http://www.eng.tau.ac.il/~bengal/BN.pdf | year = 2007 | publisher = [[John Wiley & Sons]] | isbn = 978-0-470-01861-3 | doi = 10.1002/9780470061572.eqr089 | chapter = Bayesian Networks | doi-access = free | access-date = 2007-08-27 | archive-date = 2016-11-23 | archive-url = https://web.archive.org/web/20161123041500/http://www.eng.tau.ac.il/~bengal/BN.pdf | url-status = dead }} * {{cite book |last1= Bertsch McGrayne |first1= Sharon |name-list-style= vanc |title= The Theory That Would not Die |year= 2011 |url= https://archive.org/details/theorythatwouldn0000mcgr |url-access= registration |location= New Haven |publisher= [[Yale University Press]] }} * {{cite book|last1= Borgelt|first1= Christian|last2= Kruse|first2= Rudolf|name-list-style= vanc|author2-link= Rudolf Kruse|title= Graphical Models: Methods for Data Analysis and Mining|url= http://fuzzy.cs.uni-magdeburg.de/books/gm/|date= March 2002|publisher= [[John Wiley & Sons|Wiley]]|location= [[Chichester|Chichester, UK]]|isbn= 978-0-470-84337-6}} * {{cite encyclopedia |last= Borsuk |first= Mark Edward | name-list-style = vanc | editor = Jørgensen, Sven Erik | editor-link = Sven Erik Jørgensen |editor2=Fath, Brian |encyclopedia= Encyclopedia of Ecology |title= Ecological informatics: Bayesian networks |year= 2008| publisher= Elsevier|isbn= 978-0-444-52033-3}} * {{cite book |last1=Castillo|first1=Enrique|last2=Gutiérrez |first2=José Manuel |last3=Hadi |first3=Ali S. | name-list-style = vanc |title= Expert Systems and Probabilistic Network Models |series= Monographs in computer science |year= 1997 |publisher= [[Springer Science+Business Media|Springer-Verlag]]|location=New York |isbn= 978-0-387-94858-4|pages= 481–528 |chapter= Learning Bayesian Networks}} * {{cite journal | last1 = Comley | first1 = Joshua W. | last2 = Dowe | first2 = David L. | name-list-style = vanc | title = General Bayesian networks and asymmetric languages | url = http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2003 | journal = Proceedings of the 2nd Hawaii International Conference on Statistics and Related Fields | date = June 2003 }} * {{cite book | last1 = Comley | first1 = Joshua W. | last2 = Dowe | first2 = David L. | name-list-style = vanc | year = 2005 | chapter = Minimum Message Length and Generalized Bayesian Nets with Asymmetric Languages | chapter-url = http://www.csse.monash.edu.au/~dld/David.Dowe.publications.html#ComleyDowe2005 | editor-last = Grünwald | editor-first = Peter D. | editor2-last = Myung | editor2-first = In Jae | editor3-last = Pitt | editor3-first = Mark A. | title = Advances in Minimum Description Length: Theory and Applications | series = Neural information processing series | location = [[Cambridge, Massachusetts]] | publisher = Bradford Books ([[MIT Press]]) | publication-date = April 2005 | pages = 265–294 | isbn = 978-0-262-07262-5 }} (This paper puts [[Decision tree learning|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]]). * {{cite book |last = Darwiche |first = Adnan |name-list-style = vanc |title = Modeling and Reasoning with Bayesian Networks |url = http://www.cambridge.org/9780521884389 |publisher = [[Cambridge University Press]] |year = 2009 |isbn = 978-0-521-88438-9 }} * {{Cite book|url={{google books |plainurl=y |id=mPG5RupkTX0C}}|title=Philosophy of Statistics|date=2011-05-31|publisher=Elsevier|isbn=978-0-08-093096-1|language=en|last=Dowe|first=David L.|chapter-url=http://www.csse.monash.edu.au/~dld/Publications/2010/Dowe2010_MML_HandbookPhilSci_Vol7_HandbookPhilStat_MML+hybridBayesianNetworkGraphicalModels+StatisticalConsistency+InvarianceAndUniqueness_pp901-982.pdf|chapter=Hybrid Bayesian network graphical models, statistical consistency, invariance and uniqueness|pages=[http://www.csse.monash.edu.au/~dld/Publications/2010/Dowe2010_MML_HandbookPhilSci_Vol7_HandbookPhilStat_MML+hybridBayesianNetworkGraphicalModels+StatisticalConsistency+InvarianceAndUniqueness_pp901-982.pdf 901–982]}} * {{cite book | last1 = Fenton | first1 = Norman | last2 = Neil | first2 = Martin E. | name-list-style = vanc | date = November 2007 | chapter-url = http://www.agenarisk.com/resources/apps_bayesian_networks.pdf | chapter = Managing Risk in the Modern World: Applications of Bayesian Networks | title = A Knowledge Transfer Report from the London Mathematical Society and the Knowledge Transfer Network for Industrial Mathematics. | location = [[London|London (England)]] | publisher = [[London Mathematical Society]] | access-date = 2008-10-29 | archive-date = 2008-05-14 | archive-url = https://web.archive.org/web/20080514044436/http://www.agenarisk.com/resources/apps_bayesian_networks.pdf }} * {{cite news |first1=Norman |last1=Fenton |first2=Martin E. |last2=Neil |name-list-style=vanc |title=Combining evidence in risk analysis using Bayesian Networks |url=https://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdf |work=Safety Critical Systems Club Newsletter |volume=13 |issue=4 |location=[[Newcastle upon Tyne]], England |pages=8–13 |date=July 23, 2004 |archive-url=https://web.archive.org/web/20070927153751/https://www.dcs.qmul.ac.uk/~norman/papers/Combining%20evidence%20in%20risk%20analysis%20using%20BNs.pdf |archive-date=2007-09-27 }} * {{cite book | first1 = Andrew | last1 = Gelman | first2 = John B | last2 = Carlin | first3 = Hal S | last3 = Stern | first4 = Donald B | last4 = Rubin | name-list-style = vanc | title = Bayesian Data Analysis | chapter = Part II: Fundamentals of Bayesian Data Analysis: Ch.5 Hierarchical models | chapter-url = {{google books |plainurl=y |id=TNYhnkXQSjAC|page=120}} | year = 2003 | publisher = [[CRC Press]] | isbn = 978-1-58488-388-3 | pages = 120– | url = {{google books |plainurl=y |id=TNYhnkXQSjAC}} }} * {{cite book | last = Heckerman | first = David | date = March 1, 1995 | contribution = Tutorial on Learning with Bayesian Networks | contribution-url = http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06 | editor-last = Jordan | editor-first = Michael Irwin | title = Learning in Graphical Models | series = Adaptive Computation and Machine Learning | location = [[Cambridge, Massachusetts]] | publication-date = 1998 | publisher = [[MIT Press]] | pages = 301–354 | isbn = 978-0-262-60032-3 | access-date = September 15, 2006 | archive-date = July 19, 2006 | archive-url = https://web.archive.org/web/20060719171558/http://research.microsoft.com/research/pubs/view.aspx?msr_tr_id=MSR-TR-95-06 | url-status = bot: unknown }}:Also appears as {{cite journal |date=March 1997|title= Bayesian Networks for Data Mining |journal= [[Data Mining and Knowledge Discovery]]|volume= 1| issue= 1 |pages= 79–119 |doi= 10.1023/A:1009730122752 |last1= Heckerman |first1= David|s2cid= 6294315 }} :An earlier version appears as, [[Microsoft Research]], March 1, 1995. The paper is about both parameter and structure learning in Bayesian networks. * {{cite book | last1 = Jensen | first1 = Finn V | last2 = Nielsen | first2 = Thomas D. | name-list-style = vanc | title = Bayesian Networks and Decision Graphs | url = {{google books |plainurl=y |id=cWLaBwAAQBAJ}} | edition = 2nd | series = Information Science and Statistics series | publisher = [[Springer Science+Business Media|Springer-Verlag]] | location = [[New York City|New York]] | date = June 6, 2007 | isbn = 978-0-387-68281-5 }} * {{cite journal | last1 = Karimi | first1 = Kamran | last2 = Hamilton | first2 = Howard J. | name-list-style = vanc | title = Finding temporal relations: Causal bayesian networks vs. C4. 5 | journal = Twelfth International Symposium on Methodologies for Intelligent Systems | date = 2000 | url = http://www.kamran-karimi.com/pubs/khISMIS2000.pdf }} * {{cite book | last1 = Korb | first1 = Kevin B. | last2 = Nicholson | first2 = Ann E. | name-list-style = vanc | title = Bayesian Artificial Intelligence | url = {{google books |plainurl=y |id=LxXOBQAAQBAJ}} | edition = 2nd | publisher = [[Chapman & Hall]] ([[CRC Press]]) | date = December 2010 | isbn = 978-1-58488-387-6 | series = CRC Computer Science & Data Analysis | doi = 10.1007/s10044-004-0214-5 | s2cid = 22138783 }} * {{cite journal | vauthors = Lunn D, Spiegelhalter D, Thomas A, Best N | title = The BUGS project: Evolution, critique and future directions | journal = Statistics in Medicine | volume = 28 | issue = 25 | pages = 3049–67 | date = November 2009 | pmid = 19630097 | doi = 10.1002/sim.3680 | s2cid = 7717482 }} * {{cite journal | vauthors = Neil M, Fenton N, Tailor M | title = Using Bayesian networks to model expected and unexpected operational losses | journal = Risk Analysis | volume = 25 | issue = 4 | pages = 963–72 | date = August 2005 | pmid = 16268944 | doi = 10.1111/j.1539-6924.2005.00641.x | bibcode = 2005RiskA..25..963N | s2cid = 3254505 | url = http://www.dcs.qmul.ac.uk/~norman/papers/oprisk.pdf | editor = Greenberg, Michael R. }} * {{cite journal |last= Pearl|first= Judea |author-link= Judea Pearl | name-list-style = vanc |date=September 1986|title= Fusion, propagation, and structuring in belief networks |journal= [[Artificial Intelligence (journal)|Artificial Intelligence]]|volume= 29 |issue= 3 |pages= 241–288 |doi= 10.1016/0004-3702(86)90072-X}} * {{cite book |last = Pearl |first = Judea |name-list-style = vanc |author-link = Judea Pearl |title = Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference |url = {{google books |plainurl=y |id=mn2jBQAAQBAJ}} |edition = 2nd printing |publisher = [[Morgan Kaufmann]] |location = [[San Francisco, California]] |year = 1988 |isbn = 978-0-934613-73-6 |series = Representation and Reasoning Series }} * {{cite book |last1=Pearl |first1=Judea |author-link=Judea Pearl |last2=Russell |first2=Stuart |author-link2=Stuart J. Russell |contribution=Bayesian Networks |editor-last=Arbib |editor-first=Michael A. |name-list-style=vanc |editor-link=Michael A. Arbib |title=Handbook of Brain Theory and Neural Networks |url={{google books |plainurl=y |id=Av6qWhtw0-EC}} |pages=157–160 |location=[[Cambridge, Massachusetts]] |date=November 2002 |publisher=Bradford Books ([[MIT Press]]) |isbn=978-0-262-01197-6 }} * {{Russell Norvig 2003}}. * {{cite journal | last1 = Zhang | first1 = Nevin Lianwen | last2 = Poole | first2 = David | name-list-style = vanc | title = A simple approach to Bayesian network computations | journal = Proceedings of the Tenth Biennial Canadian Artificial Intelligence Conference (AI-94). | date = May 1994 | pages = 171–178 | url = http://www.cs.ust.hk/~lzhang/paper/pspdf/canai94.pdf }} This paper presents variable elimination for belief networks. {{Refend}} == Further reading == * {{cite book | title = Bayesian Networks and BayesiaLab – A practical introduction for researchers|url={{google books |plainurl=y |id=etXXsgEACAAJ}} | first1 = Stefan | last1 = Conrady | first2 = Lionel | last2 = Jouffe | name-list-style = vanc | isbn = 978-0-9965333-0-0 | publisher = Bayesian USA | location = Franklin, Tennessee |date=2015-07-01 }} * {{cite web | url = http://pages.cs.wisc.edu/~dyer/cs540/handouts/charniak.pdf | title = Bayesian networks without tears | first = Eugene | last = Charniak | name-list-style = vanc | work = AI Magazine | date = Winter 1991 }} * {{cite book | first1 =Rudolf | last1 = Kruse | first2 = Christian | last2 = Borgelt | first3 = Frank | last3 = Klawonn | first4 = Christian | last4 = Moewes | first5 = Matthias | last5 = Steinbrecher | first6 = Pascal | last6 = Held | name-list-style = vanc | title = Computational Intelligence A Methodological Introduction |url={{google books |plainurl=y |id=etXXsgEACAAJ}}| date=2013 |publisher = Springer-Verlag | location = London | isbn = 978-1-4471-5012-1 }} * {{cite book | first1 = Christian | last1 = Borgelt | first2 = Matthias | last2 = Steinbrecher | first3 = Rudolf | last3 = Kruse | name-list-style = vanc | title = Graphical Models – Representations for Learning, Reasoning and Data Mining |url={{google books |plainurl=y |id=I8Fa-LKDpF0C}} | date = 2009 | publisher = Wiley | location = Chichester | isbn = 978-0-470-74956-2 | edition = Second }} == External links == *[http://www.niedermayer.ca/papers/bayesian/bayes.html An Introduction to Bayesian Networks and their Contemporary Applications] *[http://www.dcs.qmw.ac.uk/%7Enorman/BBNs/BBNs.htm On-line Tutorial on Bayesian nets and probability] *[https://web.archive.org/web/20170601002137/http://princesofserendib.com/ Web-App to create Bayesian nets and run it with a Monte Carlo method] *[http://robotics.stanford.edu/~nodelman/papers/ctbn.pdf Continuous Time Bayesian Networks] *[https://web.archive.org/web/20090923200511/http://wiki.syncleus.com/index.php/DANN%3ABayesian_Network Bayesian Networks: Explanation and Analogy] *[http://videolectures.net/kdd07_neapolitan_lbn/ A live tutorial on learning Bayesian networks] *[http://www.biomedcentral.com/1471-2105/7/514/abstract 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. *[http://www.labmedinfo.org/download/lmi339.pdf Hierarchical Naive Bayes Model for handling sample uncertainty] {{Webarchive|url=https://web.archive.org/web/20070928081740/http://www.labmedinfo.org/download/lmi339.pdf |date=2007-09-28 }}, shows how to perform classification and learning with continuous and discrete variables with replicated measurements. {{DEFAULTSORT:Bayesian Network}} [[Category:Bayesian networks| ]] [[Category:Graphical models]] [[Category:Causal inference]]
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