Editing Bayesian inference (section)

===Computer applications===
Bayesian inference has applications in [[artificial intelligence]] and [[expert system]]s.  Bayesian inference techniques have been a fundamental part of computerized [[pattern recognition]] techniques since the late 1950s.<ref>{{cite journal |last1=Fienberg |first1=Stephen E. |title=When did Bayesian inference become "Bayesian"? |journal=Bayesian Analysis |date=2006-03-01 |volume=1 |issue=1 |doi=10.1214/06-BA101|doi-access=free }}</ref> There is also an ever-growing connection between Bayesian methods and simulation-based [[Monte Carlo method|Monte Carlo]] techniques since complex models cannot be processed in closed form by a Bayesian analysis, while a [[graphical model]] structure ''may'' allow for efficient simulation algorithms like the [[Gibbs sampling]] and other [[Metropolis–Hastings algorithm]] schemes.<ref>{{cite book|author=Jim Albert|year=2009|title= Bayesian Computation with R, Second edition|publisher=Springer|location=New York, Dordrecht, etc.|isbn= 978-0-387-92297-3}}</ref> Recently{{when|date=September 2018}} Bayesian inference has gained popularity among the [[phylogenetics]] community for these reasons; a number of applications allow many demographic and evolutionary parameters to be estimated simultaneously.

As applied to [[statistical classification]], Bayesian inference has been used to develop algorithms for identifying [[e-mail spam]]. Applications which make use of Bayesian inference for spam filtering include [[CRM114 (program)|CRM114]], [[DSPAM]], [[Bogofilter]], [[SpamAssassin]], [[SpamBayes]], [[Mozilla]], XEAMS, and others. Spam classification is treated in more detail in the article on the [[naïve Bayes classifier]].

[[Solomonoff's theory of inductive inference|Solomonoff's Inductive inference]] is the theory of prediction based on observations; for example, predicting the next symbol based upon a given series of symbols. The only assumption is that the environment follows some unknown but computable [[probability distribution]]. It is a formal inductive framework that combines two well-studied principles of inductive inference: Bayesian statistics and [[Occam's Razor]].<ref>{{cite journal |doi= 10.3390/e13061076 |arxiv=1105.5721 |bibcode=2011Entrp..13.1076R |title=A Philosophical Treatise of Universal Induction| journal=Entropy|volume=13 |issue=6|pages=1076–1136|year=2011|last1=Rathmanner|first1=Samuel|last2=Hutter|first2=Marcus| last3=Ormerod|first3=Thomas C|s2cid=2499910 |doi-access=free }}</ref>{{rs inline|date=September 2018}} Solomonoff's universal prior probability of any prefix ''p'' of a computable sequence ''x'' is the sum of the probabilities of all programs (for a universal computer) that compute something starting with ''p''. Given some ''p'' and any computable but unknown probability distribution from which ''x'' is sampled, the universal prior and Bayes' theorem can be used to predict the yet unseen parts of ''x'' in optimal fashion.<ref>{{Cite journal |bibcode = 2007arXiv0709.1516H |title = On Universal Prediction and Bayesian Confirmation|journal = Theoretical Computer Science |volume = 384 |issue = 2007|pages = 33–48|last1 = Hutter|first1 = Marcus|last2 = He|first2 = Yang-Hui|last3 = Ormerod|first3 = Thomas C|year = 2007|arxiv = 0709.1516|doi = 10.1016/j.tcs.2007.05.016 |s2cid = 1500830}}</ref><ref>{{Cite CiteSeerX  |last1=Gács |first1=Peter |last2=Vitányi |first2=Paul M. B. |date=2 December 2010 |title=Raymond J. Solomonoff 1926-2009 |citeseerx=10.1.1.186.8268  }}</ref>