Ogled vira Bayesiansko sklepanje

'''Bayesiansko sklepanje''' je metoda statističnega sklepanja, pri katerem uporabimo [[Bayesov izrek]] za posodobitev verjetnosti za hipotezo, ko postane na voljo več dokazov ali podatkov. Bayesiansko sklepanje je pomembna tehnika v [[statistika|statistki]], zlasti v [[matematična statistika|matematični statistiki]]. Bayesiansko posodabljanje je posebno pomembno pri dinamični analizi sekvenc podatkov. Bayesiansko sklepanje je v uporabi v [[znanost]]i, [[inženirstvo|inženirstvu]], [[filozofija|filozofiji]], [[medicina|medicini]], [[šport]]u in [[pravo|pravu]]. V filozofiji teorije odločitev je Bayesiansko sklepanje tesno povezano s subjektivno verjetnostjo, pogosto imenovano Bayesianska verjetnost.

==Uvod v Bayesovo pravilo==
[[File:Bayes_theorem_visualisation.svg|thumb|300px|Geometrijska predstavitev Bayesovega izreka. V tabeli vrednosti 2, 3, 6 in 9 dajo relativne uteži vsakega ustreznega pogoja in primera. Slike označujejo celice v tabeli vključene v vsako metriko, verjetnost, da je ulomek vsake slike, ki je pobarvana. To prikazuje, da P(A<nowiki>|</nowiki>B) P(B) = P(B<nowiki>|</nowiki>A) P(A) tj. P(A<nowiki>|</nowiki>B) = {{sfrac|P(B<nowiki>|</nowiki>A) P(A)|P(B)}}. Podobno logiko lahko uporabimo, da prikažemo, da P(&not;A<nowiki>|</nowiki>B) = {{sfrac|P(B<nowiki>|</nowiki>&not;A) P(&not;A)|P(B)}} itd.]]
{{Main|Bayesov izrek}}

===Formalna razlaga===
{| class="wikitable floatright" style="font-size:100%;"
|+ [[Pogojna tabela]]
! {{diagonal split header|<br /><br />Dokazi|Hipoteza}} !! Ugaja<br />hipotezi<br />H !! Krši<br />hipotezo<br />&not;H !! rowspan="5" style="padding:0;"| !! <br />Skupno
|-
! Ima dokaze<br />E
| |'''P(H|E)·P(E)'''<br />= P(E|H)·P(H) || |'''P(&not;H|E)·P(E)'''<br />= P(E|&not;H)·P(&not;H) || '''P(E)'''
|-
! Nima dokazov<br />&not;E
| nowrap|'''P(H|&not;E)·P(&not;E)'''<br />= P(&not;E|H)·P(H) || nowrap|'''P(&not;H|&not;E)·P(&not;E)'''<br />= P(&not;E|&not;H)·P(&not;H) || nowrap|'''P(&not;E) =<br />1&minus;P(E)'''
|-
| colspan="5" style="padding:0;"|
|-
! Skupno
| &nbsp;&nbsp; P(H) || style="text-align:right;" nowrap|P(&not;H) = 1&minus;P(H) || style="text-align:center;"|1
|}
Bayesiansko sklepanje izpeljuje [[posteriorna verjetnost|posteriorno verjetnost]] kot posledico dveh predhodnikov: [[priorna verjetnost|priorne verjetnosti]] in [[funkcija verjetja|funkcije verjetja]], izpeljane iz statističnega modela za opazovane podatke. Bayesiansko sklepanje izračuna posteriorno verjetnost glede na [[Bayesov izrek]]:
<math display="block">P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)},</math>
kjer
* <math>H</math> predstavlja katerokoli ''hipotezo'', na verjetnost katere lahko vplivajo eksperimentalni podatki (ali ''dokazi''). Pogosto imamo konkurenčne hipoteze, in naloga je določiti, katera je najbolj verjetna.
* <math>P(H)</math>, ''[[priorna verjetnost]]'', je ocena verjetnosti hipoteze <math>H</math>, ''preden'' opazujemo podatke <math>E</math>, trenutne dokaze.
* <math>E</math>, ''dokazi'', ustreza novim podatkom, ki niso bili uporabljeni pri računanju priorne verjetnosti.
* <math>P(H \mid E)</math>, ''[[posteriorna verjetnost]]'', je verjetnost <math>H</math> ''pri danem'' <math>E</math>, tj. ''po tem'', ko je <math>E</math> opazovan. Zanima nas verjetnost hipoteze ''pri danih'' opazovanih dokazih.
* <math>P(E \mid H)</math> je verjetnost opazovanja <math>E</math> ''pri danem'' <math>H</math> in se imenuje ''[[funkcija verjetja|verjetje]]''. Kot funkcija <math>E</math> s fiksnim <math>H</math>, nakazuje kompatibilnost dokazov z dano hipotezo. Funkcija verjetja je funkcija dokazov <math>E</math>, medtem ko je posteriorna verjetnost funkcija hipoteze <math>H</math>.
* <math>P(E)</math> se včasih imenuje [[mejna verjetnost]] ali »modelski dokazi«. Ta dejavnik je enak za vse možne obravnavane hipoteze. (as is evidentkot je očitno na podlagi dejstva, da se hipoteza <math>H</math> ne pojavi nikjer v simbolu, za razliko od vseh drugih dejavnikov) in tako ni dejavnik pri določanju relativnih verjetnosti drugačnih hipotez.

==Viri==
{{refbegin}}
* Aster, Richard; Borchers, Brian, and Thurber, Clifford (2012). ''Parameter Estimation and Inverse Problems'', Second Edition, Elsevier. {{ISBN|0123850487}}, {{ISBN|978-0123850485}}
* {{navedi knjigo |author1 = Bickel, Peter J.  |author2 = Doksum, Kjell A. |name-list-style=amp |title = Mathematical Statistics, Volume 1: Basic and Selected Topics |edition=Second (updated printing 2007) |year=2001 |publisher=Pearson Prentice–Hall |isbn = 978-0-13-850363-5 }}
* [[George E. P. Box|Box, G.&nbsp;E.&nbsp;P.]] and [[George Tiao|Tiao, G.&nbsp;C.]] (1973) ''Bayesian Inference in Statistical Analysis'', Wiley, {{ISBN|0-471-57428-7}}
* {{navedi knjigo |author = Edwards, Ward |chapter = Conservatism in Human Information Processing |editor = Kleinmuntz, B. |title = Formal Representation of Human Judgment |publisher=Wiley |year=1968 }}
* {{cite journal |author=Edwards, Ward | quote=Chapter: Conservatism in Human Information Processing (excerpted) |editor= Daniel Kahneman |editor-link= Daniel Kahneman |editor2=Paul Slovic |editor2-link=Paul Slovic |editor3=Amos Tversky |editor3-link=Amos Tversky |title = Judgment under uncertainty: Heuristics and biases |journal=Science |volume=185 |issue=4157 |pages=1124–1131 |year=1982 |bibcode = 1974Sci...185.1124T |doi = 10.1126/science.185.4157.1124 |pmid = 17835457 | s2cid=143452957 }}
* [[Edwin Thompson Jaynes|Jaynes E.&nbsp;T.]] (2003) ''Probability Theory: The Logic of Science'', CUP. {{ISBN|978-0-521-59271-0}} ([http://www-biba.inrialpes.fr/Jaynes/prob.html Link to Fragmentary Edition of March 1996]).
* {{navedi knjigo |title=Scientific Reasoning: the Bayesian Approach| author=Howson, C. |author2=Urbach, P. |name-list-style=amp |publisher=[[Open Court Publishing Company]] |year=2005 |edition=3rd |isbn=978-0-8126-9578-6 | author-link=Colin Howson }}
* {{navedi knjigo |last1=Phillips |first1=L. D.|last2=Edwards |first2=Ward |chapter=Chapter 6: Conservatism in a Simple Probability Inference Task (''Journal of Experimental Psychology'' (1966) 72: 346-354) |title = A Science of Decision Making:The Legacy of Ward Edwards| editor=Jie W. Weiss |editor2=David J. Weiss |isbn=978-0-19-532298-9 |page=536 |date=October 2008 |publisher = Oxford University Press }}
{{refend}}

==Nadaljnje branje==
* Za celotno poročilo o zgodovini Bayesianske statistike in debate s frekventističnimi pristopi, preberi {{navedi knjigo |last=Vallverdu |first=Jordi |title=Bayesians Versus Frequentists A Philosophical Debate on Statistical Reasoning |publisher=Springer |year=2016 |location=New York |isbn=978-3-662-48638-2 }}
* {{navedi knjigo |last=Clayton |first=Aubrey |author-link=Aubrey Clayton|url=https://cup.columbia.edu/book/bernoullis-fallacy/9780231199940 |title=Bernoulli's Fallacy: Statistical Illogic and the Crisis of Modern Science |date=August 2021 |publisher=Columbia University Press |isbn=978-0-231-55335-3}}

===Osnovno===
Naslednje knjige so navedene naraščajoče glede na verjetnostno naprednost:
* Stone, JV (2013), "Bayes' Rule: A Tutorial Introduction to Bayesian Analysis",   [http://jim-stone.staff.shef.ac.uk/BookBayes2012/BayesRuleBookMain.html Download first  chapter here], Sebtel Press, England.
* {{navedi knjigo| title=Understanding Uncertainty, Revised Edition| author=Dennis V. Lindley | publisher=John Wiley | year=2013| edition=2nd | isbn=978-1-118-65012-7| author-link=Dennis V. Lindley }}
* {{navedi knjigo| title=Scientific Reasoning: The Bayesian Approach| author=Colin Howson |author2=Peter Urbach |name-list-style=amp | publisher=[[Open Court Publishing Company]]| year=2005| edition=3rd | isbn=978-0-8126-9578-6| author-link=Colin Howson }}
* {{navedi knjigo|author=Berry, Donald A.|title=Statistics: A Bayesian Perspective|url=https://archive.org/details/ssmstatisticsbay0000dona|publisher=Duxbury| year=1996|isbn=978-0-534-23476-8}}
*{{navedi knjigo|author=Morris H. DeGroot|author2=Mark J. Schervish|name-list-style=amp|title=Probability and Statistics|edition=third|isbn=978-0-201-52488-8|publisher=Addison-Wesley|year=2002|url-access=registration|url=https://archive.org/details/probabilitystati00degr_0|author-link=Morris H. DeGroot}}
* Bolstad, William M. (2007) ''Introduction to Bayesian Statistics'': Second Edition, John Wiley {{ISBN|0-471-27020-2}}
*{{navedi knjigo |author=Winkler, Robert L |title=Introduction to Bayesian Inference and Decision |publisher=Probabilistic |year=2003 |isbn=978-0-9647938-4-2 |edition=2nd  }} Updated classic textbook. Bayesian theory clearly presented.
* Lee, Peter M. ''Bayesian Statistics: An Introduction''. Fourth Edition (2012), John Wiley {{ISBN|978-1-1183-3257-3}}
* {{navedi knjigo| title = Bayesian Methods for Data Analysis, Third Edition | publisher = Boca Raton, FL: Chapman and Hall/CRC | year = 2008 | isbn = 978-1-58488-697-6|author1=Carlin, Bradley P.  |author2=Louis, Thomas A. |name-list-style=amp }}
* {{navedi knjigo| title = Bayesian Data Analysis, Third Edition | publisher = Chapman and Hall/CRC | year = 2013 | isbn = 978-1-4398-4095-5|last1=Gelman|first1=Andrew|author-link1=Andrew Gelman|last2=Carlin|first2=John B.|last3=Stern|first3=Hal S.|last4=Dunson|first4=David B.|last5=Vehtari|first5=Aki|last6=Rubin|first6=Donald B.|author-link6=Donald Rubin}}

===Vmesno ali napredno===
* {{navedi knjigo|author=Berger, James O| title=Statistical Decision Theory and Bayesian Analysis|url=https://archive.org/details/statisticaldecis0000berg| edition=Second|year=1985| publisher=Springer-Verlag|series=Springer Series in Statistics|isbn=978-0-387-96098-2| bibcode=1985sdtb.book.....B| author-link=James Berger (statistician)}}
*{{navedi knjigo|title=Bayesian Theory|publisher=Wiley|year=1994|author-link1=José-Miguel Bernardo|last1=Bernardo|first1=José&nbsp;M.|author-link2=Adrian Smith (statistician)|last2=Smith|first2=Adrian&nbsp;F.&nbsp;M.}}
* [[Morris H. DeGroot|DeGroot, Morris H.]], ''Optimal Statistical Decisions''. Wiley Classics Library. 2004. (Originally published (1970) by McGraw-Hill.) {{ISBN|0-471-68029-X}}.
* {{navedi knjigo|title=Theory of statistics|first=Mark J.|last=Schervish|publisher=Springer-Verlag|year=1995|isbn=978-0-387-94546-0}}
* Jaynes, E. T. (1998) [http://www-biba.inrialpes.fr/Jaynes/prob.html ''Probability Theory: The Logic of Science''].
* O'Hagan, A. and Forster, J. (2003) ''Kendall's Advanced Theory of Statistics'', Volume 2B: ''Bayesian Inference''. Arnold, New York. {{ISBN|0-340-52922-9}}.
* {{navedi knjigo|author=Robert, Christian P|title=The Bayesian Choice – A Decision-Theoretic Motivation|publisher=Springer|year=2001|edition=second|isbn=978-0-387-94296-4}}
* [[Glenn Shafer]] and [[Judea Pearl|Pearl, Judea]], eds. (1988) ''Probabilistic Reasoning in Intelligent Systems'', San Mateo, CA: Morgan Kaufmann.
* Pierre Bessière et al. (2013), "[http://www.crcpress.com/product/isbn/9781439880326 Bayesian Programming]", CRC Press. {{ISBN|9781439880326}}
* Francisco J. Samaniego (2010), "A Comparison of the Bayesian and Frequentist Approaches to Estimation" Springer, New York, {{ISBN|978-1-4419-5940-9}}

==Zunanje povezave==
* {{springer|title=Bayesian approach to statistical problems|id=p/b015390}}
* [http://www.scholarpedia.org/article/Bayesian_statistics Bayesian Statistics] from Scholarpedia.
* [http://www.dcs.qmw.ac.uk/%7Enorman/BBNs/BBNs.htm Introduction to Bayesian probability] from Queen Mary University of London
* [http://webuser.bus.umich.edu/plenk/downloads.htm Mathematical Notes on Bayesian Statistics and Markov Chain Monte Carlo]
* [http://cocosci.berkeley.edu/tom/bayes.html Bayesian reading list], categorized and annotated by [https://web.archive.org/web/20060711151352/http://psychology.berkeley.edu/faculty/profiles/tgriffiths.html Tom Griffiths]
* A. Hajek and S. Hartmann: [https://web.archive.org/web/20110728055439/http://stephanhartmann.org/HajekHartmann_BayesEpist.pdf Bayesian Epistemology], in: J. Dancy et al. (eds.), A Companion to Epistemology. Oxford: Blackwell 2010, 93–106.
* S. Hartmann and J. Sprenger: [https://web.archive.org/web/20110728055519/http://stephanhartmann.org/HartmannSprenger_BayesEpis.pdf Bayesian Epistemology], in: S. Bernecker and D. Pritchard (eds.), Routledge Companion to Epistemology. London: Routledge 2010, 609–620.
* [http://plato.stanford.edu/entries/logic-inductive/ ''Stanford Encyclopedia of Philosophy'': "Inductive Logic"]
*[https://web.archive.org/web/20150905093734/http://faculty-staff.ou.edu/H/James.A.Hawthorne-1/Hawthorne--Bayesian_Confirmation_Theory.pdf Bayesian Confirmation Theory] (PDF)
* [http://www.faqs.org/faqs/ai-faq/neural-nets/part3/section-7.html What is Bayesian Learning?]
* [https://causascientia.org/math_stat/DataUnkInf.html ''Data, Uncertainty and Inference''] — Informal introduction with many examples, ebook (PDF) freely available at [https://causascientia.org causaScientia]

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