View source for Bayesian statistics

{{Short description|Theory and paradigm of statistics}}
{{Bayesian statistics}}

'''Bayesian statistics''' ({{IPAc-en|ˈ|b|eɪ|z|i|ə|n}} {{respell|BAY|zee|ən}} or {{IPAc-en|ˈ|b|eɪ|ʒ|ən}} {{respell|BAY|zhən}}){{refn|{{MerriamWebsterDictionary|=2023-08-12|Bayesian}}}} is a theory in the field of [[statistics]] based on the [[Bayesian probability|Bayesian interpretation of probability]], where [[probability]] expresses a ''degree of belief'' in an [[Event (probability theory)|event]]. The degree of belief may be based on prior knowledge about the event, such as the results of previous experiments, or on personal beliefs about the event. This differs from a number of other [[Probability interpretations|interpretations of probability]], such as the [[Frequentist probability|frequentist]] interpretation, which views probability as the [[Limit of a sequence|limit]] of the relative frequency of an event after many trials.<ref name="bda">{{Cite book|last1=Gelman|first1=Andrew|title=Bayesian Data Analysis |edition=Third|last2=Carlin|first2=John B.|last3=Stern|first3=Hal S.|last4=Dunson|first4=David B.|last5=Vehtari|first5=Aki|last6=Rubin|first6=Donald B.|publisher=Chapman and Hall/CRC|year=2013|isbn=978-1-4398-4095-5|author-link1=Andrew Gelman|author-link2=John Carlin (professor)|author-link6=Donald Rubin}}</ref> More concretely, analysis in Bayesian methods codifies prior knowledge in the form of a [[prior distribution]].

Bayesian statistical methods use [[Bayes' theorem]] to compute and update probabilities after obtaining new data. Bayes' theorem describes the [[conditional probability]] of an event based on data as well as prior information or beliefs about the event or conditions related to the event.<ref name="rethinking">{{Cite book| title = Statistical Rethinking : A Bayesian Course with Examples in R and Stan |edition=2nd | publisher = Chapman and Hall/CRC | year = 2020 | isbn = 978-0-367-13991-9 |last1=McElreath|first1=Richard|author-link1=Richard McElreath}}</ref><ref>{{cite book |first=John |last=Kruschke |author-link=John K. Kruschke |title=Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan |location= |publisher=Academic Press |edition=2nd |year=2014 |isbn=978-0-12-405888-0 }}</ref> For example, in [[Bayesian inference]], Bayes' theorem can be used to estimate the parameters of a [[probability distribution]] or [[statistical model]]. Since Bayesian statistics treats probability as a degree of belief, Bayes' theorem can directly assign a probability distribution that quantifies the belief to the parameter or set of parameters.<ref name="bda" /><ref name="rethinking" />

Bayesian statistics is named after [[Thomas Bayes]], who formulated a specific case of Bayes' theorem in [[An Essay Towards Solving a Problem in the Doctrine of Chances|a paper]] published in 1763. In several papers spanning from the late 18th to the early 19th centuries, [[Pierre-Simon Laplace]] developed the Bayesian interpretation of probability.<ref>{{Cite book| title = The Theory That Would Not Die: How Bayes' Rule Cracked the Enigma Code, Hunted Down Russian Submarines, and Emerged Triumphant from Two Centuries of Controversy |edition=First | publisher = Chapman and Hall/CRC | year = 2012 | isbn = 978-0-3001-8822-6|last1=McGrayne|first1=Sharon|author-link1=Richard McElreath}}</ref> Laplace used methods now considered Bayesian to solve a number of statistical problems. While many Bayesian methods were developed by later authors, the term "Bayesian" was not commonly used to describe these methods until the 1950s. Throughout much of the 20th century, Bayesian methods were viewed unfavorably by many statisticians due to philosophical and practical considerations. Many of these methods required much computation, and most widely used approaches during that time were based on the frequentist interpretation. However, with the advent of powerful computers and new [[algorithm]]s like [[Markov chain Monte Carlo]], Bayesian methods have gained increasing prominence in statistics in the 21st century.<ref name="bda" /><ref>{{cite journal |last1=Fienberg |first1=Stephen E. |title=When Did Bayesian Inference Become "Bayesian"? |date=2006|journal=Bayesian Analysis|volume=1|issue=1|pages=1–40 |doi=10.1214/06-BA101 |doi-access=free }}</ref>

== Bayes's theorem ==
{{Main|Bayes's theorem}}
Bayes's theorem is used in Bayesian methods to update probabilities, which are degrees of belief, after obtaining new data. Given two events <math>A</math> and <math>B</math>, the conditional probability of <math>A</math> given that <math>B</math> is true is expressed as follows:<ref name="grinsteadsnell2006">{{cite book |last1=Grinstead |first1=Charles M. |last2=Snell |first2=J. Laurie |title=Introduction to probability |date=2006 |publisher=American Mathematical Society |location=Providence, RI |isbn=978-0-8218-9414-9 |edition=2nd}}</ref>

<math display="block">P(A \mid B) = \frac{P(B \mid A)P(A)}{P(B)}</math>

where <math>P(B) \neq 0</math>. Although Bayes's theorem is a fundamental result of [[probability theory]], it has a specific interpretation in Bayesian statistics. In the above equation, <math>A</math> usually represents a [[proposition]] (such as the statement that a coin lands on heads fifty percent of the time) and <math>B</math> represents the evidence, or new data that is to be taken into account (such as the result of a series of coin flips). <math>P(A)</math> is the [[prior probability]] of <math>A</math> which expresses one's beliefs about <math>A</math> before evidence is taken into account. The prior probability may also quantify prior knowledge or information about <math>A</math>. <math>P(B \mid A)</math> is the [[likelihood function]], which can be interpreted as the probability of the evidence <math>B</math> given that <math>A</math> is true. The likelihood quantifies the extent to which the evidence <math>B</math> supports the proposition <math>A</math>. <math>P(A \mid B)</math> is the [[posterior probability]], the probability of the proposition <math>A</math> after taking the evidence <math>B</math> into account. Essentially, Bayes's theorem updates one's prior beliefs <math>P(A)</math> after considering the new evidence <math>B</math>.<ref name="bda" />

The probability of the evidence <math>P(B)</math> can be calculated using the [[law of total probability]]. If <math>\{A_1, A_2, \dots, A_n\}</math> is a [[Partition of a set|partition]] of the [[sample space]], which is the set of all [[Outcome (probability)|outcomes]] of an experiment, then,<ref name="bda" /><ref name="grinsteadsnell2006" />

<math display="block">P(B) = P(B \mid A_1)P(A_1) + P(B \mid A_2)P(A_2) + \dots + P(B \mid A_n)P(A_n) = \sum_i P(B \mid A_i)P(A_i)</math>

When there are an infinite number of outcomes, it is necessary to [[Integral|integrate]] over all outcomes to calculate <math>P(B)</math> using the law of total probability. Often, <math>P(B)</math> is difficult to calculate as the calculation would involve sums or integrals that would be time-consuming to evaluate, so often only the product of the prior and likelihood is considered, since the evidence does not change in the same analysis. The posterior is proportional to this product:<ref name="bda" />

<math display="block">P(A \mid B) \propto P(B \mid A)P(A)</math>

The [[maximum a posteriori]], which is the [[Mode (statistics)|mode]] of the posterior and is often computed in Bayesian statistics using [[mathematical optimization]] methods, remains the same. The posterior can be approximated even without computing the exact value of <math>P(B)</math> with methods such as [[Markov chain Monte Carlo]] or [[variational Bayesian methods]].<ref name="bda" />

==Bayesian methods==

The general set of statistical techniques can be divided into a number of activities, many of which have special Bayesian versions.

=== Bayesian inference ===
{{Main|Bayesian inference}}
Bayesian inference refers to [[statistical inference]] where uncertainty in inferences is quantified using probability.<ref>{{Cite journal |last=Lee|first=Se Yoon|  title = Gibbs sampler and coordinate ascent variational inference: A set-theoretical review|journal=Communications in Statistics - Theory and Methods|year=2021|volume=51 |issue=6 |pages=1549–1568 |doi=10.1080/03610926.2021.1921214|arxiv=2008.01006|s2cid=220935477 }}</ref> In classical [[frequentist inference]], model [[parameter]]s and hypotheses are considered to be fixed. Probabilities are not assigned to parameters or hypotheses in frequentist inference. For example, it would not make sense in frequentist inference to directly assign a probability to an event that can only happen once, such as the result of the next flip of a fair coin. However, it would make sense to state that the proportion of heads [[Law of large numbers|approaches one-half]] as the number of coin flips increases.<ref name="wakefield2013">{{cite book |last1=Wakefield |first1=Jon |title=Bayesian and frequentist regression methods |date=2013 |publisher=Springer |location=New York, NY |isbn=978-1-4419-0924-4}}</ref>

[[Statistical models]] specify a set of statistical assumptions and processes that represent how the sample data are generated. Statistical models have a number of parameters that can be modified. For example, a coin can be represented as samples from a [[Bernoulli distribution]], which models two possible outcomes. The Bernoulli distribution has a single parameter equal to the probability of one outcome, which in most cases is the probability of landing on heads. Devising a good model for the data is central in Bayesian inference. In most cases, models only approximate the true process, and may not take into account certain factors influencing the data.<ref name="bda" /> In Bayesian inference, probabilities can be assigned to model parameters. Parameters can be represented as [[random variable]]s. Bayesian inference uses Bayes' theorem to update probabilities after more evidence is obtained or known.<ref name="bda" /><ref name="congdon2014">{{cite book |last1=Congdon |first1=Peter |title=Applied Bayesian modelling |date=2014 |publisher=Wiley |isbn=978-1119951513 |edition=2nd}}</ref> Furthermore, Bayesian methods allow for placing priors on entire models and calculating their posterior probabilities using Bayes' theorem. These posterior probabilities 
are proportional to the product of the prior and the marginal likelihood, 
where the marginal likelihood is the integral of the sampling density over the prior distribution 
of the parameters. In complex models, marginal likelihoods are 
generally computed numerically.<ref name="chib1995">{{cite journal 
|last=Chib 
|first=Siddhartha 
|title=Marginal Likelihood from the Gibbs Output 
|journal=Journal of the American Statistical Association 
|year=1995 
|volume=90 
|issue=432 
|pages=1313-1321 
|doi=10.1080/01621459.1995.10476635}}</ref>

===Statistical modeling===
The formulation of [[statistical model]]s using Bayesian statistics has the identifying feature of requiring the specification of [[prior distribution]]s for any unknown parameters. Indeed, parameters of prior distributions may themselves have prior distributions, leading to [[Bayesian hierarchical modeling]],<ref name="KruschkeVanpaemel2015">{{cite book |last1=Kruschke|first1=J K|author-link1=John K. Kruschke |last2=Vanpaemel |first2=W |chapter=Bayesian Estimation in Hierarchical Models |pages=279–299 |title=The Oxford Handbook of Computational and Mathematical Psychology |editor-last1=Busemeyer |editor-first1=J R |editor-last2=Wang |editor-first2=Z |editor-last3=Townsend |editor-first3=J T |editor-last4=Eidels |editor-first4=A |year=2015 |publisher=Oxford University Press |url=https://jkkweb.sitehost.iu.edu/articles/KruschkeVanpaemel2015.pdf}}</ref><ref name=":bmdl">Hajiramezanali, E. & Dadaneh, S. Z. & Karbalayghareh, A. & Zhou, Z. & Qian, X. Bayesian multi-domain learning for cancer subtype discovery from next-generation sequencing count data. 32nd Conference on Neural Information Processing Systems (NIPS 2018), Montréal, Canada. {{ArXiv|1810.09433}}</ref><ref>{{Cite journal |last1=Lee|first1=Se Yoon |first2=Bani|last2=Mallick|  title = Bayesian Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas|journal=Sankhya B|year=2021|volume=84 |pages=1–43 |doi=10.1007/s13571-020-00245-8|doi-access=}}</ref> also known as multi-level modeling. A special case is [[Bayesian networks]].

For conducting a Bayesian statistical analysis, best practices are discussed by van de Schoot et al.<ref name="vandeShootEtAl2021">{{cite journal|last1=van de Schoot |first1=Rens|last2=Depaoli |first2=Sarah |last3=King |first3=Ruth |last4=Kramer |first4=Bianca |last5=Märtens |first5=Kaspar |last6=Tadesse |first6=Mahlet G. |last7=Vannucci |first7=Marina |last8=Gelman |first8=Andrew |last9=Veen |first9=Duco |last10=Willemsen |first10=Joukje |last11=Yau |first11=Christopher |title=Bayesian statistics and modelling|journal=Nature Reviews Methods Primers|date=January 14, 2021|volume=1|number=1|pages=1–26|doi=10.1038/s43586-020-00001-2|hdl=1874/415909 |s2cid=234108684 |url=https://osf.io/wdtmc/|hdl-access=free }}</ref> 

For reporting the results of a Bayesian statistical analysis, Bayesian analysis reporting guidelines (BARG) are provided in an open-access article by [[John K. Kruschke]].<ref name="Kruschke2021BARG">{{cite journal|last=Kruschke|first=J K|author-link=John K. Kruschke|title=Bayesian Analysis Reporting Guidelines|journal=Nature Human Behaviour|date=Aug 16, 2021|volume=5|issue=10 |pages=1282–1291|doi=10.1038/s41562-021-01177-7|pmid=34400814 |pmc=8526359 }}</ref>

===Design of experiments===
The [[Bayesian design of experiments]] includes a concept called 'influence of prior beliefs'. This approach uses [[sequential analysis]] techniques to include the outcome of earlier experiments in the design of the next experiment. This is achieved by updating 'beliefs' through the use of prior and [[posterior distribution]]. This allows the design of experiments to make good use of resources of all types. An example of this is the [[multi-armed bandit problem]].

===Exploratory analysis of Bayesian models===

Exploratory analysis of Bayesian models is an adaptation or extension of the [[exploratory data analysis]] approach to the needs and peculiarities of Bayesian modeling. In the words of Persi Diaconis:<ref>Diaconis, Persi (2011) Theories of Data Analysis: From Magical Thinking Through Classical Statistics. John Wiley & Sons, Ltd 2:e55 {{doi|10.1002/9781118150702.ch1}}</ref> 
{{Quote|Exploratory data analysis seeks to reveal structure, or simple descriptions in data. We look at numbers or graphs and try to find patterns. We pursue leads suggested by background information, imagination, patterns perceived, and experience with other data analyses}}

The [[Bayesian inference|inference process]] generates a posterior distribution, which has a central role in Bayesian statistics, together with other distributions like the posterior predictive distribution and the prior predictive distribution. The correct visualization, analysis, and interpretation of these distributions is key to properly answer the questions that motivate the inference process.<ref>{{cite journal |doi=10.21105/joss.01143 |title=ArviZ a unified library for exploratory analysis of Bayesian models in Python |year=2019 |last1=Kumar |first1=Ravin |last2=Carroll |first2=Colin |last3=Hartikainen |first3=Ari |last4=Martin |first4=Osvaldo |journal=Journal of Open Source Software |volume=4 |issue=33 |page=1143 |bibcode=2019JOSS....4.1143K |doi-access=free |hdl=11336/114615 |hdl-access=free }}</ref>

When working with Bayesian models there are a series of related tasks that need to be addressed besides inference itself:

* Diagnoses of the quality of the inference, this is needed when using numerical methods such as [[Markov chain Monte Carlo]] techniques
* Model criticism, including evaluations of both model assumptions and model predictions
* Comparison of models, including model selection or model averaging
* Preparation of the results for a particular audience

All these tasks are part of the Exploratory analysis of Bayesian models approach and successfully performing them is central to the iterative and interactive modeling process. These tasks require both numerical and visual summaries.<ref>{{cite journal |arxiv=1709.01449 |doi=10.1111/rssa.12378 |title=Visualization in Bayesian workflow |year=2019 |last1=Gabry |first1=Jonah |last2=Simpson |first2=Daniel |last3=Vehtari |first3=Aki |last4=Betancourt |first4=Michael |last5=Gelman |first5=Andrew |s2cid=26590874 |journal=Journal of the Royal Statistical Society, Series A (Statistics in Society) |volume=182 |issue=2 |pages=389–402 }}</ref><ref>{{cite journal |arxiv=1903.08008 |last1=Vehtari |first1=Aki |last2=Gelman |first2=Andrew |last3=Simpson |first3=Daniel |last4=Carpenter |first4=Bob |last5=Bürkner |first5=Paul-Christian |title=Rank-Normalization, Folding, and Localization: An Improved Rˆ for Assessing Convergence of MCMC (With Discussion) |journal=Bayesian Analysis |year=2021 |volume=16 |issue=2 |page=667 |doi=10.1214/20-BA1221 |bibcode=2021BayAn..16..667V |s2cid=88522683 }}</ref><ref name="Martin2018">{{cite book|url=https://books.google.com/books?id=1Z2BDwAAQBAJ|title=Bayesian Analysis with Python: Introduction to statistical modeling and probabilistic programming using PyMC3 and ArviZ|last1=Martin|first1=Osvaldo|date=2018|publisher=Packt Publishing Ltd|isbn=9781789341652|language=en}}</ref>

==See also==
* [[Bayesian epistemology]]
* For a list of mathematical logic notation used in this article
** [[Notation in probability and statistics]]
** [[List of logic symbols]]

==References==
{{Reflist}}

==Further reading==
*{{cite book |first1=José M. |last1=Bernardo |authorlink=José-Miguel Bernardo |first2=Adrian F. M. |last2=Smith |authorlink2=Adrian Smith (statistician) |title=Bayesian Theory |location=New York |publisher=Wiley |year=2000 |isbn=0-471-92416-4 }}
*{{cite book |first1=William M. |last1=Bolstad |first2=James M. |last2=Curran |title=Introduction to Bayesian Statistics |location= |publisher=Wiley |edition=3rd |year=2016 |isbn=978-1-118-09156-2 }}
*{{cite book |first=Allen B. |last=Downey |authorlink=Allen B. Downey |title=Think Bayes: Bayesian Statistics in Python |location= |publisher=O'Reilly |edition=2nd |year=2021 |isbn=978-1-4920-8946-9 |url=http://greenteapress.com/wp/think-bayes/ }}
*{{cite book |first=Peter D. |last=Hoff |title=A First Course in Bayesian Statistical Methods |location=New York |publisher=Springer |edition=2nd |year=2009 |isbn=978-1-4419-2828-3 }}
*{{cite book |first=Peter M. |last=Lee |title=Bayesian Statistics: An Introduction |location= |publisher=Wiley |edition=4th |year=2012 |isbn=978-1-118-33257-3 }}
*{{cite book |last=Robert |first=Christian P. |authorlink=Christian Robert |title=The Bayesian Choice : From Decision-Theoretic Foundations to Computational Implementation |location=New York |publisher=Springer |edition=2nd |year=2007 |isbn=978-0-387-71598-8 }}
*''Johnson, Alicia A.; Ott, Miles Q.; Dogucu, Mine. (2022)'' [https://www.bayesrulesbook.com/ Bayes Rules! An Introduction to Applied Bayesian Modeling]. Chapman and Hall <nowiki>ISBN 9780367255398</nowiki>

==External links==
{{Wikiversity|Bayesian statistics}}
* {{cite web| author=Theo Kypraios| title=A Gentle Tutorial in Bayesian Statistics| url=https://kupdf.com/download/a-gentle-tutorial-in-bayesian-statisticspdf_59b0ed86dc0d602e3b568edc_pdf| type=PDF| access-date=2013-11-03}}
* {{cite book| author=Jordi Vallverdu | title = Bayesians Versus Frequentists A Philosophical Debate on Statistical Reasoning | url=https://www.springer.com/gp/book/9783662486368}}
* [http://www.scholarpedia.org/article/Bayesian_statistics Bayesian statistics] [[David Spiegelhalter]], Kenneth Rice [[Scholarpedia]] 4(8):5230. [[doi:10.4249/scholarpedia.5230]]
* [http://bayesmodels.com/ Bayesian modeling book] and examples available for downloading.
* {{cite web| author=Rens van de Schoot | title = A Gentle Introduction to Bayesian Analysis | url=https://www.statmodel.com/download/introBayes.pdf}}
* [https://marketing.dynamicyield.com/bayesian-calculator/ Bayesian A/B Testing Calculator] Dynamic Yield

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[[Category:Bayesian statistics| ]]