Editing Bayesian statistics (section)

=== 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>