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In Bayesian statistics, the posterior probability is the probability of the parameters given the evidence , and is denoted (|). It contrasts with the likelihood function , which is the probability of the evidence given the parameters: p ( X | θ ) {\displaystyle p(X|\theta )} .
An informative prior expresses specific, definite information about a variable. An example is a prior distribution for the temperature at noon tomorrow. A reasonable approach is to make the prior a normal distribution with expected value equal to today's noontime temperature, with variance equal to the day-to-day variance of atmospheric temperature, or a distribution of the temperature for ...
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. [11]
Update the point with least likelihood with some Markov chain Monte Carlo steps according to the prior, accepting only steps that keep the likelihood above . end return Z {\displaystyle Z} ; At each iteration, X i {\displaystyle X_{i}} is an estimate of the amount of prior mass covered by the hypervolume in parameter space of all points with ...
The posterior probability of a model depends on the evidence, or marginal likelihood, which reflects the probability that the data is generated by the model, and on the prior belief of the model. When two competing models are a priori considered to be equiprobable, the ratio of their posterior probabilities corresponds to the Bayes factor .
Given a vector of parameters to determine, a prior probability () over those parameters and a likelihood (,) for making observation , given parameter values and an experiment design , the posterior probability can be calculated using Bayes' theorem
where (|) denotes the posterior, (|) the likelihood, () the prior, and () the evidence (also referred to as the marginal likelihood or the prior predictive probability of the data). Note that the denominator p ( D ) {\displaystyle p(D)} is normalizing the total probability of the posterior density p ( θ | D ) {\displaystyle p(\theta |D)} to ...
The likelihood ratio is central to likelihoodist statistics: the law of likelihood states that the degree to which data (considered as evidence) supports one parameter value versus another is measured by the likelihood ratio.