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The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. [1]
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 .
In a Bayesian setting, this comes up in various contexts: computing the prior or posterior predictive distribution of multiple new observations, and computing the marginal likelihood of observed data (the denominator in Bayes' law). When the distribution of the samples is from the exponential family and the prior distribution is conjugate, the ...
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]
Bayesian-specific workflow stratifies this approach to include three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function ; and (b)–(iii) making a posterior inference. The resulting posterior inference can be used ...
Step 5: The posterior distribution is approximated with the accepted parameter points. The posterior distribution should have a non-negligible probability for parameter values in a region around the true value of in the system if the data are sufficiently informative. In this example, the posterior probability mass is evenly split between the ...
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 ...
The likelihood ratio is a statistic derived by comparing the odds that the evidence (expert testimony of a match) would be found if the defendant was the source with the odds that it would be found if defendant was not the source. If it is ten times more likely that the testimony of a match would occur if the defendant was the source than if ...