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The posterior probability distribution of one random variable given the value of another can be calculated with Bayes' theorem by multiplying the prior probability distribution by the likelihood function, and then dividing by the normalizing constant, as follows:
A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the relative proportions of voters who will vote for a particular politician in a future election.
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 Bayesian probability theory, if, given a likelihood function (), the posterior distribution is in the same probability distribution family as the prior probability distribution (), the prior and posterior are then called conjugate distributions with respect to that likelihood function and the prior is called a conjugate prior for the likelihood function ().
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]
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 ...
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 ...