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In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest ...
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 .
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 ...
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 ...
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 ().
Bayesian linear regression is a type of conditional modeling in which the mean of one variable is described by a linear combination of other variables, with the goal of obtaining the posterior probability of the regression coefficients (as well as other parameters describing the distribution of the regressand) and ultimately allowing the out-of-sample prediction of the regressand (often ...
Bayesian statistics; Posterior = Likelihood × Prior ÷ Evidence: ... Figure 1. The likelihood function ... One example occurs in 2×2 tables, ...
for the approximate posterior over and the approximate log marginal likelihood respectively. The main weaknesses of Laplace's approximation are that it is symmetric around the mode and that it is very local: the entire approximation is derived from properties at a single point of the target density.