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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]
In Bayesian statistics, the posterior predictive distribution is the distribution of possible unobserved values conditional on the observed values. [1] [2]Given a set of N i.i.d. observations = {, …,}, a new value ~ will be drawn from a distribution that depends on a parameter , where is the parameter space.
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 ...
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 ().
Following Bayes' Rule, the likelihood when seen as a conditional density can be multiplied by the prior probability density of the parameter and then normalized, to give a posterior probability density.
The posterior probability of given is high, indicating strong evidence that is in better agreement with than . These results can occur at the same time when H 0 {\displaystyle H_{0}} is very specific, H 1 {\displaystyle H_{1}} more diffuse, and the prior distribution does not strongly favor one or the other, as seen below.
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 ...
A conjugate prior is defined as a prior distribution belonging to some parametric family, for which the resulting posterior distribution also belongs to the same family. This is an important property, since the Bayes estimator, as well as its statistical properties (variance, confidence interval, etc.), can all be derived from the posterior ...