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Posterior probability is a conditional probability conditioned on randomly observed data. Hence it is a random variable. For a random variable, it is important to summarize its amount of uncertainty. One way to achieve this goal is to provide a credible interval of the posterior probability. [11]
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.
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 ().
The posterior probability of a tree will be the probability that the tree is correct, given the prior, the data, and the correctness of the likelihood model. MCMC methods can be described in three steps: first using a stochastic mechanism a new state for the Markov chain is proposed. Secondly, the probability of this new state to be correct is ...
Credible intervals are typically used to characterize posterior probability distributions or predictive probability distributions. [1] Their generalization to disconnected or multivariate sets is called credible region. Credible intervals are a Bayesian analog to confidence intervals in frequentist statistics. [2]
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 ...
An estimation procedure that is often claimed to be part of Bayesian statistics is the maximum a posteriori (MAP) estimate of an unknown quantity, that equals the mode of the posterior density with respect to some reference measure, typically the Lebesgue measure.
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.