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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]
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 ().
Given a vector of parameters to determine, a prior probability () over those parameters and a likelihood (,) for making observation , given parameter values and an experiment design , the posterior probability can be calculated using Bayes' theorem
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 ...
This shows that the posterior predictive distribution of a series of observations, in the case where the observations follow an exponential family with the appropriate conjugate prior, has the same probability density as the compound distribution, with parameters as specified above.
The expert knowledge is represented by some (subjective) prior probability distribution. These data are incorporated in a likelihood function. The product of the prior and the likelihood, when normalized, results in a posterior probability distribution that incorporates all the information known to date. [8]
Given data and parameter , a simple Bayesian analysis starts with a prior probability (prior) () and likelihood to compute a posterior probability () (). Often the prior on θ {\displaystyle \theta } depends in turn on other parameters φ {\displaystyle \varphi } that are not mentioned in the likelihood.