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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 contrast, Bayesian posterior expectations are invariant under reparameterization. As an example of the difference between Bayes estimators mentioned above (mean and median estimators) and using a MAP estimate, consider the case where there is a need to classify inputs x {\displaystyle x} as either positive or negative (for example, loans as ...
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 ...
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 distribution. Conjugate priors are especially useful for sequential estimation, where the posterior of the current measurement is used as the prior in the next measurement.
Step 5: The posterior distribution is approximated with the accepted parameter points. The posterior distribution should have a non-negligible probability for parameter values in a region around the true value of in the system if the data are sufficiently informative. In this example, the posterior probability mass is evenly split between the ...
The nonlinear equation describing observation of position of a free particle, which is a special case of the posterior Belavkin equation of the diffusion type, was also obtained by Diosi [6] and appeared in the works of Gisin, [7] Ghirardi, Pearle and Rimini, [8] although with a rather different motivation or interpretation.