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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.
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 ...
The use of an improper prior means that the Bayes risk is undefined (since the prior is not a probability distribution and we cannot take an expectation under it). As a consequence, it is no longer meaningful to speak of a Bayes estimator that minimizes the Bayes risk. Nevertheless, in many cases, one can define the posterior distribution
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 as either positive or negative (for example, loans as risky or safe).
One may also define an interval for which the mean is the central point, assuming that the mean exists. γ {\displaystyle \gamma } -Smallest Credible Regions ( γ {\displaystyle \gamma } -SCR) can easily be generalized to the multivariate case, and are bounded by probability density contour lines . [ 4 ]
Bayesian hierarchical modelling is a statistical model written in multiple levels (hierarchical form) that estimates the parameters of the posterior distribution using the Bayesian method. [1]
In general, with a normally-distributed sample mean, Ẋ, and with a known value for the standard deviation, σ, a 100(1-α)% confidence interval for the true μ is formed by taking Ẋ ± e, with e = z 1-α/2 (σ/n 1/2), where z 1-α/2 is the 100(1-α/2)% cumulative value of the standard normal curve, and n is the number of data values in that ...