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In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest ...
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.
where g(r) represents the prior probability density distribution of r, which lies in the range 0 to 1. The prior probability density distribution summarizes what is known about the distribution of r in the absence of any observation. We will assume that the prior distribution of r is uniform over the interval [0, 1]. That is, g(r) = 1. (In ...
The following loss function is trickier: it yields either the posterior mode, or a point close to it depending on the curvature and properties of the posterior distribution. Small values of the parameter K > 0 {\displaystyle K>0} are recommended, in order to use the mode as an approximation ( L > 0 {\displaystyle L>0} ):
The inference process generates a posterior distribution, which has a central role in Bayesian statistics, together with other distributions like the posterior predictive distribution and the prior predictive distribution. The correct visualization, analysis, and interpretation of these distributions is key to properly answer the questions that ...
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 probability distribution function (and thus likelihood function) for exponential families contain products of factors involving exponentiation. The logarithm of such a function is a sum of products, again easier to differentiate than the original function.
This particular distribution is known as the flat Dirichlet distribution. Values of the concentration parameter above 1 prefer variates that are dense, evenly distributed distributions, i.e. all the values within a single sample are similar to each other. Values of the concentration parameter below 1 prefer sparse distributions, i.e. most of ...