This is the case when conjugate priors are used.

They arise particularly in the use of conjugate priors.

Finally, the effect of the conjugate priors on distances and flatness is considered.

Conjugate priors are often very flexible and can be very convenient.

All members of the exponential family have conjugate priors.

Exponential families have conjugate priors, an important property in Bayesian statistics.

Following are some examples of conjugate priors.

In parameterized form, the prior distribution is often assumed to come from a family of distributions called conjugate priors.

Conjugate priors.

Further, conjugate priors may give intuition, by more transparently showing how a likelihood function updates a prior distribution.

The above formula reveals why it is more convenient to do Bayesian analysis of conjugate priors for the normal distribution in terms of the precision.

Conjugate priors are especially useful for sequential estimation, where the posterior of the current measurement is used as the prior in the next measurement.

When a family of conjugate priors exists, choosing a prior from that family simplifies calculation of the posterior distribution.

This partition induces a variational update or step for each marginal density-that is usually solved analytically using conjugate priors.

One of which is that all members have conjugate prior distributions - whereas very few other distributions have conjugate priors.

Again, these conjugate priors are not all NEF-QVF.

Subjective probability, prior and posterior distributions, conjugate priors, treatment of nuisance parameters, stable estimation and noninformative priors.

It is a typical characteristic of conjugate priors that the dimensionality of the hyperparameters is one greater than that of the parameters of the original distribution.

Hyperpriors, like conjugate priors, are a computational convenience - they do not change the process of Bayesian inference, but simply allow one to more easily describe and compute with the prior.

Using a single conjugate prior may be too restrictive, but using a mixture of conjugate priors may give one the desired distribution in a form that is easy to compute with.

In both eigenfunctions and conjugate priors, there is a finite-dimensional space which is preserved by the operator: the output is of the same form (in the same space) as the input.

Laplace also introduced primitive versions of conjugate priors and the theorem of von Mises and Bernstein, according to which the posteriors corresponding to initially differing priors ultimately agree, as the number of observations increases.

In such a case, the weights are typically viewed as a K-dimensional random vector drawn from a Dirichlet distribution (the conjugate prior of the categorical distribution), and the parameters will be distributed according to their respective conjugate priors.

Conjugate priors are analogous to eigenfunctions in operator theory, in that they are distributions on which the "conditioning operator" acts in a well-understood way, thinking of the process of changing from the prior to the posterior as an operator.

If the likelihood and its prior take on simple parametric forms (such as 1- or 2-dimensional likelihood functions with simple conjugate priors), then the empirical Bayes problem is only to estimate the marginal and the hyperparameters using the complete set of empirical measurements.