Which is the best description of a conjugate prior?

Which is the best description of a conjugate prior?

A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise numerical integration may be necessary. Further, conjugate priors may give intuition, by more transparently showing how a likelihood function updates a prior distribution.

Which is the conjugate prior for a gamma distribution?

I want conjugate prior for β and the posterior. ( − β ∑ i y i) p ( β). Therefore, the conjugate prior for β would be gamma ( α 0, β 0). ( − β ( ∑ i y i + β 0)). So, the posterior is gamma ( α 0, ∑ i y i + β 0). However, Wikipedia says the posterior should be gamma ( α 0 + n α, ∑ i y i + β 0).

How are hyperparameters used in conjugate prior distributions?

In general, for nearly all conjugate prior distributions, the hyperparameters can be interpreted in terms of pseudo-observations. This can help both in providing an intuition behind the often messy update equations, as well as to help choose reasonable hyperparameters for a prior.

How to define conjugate priors for an exponential family?

For exponential families the likelihood is a simple standarized function of the parameter and we can define conjugate priors by mimicking the form of the likelihood. Multiplication of a likelihood and a prior that have the same exponential form yields a posterior that retains that form.

When does the posterior have the same algebraic form as the prior?

For certain choices of the prior, the posterior has the same algebraic form as the prior (generally with different parameter values). Such a choice is a conjugate prior . A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise numerical integration may be necessary.

When is a posterior distribution called a conjugate distribution?

In Bayesian probability theory, if the posterior distributions p ( θ | x) are in the same probability distribution family as the prior probability distribution p (θ), the prior and posterior are then called conjugate distributions, and the prior is called a conjugate prior for the likelihood function.