In this section we introduce the idea of a conjugate prior. The basic idea is as follows. Given a likelihood p(x|θ), we choose a family of prior distributions ...

27 февр. 2016 г. · The conjugate prior is a gamma distribution on θ>0, this is given as example on p46 og Gelman et.al.: "Bayesian Data Analysis" (Third edition). self study - What is the posterior distribution of θ? Is the Gamma ... Exponential family and conjugate priors - Cross Validated Другие результаты с сайта stats.stackexchange.com

The prior distribution π(θ)∝C(θ)b0exp{η(θ)⊤a0} π ( θ ) ∝ C ( θ ) b 0 exp ⁡ { η ( θ ) ⊤ a 0 } is conjugate to the exponential family (equation (4.4)). π(θ|y ...

Under general conditions, any exponential family has a conjugate prior, with p.d.f. pn0,t0 (θ) ∝ exp n0t0ϕ(θ) − n0κ(θ) 1(θ ∈ Θ) for the values of n0 > 0 and ...

A conjugate prior is an algebraic convenience, giving a closed-form expression for the posterior; otherwise, numerical integration may be necessary.

Definition: Given a prior distribution and a likelihood function, if the corresponding posterior distribution has the same functional form as the prior.

If f(x|θ) is an exponential family, with density as in Definition 3, then a conjugate prior distribution for θ exists. Theorem 9. The prior distribution p(θ) ∝ ...

4 нояб. 2024 г. · This section explains the theory of conjugate priors for exponential families of distributions, which is due to Diaconis and Ylvisaker (1979).

We characterize conjugate prior measures on Θ through the property of linear posterior expectation of the mean parameter of X:E{E(X|θ)|X=x}=ax+b X : E { E ( X ...

If the prior and the posterior belong to the same parametric family, then the prior is said to be conjugate for the likelihood.