11
$\begingroup$

I have been reading about noninformative priors. Two of the most popular priors of this kind seem to be the Jeffreys prior and the reference prior. The Jeffreys prior has a clear construction, being the square root of the determinant of the Fisher information matrix. However, the construction of the reference prior doesn't seem intuitive at all to me. Is there an intuitive way of calculating the reference prior or does one have to engulf the whole paper THE FORMAL DEFINITION OF REFERENCE PRIORS in order to understand it?

$\endgroup$

1 Answer 1

1
$\begingroup$

The reference prior improves upon Jeffrey's prior technique for finding a multiparameter prior by decomposing the problem into a series of conditional lower dimensional problems, for which reasonable noninformative priors can be computed. The goal is to obtain a noninformative prior. It requires further proof to show that the reference prior for a specific setting results in a proper posterior distribution (https://www.jstor.org/stable/3085905).

I will give you an example in terms of a linear regression model with a patterned variance-covariance matrix and normal errors. Let $\boldsymbol{\theta} = (\boldsymbol{\beta}, \boldsymbol{\phi})$, where $\boldsymbol{\beta}$ represents the parameters in the mean function and $\boldsymbol{\phi}$ represents the parameters in the variance function. Given $\boldsymbol{\phi}$, a noninformative prior for $\boldsymbol{\beta}$ would be proportional to 1 (i.e. uniform over the real line). Thus we can decompose the prior and construct the reference prior, $\pi^R\left(\boldsymbol{\theta}\right)=\pi^R\left(\boldsymbol{\beta}|\boldsymbol{\phi}\right)\pi^R\left(\boldsymbol{\phi}\right),$ where $\pi^R\left(\boldsymbol{\beta}|\boldsymbol{\phi}\right)=1$. Next, $\pi^R\left(\boldsymbol{\phi}\right)$ is computed using the Jefferys-rule prior, but for the marginal model defined via the integrated likelihood \begin{eqnarray*} L^1 \left(\boldsymbol{\phi}\right) = \int_{\mathbb{R}^p} L(\boldsymbol{\theta}) \pi^R\left(\boldsymbol{\beta}|\boldsymbol{\phi}\right) d \boldsymbol{\beta}. \end{eqnarray*} Note, that a closed-form solution for $L^1 \left(\boldsymbol{\phi}\right)$ exists for this model so that obtaining the reference prior is not difficult. The difficulty is in proving that the posterior density will always be proper and hence you have an "automatic" noninformative prior at your disposal to perform Bayesian analysis.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.