The modified power prior


knitr::opts_chunk$set(echo=TRUE, fig.width=8, fig.height=2)

I want to write a paper about the hedging hyperprior. Here is a brief description of an approach it is comparable to, the modified power prior.

Informative prior distributions

Justify the use of informative priors.

The power prior

The power prior was developed to combat the excessive influence of an informative prior when the data conflicts with that prior.

Instead of the standard Bayesian approach where you multiply the Likelihood (L) by the prior (g), you use

\(h(\theta|\mathbf{X}) \propto L(\mathbf{X}|\theta)g(\theta)^\tau g_0(theta)\)

where \(g_0\) is a non-informative prior and \(\tau\) is between 0 and 1.

For \(\tau\)=1, the prior is used at full strength. When \(\tau\)=0, the informative prior effectively disappears, leaving you with just the non-informative prior. For values of \(\tau\) between 0 and 1, the informative prior is weakened, but still exerts some pull on the posterior distribution.

For intermediate values of \(\tau\), the formulation is not consistent with Bayes rule and you need to include a “fudge factor” of

\(\int g(\theta)^{\tau}g_0(\theta)d\theta\)

A simpler approach is to include \(\tau\) in the prior distribution and then place a hyperprior on \(\tau\).

The parameter \(\alpha\) in an inverse gamma distribution can be thought of as a prior sample size. Large values of \(\alpha\) produce informative priors that are equivalent to adding \(\alpha\) pseudo-data values. If you multiply \(\alpha\) by \(\tau\), you can reduce the amount of pseudo data being added.

The hedging hyperprior