Priors and Reference Analysis

In the fully Bayesian framework the choice of prior is critical to the inference we perform. In group statistics for FMRI the number of observations we have is typically so small as to make the influence of the priors significant. As we have no prior information we want the priors we use to be in some sense ``non-informative'', i.e. we want to ``let the data speak for itself''. Reference priors are priors which attempt to reflect such prior ignorance. For an overview see (17,2).

An intuitive approach would be to choose the prior of $\theta$ to be $\pi(\theta)=1$. However, the resulting posteriors can change significantly depending on the parameterisation used. This is because a constant prior for one parameter will not typically transform into a constant prior for another. To overcome this reparameterisation problem the Jeffreys prior was introduced for one-dimensional problems (17):

$$\pi(\theta)\propto \det(H(\theta))^{1/2}$$ (7)

where $H(\theta)$ is the Fisher information. However, this has difficulties dealing with multi-dimensional problems, i.e. $\vec{\Theta}=(\theta_1\ldots \theta_m)$. The Berger-Bernardo method (2) of reference analysis overcomes this by determining reference priors using information-theoretical ideas which maximise the amount of expected ``information'' from the data. See appendix 10.5 for the derivation of the reference priors used in this paper.

The use of reference priors can be justified by consideration of the information theory that underpins them (2). However, whilst in this paper the null hypothesis frequentist inference is generally unknown, it is interesting to note that the Berger-Bernardo reference priors give the same inference as frequentist null hypothesis testing for cases of GLM inference for which the frequentist null hypothesis test is known. For example, in frequentist inference on the GLM we typically examine the probability of attaining statistics for linear combinations (contrasts) of regression parameters under the null hypothesis. In cases where such null hypothesis frequentist inference for the GLM is known, the Berger-Bernardo reference priors give the same probabilities when we test the probability that a contrast is greater than zero.