Dear Sam and list,
Sam Gershman wrote:
> I have a comment about the use of marginal likelihood (or a variational
> approximation thereof) to select priors, as was used in this paper:
>
> Penny et al (2007). Bayesian comparison of spatially regularised general
> linear models. Human Brain Mapping 28(4):275-293.
>
> In this paper, the authors use the free energy lower bound on the marginal
> likelihood to compare different spatial priors for the AR coefficients.
>
> It's my understanding that the marginal likelihood, while it can be used to
> compare alternative likelihood functions, serves no purpose in comparing
> priors.
That's not my understanding.
From an orthodox Bayesian perspective, the choice of prior should
> not depend on the data likelihood, because all prior knowledge should be
> specified before looking at the data.
I agree.
However, my perspective and that taken by other developers of Bayesian
methods in SPM (and in other communities eg. stats/machine learning) is
an *empirical* Bayesian one, rather than an orthodox Bayesian one.
See eg. http://en.wikipedia.org/wiki/Empirical_Bayes_method
In this framework, one is allowed to estimate parameters of the prior
distributions. The form of the prior is fixed, but some/all of its
parameters may be updated dependent on data.
(see also next email)
Best,
Will.
--
William D. Penny
Wellcome Trust Centre for Neuroimaging
University College London
12 Queen Square
London WC1N 3BG
Tel: 020 7833 7475
FAX: 020 7813 1420
Email: [log in to unmask]
URL: http://www.fil.ion.ucl.ac.uk/~wpenny/
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