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Earlier this month, Daniel Sabanés Bové and Leo Held posted a paper about g-priors on arXiv. While I glanced at it for a few minutes, I did not have the chance to get a proper look at it till last Sunday. The g-prior was first introduced by the late Arnold Zellner for (standard) linear models, but they can be extended to generalised linear models (formalised by the late John Nelder) at little cost. In Bayesian Core, Jean-Michel Marin and I do centre the prior modelling in both linear and generalised linear models around g-priors, using the naïve extension for generalised linear models, $beta sim mathcal{N}(0,g sigma^2 (mathbf{X}^text{T}mathbf{X})^{-1})$

as in the linear case. Indeed, the reasonable alternative would be to include the true information matrix but since it depends on the parameter $beta$ outside the normal case this is not truly an alternative. Bové and Held propose a slightly different version $beta sim mathcal{N}(0,g sigma^2 c (mathbf{X}^text{T}mathbf{W}mathbf{X})^{-1})$

where W is a diagonal weight matrix and c is a family dependent scale factor evaluated at the mode 0. As in Liang et al. (2008, JASA) and most of the current literature, they also separate the intercept $beta_0$ from the other regression coefficients. They also burn their “improperness joker” by choosing a flat prior on $beta_0$, which means they need to use a proper prior on g, again as Liang et al. (2008, JASA), for the corresponding Bayesian model comparison to be valid. In Bayesian Core, we do not separate $beta_0$ from the other regression coefficients and hence are left with one degree of freedom that we spend in choosing an improper prior on g instead. (Hence I do not get the remark of Bové and Held that our choice “prohibits Bayes factor comparisons with the null model“. As argued in Bayesian Core, the factor g being an hyperparameter shared by all models, we can use the same improper prior on g in all models and hence use standard Bayes factors.) In order to achieve closed form expressions, the authors use Cui and George ‘s (2008) prior $pi(g) propto (1+g)^{1+a}exp{-b/(1+g)}$

which requires the two hyper-hyper-parameters a and b to be specified.

The second part of the paper considers computational issues. It compares the ILA solution of Rue, Martino and Chopin (2009, Series B) with an MCMC solution based on an independent proposal on g resulting from linear interpolations (?). The marginal likelihoods are approximated by Chib and Jeliazkov (2001, JASA) for the MCMC part. Unsurprisingly, ILA does much better, even with a 97% acceptance rate in the MCMC algorithm.

The paper is very well-written and quite informative about the existing literature. It also uses the Pima Indian dataset  (The authors even dug out a 1991 paper of mine I had completely forgotten!) I am actually thinking of using the review in our revision of Bayesian Core, even though I think we should stick to our choice of including $beta_0$ within the set of parameters…

Filed under: Books, R, Statistics Tagged: Arnold Zellner, Bayesian Core, Bayesian model choice, g-prior, GLMs, ILA, MCMC        