Marginal likelihoods for the generic0 model

[Geir Storvik]

As stated in the documentation of the generic0 model, there is a missing term in the calculation of the marginal likelihood, corresponding to the determinant of the structural part of the precision matrix. In cases with singular matrices, but with constraints making the constrained covariance matrix proper, it seems like using the non-zero eigenvalues of the constrained covariance matrix gives the right correction. But what if the constraints still lead to some non-identifiability issues but these are "solved" through the data, that is the posterior covariance matrix is valid. Which correction should be used in that case?

Geir

[Håvard Rue]

On Mon, 2023-03-06 at 01:22 -0800, Geir Storvik wrote:
> As stated in the documentation of the generic0 model, there is a
> missing term in the calculation of the marginal likelihood,
> corresponding to the determinant of the structural part of the
> precision matrix. In cases with singular matrices, but with
> constraints making the constrained covariance matrix proper, it seems
> like using the non-zero eigenvalues of the constrained covariance
> matrix gives the right correction. 

that is correct.

> But what if the constraints still lead to some non-identifiability
> issues but these are "solved" through the data, that is the posterior
> covariance matrix is valid. Which correction should be used in that
> case?

In general, you could do little else than including the non-singular
part of the normalizing constant (ie the determinant) for the prior.

Issues will appear if you compare two models where this singularity is
different or has different dimension. like comparing

        y ~ 1 + x
        
with

        y ~ 1 + x + xx

with constant priors for all fixed effects, then a constant prior for
the intercept and for 'x', will  cancel in comparison, the choice for
the prior for 'xx' will matter.




--
Håvard Rue
hr...@r-inla.org