Reparametrization from x to eta

[Geir Storvik]

Most  presentations of the inla procedure state that the likelihood introduce extra terms on the diagonal. However, when data depend on fixed effects and/or several random effects, there will also be some interaction effects included. It is stated in Rue and Held (2005, page 174) that this is dealt with through a re-parametrization. Is this how it still is implemented in INLA? 

[Håvard Rue]

There was some technical issues when replying to this msg, so I retry
and amend the response with some more details.

In the classic formulation, then the introduction of the
linear.predictor into the latent field, will do this
'reparametersisation', and all interactions is on that level, while the
likelihood for each observation only depend on one linear.predictor.

In the new formulation, this is different which is described in the
paper


@Article{art703,
author = {J. {van Niekerk} and E. Krainksi and D. Rustand and H. Rue},
title = {A new avenue for Bayesian inference with {INLA}},
journal = CSDA,





On Fri, 2023-03-03 at 11:39 +0300, Helpdesk (Haavard Rue) wrote:
> with new default procedure, inla.mode="compact" (or earlier
> "experimental"), then this is different; see attached

> > --
> > You received this message because you are subscribed to the Google
> > Groups "R-inla discussion group" group.
> > To unsubscribe from this group and stop receiving emails from it,
> > send
> > an email to r-inla-discussion...@googlegroups.com.
> > To view this discussion on the web, visit
> > https://groups.google.com/d/msgid/r-inla-discussion-group/245a019b-6ac2-4426-a3eb-61dadee3542bn%40googlegroups.com
> > .
>

--
Håvard Rue
hr...@r-inla.org

[FredrikLAa]

Hi,

In the classical way of estimating models with INLA, in the Gaussian approx of pi(Latent field|hyperpar, data) how is the log-likelihood approximated? Is it like described on the cited page by Geir, i.e a "quadratic taylor" expansion which is evaluated at the mode of the latent field. 

Here latent field = linear predictor with noise, fixed effects +++

Best
Fredrik