Threshold-like behaviour with pc priors in SPDE model ?

[Francois R]

Hello INLA people

I have been using INLA for spatial models for a little while and I often have had the impression that my choices of pc priors had very little impacts on the estimation of the hyperparameters of the spatial field when using SPDE model. Often, I have to use extreme priors to significantly change the estimation of hyperparameters. This could be due to the amount of data overwhelming the priors, but in most cases I have few data and the estimation of parameters such as the range or the sd of the spatial field should at least be moderatly influenced by the priors and associated with relatively large uncertainties.

I'm currently trying to build spatiotemporal models for zero-inflated seasonal count data using a negative binomial distribution and an SPDE model with an AR1 weekly component. In some cases, I'm suspecting confounding between the weekly spatial fields and a temporal fixed effect used to capture the seasonal trend within seasons (julian date with a quadratic effect). The weekly spatial field sometimes takes up what should be attributed to the julian date. One of the possible solution to prevent that could be to use stronger priors on the spatial field to limit its flexibility. I thus tried using smaller sd for the field while keeping the range constant. This didn't seem to change the situation much unless I used very small sd or fixed it to some small value. 

To better understand what was going on, I studied the relation between the posterior mean sigma and the prior for sigma and this is what I found (see image below). All other priors were kept at the same values. The range was set to prior.range = c(5, 0.01) (a 1% chance the range if inferior to 5km) and sigma was set to prior.sigma  = c(sigma, 0.01) ( a 1% chance sd is superior to sigma). 

There is a sharp change in the posterior mean sigma at around sigma = 0.04 and on each side of this change, the posterior mean sigma are quite similar. Importantly, for any reasonable sigma value I would use, the posterior mean sigma is roughly the same (~ 1.3). In this case, I expect that my fixed effects should explain only a small fraction of the counts and that the variance of the spatial field should be quite large. Hence, anything below c(1, 0.01) would appear highly unreasonable (even on the scale of the linear predictor) and thus all reasonable sigma seem to lead to almost the same posterior sigma.

Is this expected behaviour for a pc prior on the sd of a spatial field? Are pc prior more prone to threshold effects compared to standard priors? Perhaps the impact of pc priors is more related to what the data says compared to what the modeler thinks? Or could this indicate some problem with the model?

Thanks in advance for the input

François Rousseu