SPDE priors and overfitting

[lere...@gmail.com]

Dear inla-group,
When using SPDE models, default priors may result in overparametrized models (high number of effective parameters, low number of equivalent replicates).
I wonder how to define 'better' priors alleviating this undesirable effect. I expect this can be achieved by changing parameters within the inla.spde2.matern() function.
Without apriori knowledge, I expect that range (prior.range.nominal) and variance (prior.variance.nominal) parameters (or alternatively tau and kappa mean parameters) should be let to default ones.
It remains the precision parameter of thetas (theta.prior.prec). Is testing different values for this parameter and looking at posterior results (e.g. number of effective parameters and cpo) a good solution to identify 'better' priors?
If so, Is precision for tau and kappa should be the same for estimate a stationnary field?
Best regards,
Kévin

[Geir-Arne Fuglstad]

Hi Kévin,

Ingebrigtsen, et al. (http://www.sciencedirect.com/science/article/pii/S2211675313000377 and http://arxiv.org/pdf/1412.2798v2.pdf) discusses some of the issues you bring up: mean and variance for Gaussian prior for the stationary part and how to select the variances for the priors on the non-stationarity parameters.

However, recently, there has been a recent drive towards developing new priors based on "Penalised Complexity" (http://arxiv.org/pdf/1403.4630v4.pdf) in the INLA group and this is currently being extended also to spatial and non-stationary spatial models (http://arxiv.org/pdf/1503.00256v1.pdf). The prior for the stationary model can be specified without too much hassle, but the documentation for doing that is currently lacking, and the prior for the non-stationary model is work-in-progress and does not have an easy interface. If you are interested in exploring this avenue, you can contact me.

As for the second part on evaluating the different priors, I think it will depend on the goal of the analysis. If the choice of prior leads to better cpo scores, the predictive distributions are better and one could argue that if predictions are the main goal, the prior is 'better'. But if the goal is to decide whether non-stationarity is present or whether a certain covariate is affecting non-stationarity, I'm not sure if it would be appropriate to base the selection of the prior on the predictive properties of the model. First of all, it is not completely Bayesian and, secondly, using the data twice might lead to biases.

-- 

Geir-Arne

[Finn Lindgren]

Geir-Arne's comments are good.

Further to that, there are some simple things that can be done better than the defaults.

The default prior range is usually ok, but the default prior for the variance can easily be completely inappropriate, as it depends on the scale of the latent component (and not necessarily the scale of the data itself).  The minimum approach that can take that into account is included here:

  http://www.r-inla.org/examples/tutorials/spde-tutorial-from-jss

For example, the prior median "sigma0" in the examples should be set to something more sensible than the default, 1.

In retrospect, I wish I had let the code give an error (or at least a warning) if the user doesn't specify any prior distribution parameters, as that isn't very Bayesian!

When the newer approaches are mature enough to use on all spde models, we may consider not allowing any preset "default" priors at all.

Finn

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