# CAR model within Poisson Process fails

Hi,

I have a 100 locations in space. I simulated some data from a temporal Poisson process with intensity function lambda_j for each (fixed) location j.
I want to include a term s_j |s_{-j} ~ N(0, (sigmaD(I-alphaB))^-1) which is coming from a CAR prior (see the sparse representation) into the intensity function lambda_j, i.e. lamda_j(t) = tau + exp(s_j) Sum_{j \in N(t)} exp(psi - kappa) with parameters psi, kappa and tau (and sigma and alpha from the GMRF)

The code without the s_j runs through and gives great mixing and estimates of the remaining parameters. Thus, I conclude that my PP implementation is correct. On the other hand, simulating from a GMRF with the above precision matrix and estimating its parameters sigma and alpha again is working great as well.
Hence, both parts separately work well.

Combining both parts I have tried different combinations of fixing and estimating of parameters.
I found out that it works well as long as I fix the s_j ’ s. As soon as I try to estimate them, I either get a mass matrix contains 0’s on the diagonal error, or in the case when fixing all parameters but the s_j’s, the estimates are just very wrong (between 300 and 700, real values lie between -6 and 4) and mixing is almost non-existent.

Can anyone help me in that matter? How can I get decent estimates for the s_j’s?

You’d have to supply some code that illustrates the issue for anyone to be able to help you. Its impossible to diagnose otherwise.