Hello everyone,

I am working on a Multinomial spatial classification problem, where we have considered a CAR prior on the regression coefficients \beta s. I have attached the stan code below. While the code is syntactically alright and it runs fine, the chains are not mixing properly. The data has 3 classes. M=336 and N=30K+.

Now I know, there might be several reasons for the chains not mixing, but I feel like I am also not specifying the model correctly. I was going through the STAN manual, and I saw a point about the K vs K-1 specification of the \beta s. However, I feel like the K-1 specification might not be appropriate given the CAR prior. Is there a workaround on this? Any other suggestion to improve the MCMC sampling is also welcome. Thank you in advance.

The CAR prior has been set up as per suggestion in Banerjee,Carlin, and Gelfand.

```
data {
int<lower=1> N; // Number of data
int<lower=1> M; // Number of covariates
matrix[N, M] X;
int<lower=1,upper=3> y[N];
matrix [M,M] DW; // Diagonal matrix, entries equal the number of neighbors
matrix [M,M] W; // Neighborhood matrix
real lmin; // minimum eigenvalue of DW^{-1/2}W DW^{-1/2}
real lmax; // maximum eigenvalue of DW^{-1/2}W DW^{-1/2}
}
transformed data{
vector[M] zeros=rep_vector(0, M);
}
parameters {
vector[M] beta;
real alpha;
}
transformed parameters{
matrix[M,M] prec= inverse(DW-alpha*W);
}
model {
beta ~ multi_normal(zeros,prec);
alpha ~ uniform(1/lmin,1/lmax);
y ~ categorical_logit(X*beta);
}
```