Dear Stan community,

After finding out that the ICAR priors can be implemented in Stan (Morris et al.,2019), I was wondering whether it’s possible to use RStan for disaggregating areal data, namely counts.

For the sake of discussion, let \mathcal{D} be the study area with spatial support consisting of M non-overlapping regions, \left\{ \mathcal{R}_{m} \right\}_{m=1}^{M}, where y_{m} denotes the observed count of \mathcal{R}_{m}.

The ultimate goal is to estimate the \textit{latent} counts, say \widetilde{Y}_{n}, corresponding to N non-overlapping regions comprising a finer and misaligned spatial support, \left\{ \mathcal{S}_{n} \right\}_{n=1}^{N}.

Within a hierarchical framework, if \widetilde{Y}_{n} is assumed to have a Poisson distribution with intensity rate \lambda_{n}:

\widetilde{Y}_{n}\mid \lambda_{n} \sim Poisson (A_{n}\lambda_{n}),

where A_{n} is the area of \mathcal{S}_{n} and \log(\lambda_{n})=Z_{n1}\beta_{1}+\ldots+Z_{nP}\beta_{P}+U_{n}

with Z_{np} the measurement of covariate Z_{p} and U_{n} the spatial effect associated to \mathcal{S}_{n}, then

{Y}_{m}\mid\boldsymbol{\lambda} \sim Poisson(\sum_{n:\mathcal{S}_{n}\cap\mathcal{R}_{m}\neq\emptyset}A_{mn}\lambda_{n}),

where A_{mn} is the area of \mathcal{S}_{n}\cap\mathcal{R}_{m}.

Further, the prior for the set of spatial effects \mathbf{U} is an IGMRF (Intrinsic Gaussian Markov Random Field) with `precision matrix’ \kappa\mathbf{W}, where \mathbf{W} is the neighborhood structure of regions \mathcal{S}_{n}. A fully Bayesian hierarchical model is set by specifying a multivariate normal prior for the fixed effects \boldsymbol{\beta} and a Gamma hyperprior for the precision parameter \kappa.

Is it possible to fit this model via the stan function? I should mention that I’m new to Stan, so any suggestion about how to use Stan for estimating latent variables when the number of observations is smaller than the number of latent variables is appreciated in advance.

Best wishes,

Roman.