Fitting the Besag, York, and Mollie spatial autoregression model with discrete data

Rudy Banerjee writes:

I am trying to use the Besag, York & Mollie 1991 (BYM) model to study the sociology of crime and space/time plays a vital role. Since many of the variables and parameters are discrete in nature is it possible to develop a BYM that uses an Integer Auto-regressive (INAR) process instead of just an AR process?

I’ve seen INAR(1) modeling, even a spatial INAR or SINAR paper but they seem to be different that the way BYM is specified in the Bayes framework.

Does it even make sense to have a BYM that is INAR? I can think of discrete jumps in independent variables that affect the dependent variable in discrete jumps. (Also, do these models violate convexity requirements often required for statistical computing?)

My reply:

1. To see how to fit this sort of model in a flexible way, see this Stan case study, Spatial Models in Stan: Intrinsic Auto-Regressive Models for Areal Data, from Mitzi Morris.

2. Rather than trying to get cute with your discrete modeling, I’d suggest a simple two-level approach, where you use an underlying continuous model (use whatever space-time process you want, BYM or whatever) and then you can have a discrete data model (for example, negative binomial, that is, overdispersed Poisson) on top of that.

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