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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.


  1. harryq says:

    I do the sort of analyses that Andrew refers to in point #2 all the time (usually Poisson-distributed counts, but occasionally with individual-level binary data from surveys), but my code is nowhere near being fit for dissemination. Instead, I think Duncan Lee’s CARBayesST package should do ~everything you’d want to do.

    I’m sure modifying Mitzi’s Stan code to fit similar types of space-time models would be relatively easy to do, too, but I haven’t tried.

    • Andrew says:


      I followed your link. I like the functionality but I’m not so thrilled with the use of Gibbs and Metropolis because of the usual concerns about speed, convergence, and effort required to expand or generalize the model. But, as you said, it should be easy enough for users to express these models in Stan as needed, when the Gibbs/Metropolis implementation runs into trouble.

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