Regularized Horseshoe Prior for Poisson Regression with Offset

Modeling
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1

I am trying to follow Piironen and Vehtari (2017) to implement a regularized horseshoe prior for a Poisson regression model with an offset. I am modelling the number of events in a location with population size as an offset so that the model predicts the rate of events (that is, events per 100,000 residents) rather than the number of events.

brm(
    n_events ~ 1 + offset(log(pop_n_total / 1e5)) + ...,
    family = poisson,
    prior = set_prior(horseshoe(scale_global = tau_0), class = "b"),
    ...
)

Piironen and Vehtari (2017) recommend using formula 3.12 to set the scale_global of the regularized horseshoe prior: \tau_0 = \frac{p_0}{D - p_0} \frac{\sigma}{\sqrt{n}}, where p_0 is the number of non-zero regression coefficients, D is the number of predictor variables, and \sigma^2 is the pseudo variance for the corresponding regression model.

Table 1 defines \sigma^2 = \mu^{-1} with \mu = \bar{y} for Poisson regression with a log link function. What should I use as the pseudo variance \sigma^2 for a Poisson model with an offset? I don’t know enough math to figure this out myself—but using \mu = \bar{y} doesn’t seem right.

Any advice would be much appreciated. Thank you!

2

I think if you standardise the offset to have mean 0 and SD 1 it should work like that. But see also the Yanchenko paper for R2D2 for GLMs where I think they discuss offsets.

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