I am using
brms in R to fit a generalized linear mixed model using a Poisson distribution with log link.
The model takes count data that ranges from 0 to 3 as response variable, and two z-transformed (standardized) continuous predictors and their interaction as population-level (fixed) effects.
I wanted to use priors for the population-level effects that do not give importance to completely implausible values for the coefficients (I guess this can be called weakly informative priors?). Kind of the same aim advocated by Gabry et al. (2019), but without the data-visualization part.
For example, if this was a linear model, I would set the prior for all fixed effects as a normal distribution with
mean = 0 and
sd = 1.5, meaning that I am pretty (95%) sure that a one standard-deviation increase/decrease in the predictors will not be associated to more than a 3-unit increase/decrease in the response variable — after all, this is the entire range of my response variable!
I am having a hard time applying the same reasoning to a Poisson log model, because a one-unit increase in the predictor is associated to a multiplicative (not additive) change in the response variable. Following the line of thought above, absolutely any value would be plausible, from minus infinity (ln(0/3) = -\infty) to infinity (ln(3/0) = \infty).
My question is: how can I choose a reasonable prior in this case? Or do I — and, by extension, anyone fitting Poisson log models with a response variable that includes zeroes — have no choice but to set a flat prior?