Do you specify priors according to the link function's transformed space?

Question

Suppose I'm developing a model where the response variable is weight measured in pounds and is Gamma distributed. I would like to specify a prior on my intercept coefficient using other information from previous studies.

Since I am assuming the data generating process is Gamma distributed, I will be using a log-link for my model.

My question is, do I specify my prior in terms of the original measurement, (pounds), or in terms of the transformed space, log(pounds)?

$$ \alpha \sim N(300, 20) $$

or

$$ \alpha \sim N(5.7, 3) $$

Answer 1

If $\alpha$ is the intercept, then I think it should be specified on the log pounds scale. The reason is pretty clear. Assume all the other covariates are mean centered, and then consider the linear predictor for the "average" observation. The linear predictor is

$$\eta = \alpha$$

And so the mean of the gamma is then

$$\mu = \exp(\eta) = \exp(\alpha)$$

If $\alpha$ is on the log pounds scale, then $\exp(\alpha)$ is on the pounds scale, and so to is the mean of the gamma.