Hi everyone in the Stan community!
I’m struggling to wrap my head around the usage of informative priors in rstanarm functions (in a case where I want exceptionally specific prior information to be included). I’m afraid that the more I think about it without expert feedback, the more I will get lost.
If I understand correctly, stan_glm runs models on transformed data. However, I’m still not 100% sure if it’s standardization or mean-centering (the part of the rstanarm manual on the prior_intercept statement got me confused there). Either way, I’m not sure how to express specific priors correctly.
To clarify my issue, here’s an example: Consider a simple linear regression model
y = a + bx
, where I’d like to apply specific unstandardized normal priors for both parameters a and b, let’s say
a \sim \mathcal{N}(100, 1) and
b \sim \mathcal{N}(5, 2)
How would I express these two specific priors within the prior and prior_intercept statement of stan_glm function? Given that stan_glm requires the expression of priors on a standardized scale, my first idea was to convert the parameters according to
b \hspace{1pt} '= b \hspace{1pt} \frac{\sigma_x}{\sigma_y} and
a \hspace{1pt} '= \frac{a + b \hspace{1pt} ' \hspace{1pt} \frac{\sigma_y}{\sigma_x} \hspace{1pt} \overline{x} - \overline{y}}{\sigma_y}
, where a’ and b’ represent the standardized parameters I could use in the prior arguments of stan_glm.
However, I can’t figure out how to continue from there. If anything, these formulas would only apply to the location argument of my specific priors ? But what about the scale argument?
It would be very appreciated if somebody could help me along on this topic. How do you convert a specific unstandardized normal prior to a scale that fits the requirements of the stan_glm prior arguments? What are the respecitive prior expressions for a and b in the fictional example above?
Thank you for your help,
Beni