I have a general question about modelling prior distribution for
sd class (group-level).
When specifying a prior distribution, P. Bürkner (1) wrote:
Each group-level effect of each grouping factor has a standard deviation parameter, which is restricted to be non-negative and, by default, has a half Student-t prior with 3 degrees of freedom and a scale parameter that is minimally 10.
I am confused about the part in bold. Now, If I specify a Student t Distribution with the same proprieties as above:
plot( density ( LaplacesDemon::rst(1000, mu = 0, sigma = 10, nu = 3)))
The prior distribution is not bounded to only positive values, And I don’t understand why it is not the case, even if posteriors of
sd class are supposed to be positive values (am I right?)… But does
[…] which is restricted to be non-negative […].
means that priors are only gonna be those in the right side of the above distribution plot ?
I am confused in how this works, and would appreciate any information about the technicalities around this. I am not sure either If I interpret correctly the quote above, so please don’t hesitate to tell me if I am wrong, I have some Bayesian knowledge, but I am far from being an expert.
I have read information on several sites, including
Stan's help page on prior distributions, Gelman’s page on informative priors and other sources, but did not find a satisfactory answer to my question.
I work on behavioural traits (vigilance) and fitness (survival) in an ungulate. I investigated the drivers of several vigilance traits (which are count data, y1, y2, y3) through generalized linear models with negative-binomial distribution using
glmmTMB. Each of these models include
id (identity of the animal) and
observer as random effects, as well as a bunch of population-effects. From that part of my analyses, I saw that the variance of
observer random effect is very small (actually,close to 0). I also use results from these analyses to specify prior information on population level-effects.
I am now trying to study the correlations at the individual level (group-level=
id) between these behavioural traits (y1,y2,y3) and survival (0-1 data, Bernoulli, y4). I chose to do a Multivariate model and to compute the correlations between shared group levels of different formulas. I followed what is written by P. Bürkner (2) :
Then, however, specifying group-level effects of the same grouping factor to be correlated across formulas becomes complicated. The solution implemented in brms (and currently unique to it) is to expand the | operator into |< ID >|, where < ID > can be any value. Group-level terms with the same ID will then be modelled as correlated if they share same grouping factor(s).
What I am doing now is that I try to specify a prior distribution for a random effect (observer effect). I know from earlier analyses that the variance is small, but to be consistent, I still want to include this group-level effect in the formulas of my multivariate analysis. I am now trying to specify a prior distribution that will “help”
observer group-level effect to stay close to zero and improves sampling speed + convergence.
(1) Bürkner, P.-C. (2017). brms : An R package for bayesian multilevel models using Stan. Journal of Statistical Software , 80 , 1–28. doi: 10.18637/jss.v080.i01
(2) Bürkner, P.-C. (2018). Advanced Bayesian multilevel modeling with the R package brms. The R Journal , 10 , 395–411. doi: 10.32614/RJ-2018-017