Hello,

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-negativeand, 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.

*Thanks!*

## Question background:

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 `year`

, `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.

**References**:

(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