Hi, I am fitting a simple model with random effect as shown below:

`alpha_i`

in the model represents random effect for different genes (`i`

here represents genes). I assume `alpha_i`

follows a normal distribution with a hierarchical prior `sigma_gene`

. The ultimate goal is to estimate the posterior distribution of `sigma_gene`

. Other variables are not related to this question so I skipped the introduction of them.

Here is how I specified prior for this model:

```
bpriors <- prior(normal(0, sigma_global_sd), class = "b", coef = "Intercept") +
prior(constant(1), class = "b", coef = "sc.prop") +
prior(normal(0, sigma_gene_sd), class = "sd") +
prior("target += cauchy_lpdf(sigma_global_sd | 0, 10)", check = FALSE) +
prior("target += cauchy_lpdf(sigma_gene_sd | 0, 10)", check = FALSE) +
prior(gamma(0.01, 0.01), class = "shape") +
prior(beta(1, 1), class = "zi")
stanvars <- stanvar(scode = "real<lower=0> sigma_global_sd; real<lower=0> sigma_gene_sd;",
block = "parameters")
```

`sigma_gene_sd`

in the code represents `sigma_gene`

in the model above. Below is the brms result after fitting the model:

Based on the result and the data I have, I am pretty sure that the distribution of `sd_gene__Intercept`

represents the estimate posterior distribution of the hyperparameter `sigma_gene`

in the model, i.e., the posterior distribution of the variance of the normal distribution of the random effect. Then my first question is, if this is correct, then what is the distribution for `sigma_gene_sd`

? What is the difference between `sd_gene__Intercept`

and `sigma_gene_sd`

?

Another question is that, if I run `prior_summary`

on the model, here are the list of parameters and their corresponding priors:

What is the difference between the three parameters in the red square? Are they three different parameters or they represent the same thing?

Thank you!