Prior for pameter above 0 with random effect

Modeling
1

I am fitting a non-linear gaussian model with brms:

mod_full_k <- bf(Y|se(Y_std_error, sigma = TRUE) ~ (W*(Q+(k*V)))/
       (exp(logI0)*(M^b+(VM/2)*b*(b-1)*M^(b-2))),
                 logI0 ~ 1,
                 b ~ 1,
                 k~1+(1|site),
                 nlf(sigma ~ int+W^a),
                 int ~ 1,
                 a ~ 1,
                 nl = T,
                 family = gaussian())

The parameter k can only have positive values and therefore I defined the priors as:

prior <- c(
  prior(normal(10,2), nlpar = "logI0", lb = 0), 
  prior(beta(0.7*4,(1-0.7)*4), nlpar = "b", lb = 0, ub = 1), 
  prior(lognormal(-1.5,0.7), nlpar = "k", lb = 0),
  prior(exponential(3), class = "sd", nlpar = "k"),
  prior(normal(0,0.5), class = "b", nlpar = "a", lb = 0),
  prior(normal(0,1), class = "b", nlpar = "int", lb = 0),
  prior(exponential(3), class = "sd", nlpar = "logI0")
)

I don’t get any negative draws for the k_intercept parameter, but when I add the sd parameter to get the posterior, some draws are negative and the 95% CI overlap with zero for some parameters. How can I specify the priors and model so all k posterior (k + sd_k) are positive? Before I had:

inv_logit(k) ~ normal(0,1.5)

and this worked well, but then I read a paper and realised k can have values above 1. I thought of log transforming k such that:

log(k) ~ normal(0,1)

or something like that, but then any draws (with sd) with identity value between 0 and 1 will result in negative values.

2

If log(k) is normally distributed, then k is strictly positive. k can still take values between 0 and 1 because the Gaussian random effect distribution for log(k) admits negative values.

3

Thank you! I ended up specifying my model as:

bf(Y|se(Y_std_error, sigma = TRUE) ~ (W*(Q+(exp(logk)*V)))/
       (exp(logI0)*(M^b+(VM/2)*b*(b-1)*M^(b-2))),
                 logI0 ~ 1,
                 b ~ 1,
                 logk~1+(1|site),
                 nlf(sigma ~ int+W^a),
                 int ~ 1,
                 a ~ 1,
                 nl = T,
                 family = gaussian())