Adding a logit link to brms mixture model

Modeling
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1

Hi! I am fitting a mixture model with brms and using a mixture of two normal distributions. How can I set my theta2 parameter (mixing parameter) to have a logit link? My current code is below:

mix <- mixture(gaussian(), gaussian())
prior <- c(
  prior(normal(0, 2), Intercept, dpar = mu1),
  prior(normal(0, 4), Intercept, dpar = mu2)
)

model <- brm(bf(y ~ 1,
              mu1 ~ 1 + x + (1|id),
              mu2 ~ 1 + x + (1|id),
              theta2 ~ z),
           data = dat,
           family = mix,
           prior = prior,
           init = 0,
           chains = 2)

This provides results where the links are: mu1 = identity; sigma1 = identity; mu2 = identity; sigma2 = identity; theta1 = identity; theta2 = identity. I would like to model theta2 as if in a logistic regression but am not sure where I can add this to the family argument. Thank you!

2

Hopefully others can confirm this, but when you model mixing proportions like you have in brms, I believe you are already modelling the probabilities in log-odds. No need to specify this specifically, although maybe this could show up more explicitly?