I was working through the brms mixture model examples provided here and I had some difficulty understanding the interpretation of the parameters predicting group membership. Straight from the example, here is the toy data and model.

```
## simulate some data
set.seed(1234)
dat <- data.frame(
y = c(rnorm(200), rnorm(100, 6)),
x = rnorm(300),
z = sample(0:1, 300, TRUE)
)
## predict the mixing proportions
fit4 <- brm(bf(y ~ x + z, theta2 ~ x),
dat, family = mix, prior = prior,
inits = 0, chains = 2)
summary(fit4)
```

When I look at `summary(fit4)`

this is what I see:

```
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
mu1_Intercept 0.01 0.11 -0.21 0.24 3348 1.00
mu2_Intercept 5.94 0.13 5.68 6.20 3457 1.00
theta2_Intercept -0.72 0.18 -1.07 -0.39 4568 1.00
mu1_x 0.06 0.07 -0.08 0.20 3829 1.00
mu1_z -0.11 0.14 -0.41 0.16 3533 1.00
mu2_x -0.05 0.10 -0.26 0.15 3952 1.00
mu2_z 0.46 0.18 0.11 0.81 3883 1.00
theta2_z 0.04 0.25 -0.44 0.52 3855 1.00
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sigma1 1.04 0.05 0.94 1.15 3861 1.00
sigma2 0.90 0.07 0.78 1.05 3407 1.00
```

So, now I’m trying to figure out the interpretation of `theta2_Intercept`

and `theta2_z`

. My best guess is that the model predicts a group membership for Group 2 (i.e., `theta2`

) and that if I regress this predicted group membership on `z`

I’ll be able to recover the parameters from the model. So, I tried that:

```
## compute the membership probabilities
ppm <- pp_mixture(fit4)
## extract point estimates for each observation
head(ppm[, 1, ])
# Get theta2 membership and put it in data frame
dat$theta2 <- ppm[, 1, 2]
# regress theta2 on x
fit5 <- brm(theta2 ~ z, dat, chains = 2)
summary(fit5)
```

These are the results for `summary(fit5)`

:

```
Population-Level Effects:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
Intercept 0.33 0.04 0.25 0.41 2174 1.00
z 0.01 0.05 -0.10 0.11 2539 1.00
Family Specific Parameters:
Estimate Est.Error l-95% CI u-95% CI Eff.Sample Rhat
sigma 0.47 0.02 0.44 0.51 2131 1.00
```

As you can see, the results for `summary(fit5)`

doesn’t at all recover the parameters predicting `theta2`

from `fit4`

.

So my question is where did I go wrong and how should I interpret the parameters predicting `theta2`

from `fit4`

?