Predicting group membership in mixture model using brms

Interfaces brms
1

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?

2

This is not a reproducible example. You failed to specify the prior for your model in the example:

mix <- mixture(gaussian, gaussian)
prior <- c(
  prior(normal(0, 7), Intercept, dpar = mu1),
  prior(normal(5, 7), Intercept, dpar = mu2)
)
3

You’re right. Thanks for the addition.

I’m still looking for help with my problem. If anyone can assist, I’d appreciate it.