# How to reparameterize a Gaussian mixture model?

**URL:** <https://discourse.mc-stan.org/t/how-to-reparameterize-a-gaussian-mixture-model/12792>\
**Category:** Modeling\
**Created:** [January 24, 2020, 8:48am UTC](https://discourse.mc-stan.org/t/how-to-reparameterize-a-gaussian-mixture-model/12792 "2020-01-24T08:48:56Z")\
**Posts on this page:** 1\
**Showing post:** 12

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**Author:** ![Max\_Mantei](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/max_mantei/32/7912_2.png) [@Max\_Mantei](https://discourse.mc-stan.org/u/Max_Mantei)\
**Post date:** [January 24, 2020, 2:03pm UTC](https://discourse.mc-stan.org/t/how-to-reparameterize-a-gaussian-mixture-model/12792/12 "2020-01-24T14:03:47Z")

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Maybe a non-centered parameterization of the mixture components works better?

```stan
// Guassian mixture model
data {
  int<lower=1> N; // number of data points
  real y[N]; // observed values
  real uncertainties[N]; // uncertainties
}
parameters {
  real<lower=0,upper=1> p; // mixing proportion
  ordered[2] mu;             
  real<lower=0> sigma; // spread of mixture components
  vector[N] z1;
  vector[N] z2;
}
transformed parameters {
  vector[N] y_mix1 = mu[1] + sigma*z1;
  vector[N] y_mix2 = mu[2] + sigma*z2; // I corrected this in an edit!!
}
model {
  sigma ~ normal(0, 0.1);
  mu[1] ~ normal(-1.4, 1);
  mu[2] ~ normal(-1.0, 1);
  p ~ beta(2, 2);
  
  z1 ~ std_normal();
  z2 ~ std_normal();

  for (n in 1:N) {
    target += log_mix(p,
                      normal_lpdf(y[n] | y_mix1[n], uncertainties), // corrected (see edit2 below)
                      normal_lpdf(y[n] | y_mix2[n], uncertainties)); // corrected (see edit2 below)
  }
}
generated quantities{
  vector[N] y_true;
  
  for (n in 1:N)
    y_true[n] = bernoulli_rng(p) == 0 ? y_mix1[n] : y_mix2[n];

}

```

Can’t make anything of the results though… Do they make sense?

_Edit: I corrected something in the code above (see comment in code): the auxillary variable for y\_mix2 should be be z2…_

_Edit2: As pointed out by @Evgenii, `y_mix1` and `y_mix2` need to be indexed as well in the `log_mix` call \rightarrow changed to `y_mix1[n]` and `y_mix2[n]`._

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