Ordered Simplex

Modeling
1

I am interested in an ordered simplex - I have a mixture model, and am getting switching of parameters for different threads, but believe an ordered simplex would solve that issue. Is it possible to enforce this ordering? Is enforcing an ordering on a simplex a bad idea on a mixture model?

2

It is quite possible to use an ordered simplex if you do

data {
  int<lower=1> K;
  vector<lower=0>[K] alpha;
}
parameters {
  positive_ordered[K] lambda;
}
transformed parameters {
  simplex[K] pi = lambda / sum(lambda);
}
model {
  // implies pi ~ dirichlet(alpha)
  target += gamma_lpdf(lambda | alpha, 1);
  // use pi in your likelihood
}

Opinions differ as to whether this is a disaster or works perfectly fine.

3 Likes
3

Thank you - this is a creative approach. I will give it a try to see if it leads to disaster ;)

4

Why wouldn’t this work? It’s identified through the prior.

Of course, the prior will matter here for fitting.

A simpler alternative

parameters {
  positive_ordered[K] pi_raw;
  ...
transformed parameters {
  ordered[K] pi = softmax(pi_raw);

won’t work because the pi_raw parameters won’t be identified. And you can’t just add a zero value to identify if you want to preserve orderedness. You can’t just put a prior on pi here without including the log Jacobian of the softmax transform, either.

5

Something about when you have so much data that the posterior distribution of pi has very little variance, then NUTS has to take many very small steps to navigate through the values of lambda that is consistent with both that value of pi and the gamma prior on lambda.