Truncated Multivariate normal distrbution

[nikuehn]

Hi,

sorry for warmng up something that has been discussed before, but I’m a bit dense when it comes to things like truncation and possible Jacobian adjustments.
I am aware of this post

How to truncate a multivariate normal distribution? (sidebar, how to access old stan mailing lists posts?)

Hello! I am attempting to build a dynamic linear model where the coefficients are constrained to be above 0 for domain-specific reasons. I would like to model the covariance between the coefficients (e.g. if one goes up, does that make another more likely to increase? decrease?) Stan does not allow the truncation of multivariate distributions. However, it seems that I am not the first person that has asked this question [38] But when I click on that link I get the following warning: [18] I…

and this

Simulate from truncated multivariate normal distribution in Stan

Hello Stan team, I am looking for a way to simulate a truncated Multivariate normal distribution in STAN, i.e. something like multi_normal_rng(mu, Sigma), but with constraints on the bounds, in that each underlying univariate variable has its own bounded domain (or more specifically, some have 0 upper bound and -inf lower bound, and some have 0 lower bound and +inf upper bound). And of course Sigma is not diagonal :-) I would need to do it generated_quantities() block. Anyone having brillian…

and it’s really helpful. Now, the following is a valid stan program.

data {
  int D;
  vector[D] ytrunc;
  cov_matrix[D] S;
}

transformed data {
  vector[D] mu = rep_vector(0, D);
  cholesky_factor_cov[D] L = cholesky_decompose(S);
}

parameters {
  vector<upper=ytrunc>[D] Y;
}

model {
  Y ~ multi_normal_cholesky(mu, L);
}

The model compiles and runs, and sampled Y values are not larger than ytrunc. I wonder if this model would be a correct implementation of a truncated multivariate normal?
I ran the model above, as well as a model with make_stuff as in the linked posts, and it seems there are subtle differences, and I would like to understand what the underlying difference is.
Sorry if this is totally clear and answered somewhere in the documentation.

[edm]

For your model, the density underlying multi_normal_cholesky() is still the usual, untruncated mvn density. It is not normalized to account for the truncation, which could cause inference problems if this was embedded in a larger model.

Looking at the tMVN document from your first link, I think that the Stan model you specified is similar to the rejection sampling approach mentioned on p. 5. If you are only looking to draw samples of Y without evaluating the underlying density, then the document shows that rejection sampling is less efficient than the more complicated approaches.