Conditional non-centering improves terminal-drift sampling in mixed Gaussian–Bernoulli models

[yuhangxoox]

Hi Erik and Scott,

Following our email discussion, I am opening this issue to document the terminal-drift sampling behaviour we encountered in mixed Gaussian–Bernoulli models and the conditional non-centering that substantially improved it.

Motivation

Our application contains Gaussian traits together with a Bernoulli-logit trait. During diagnosis, the sampling problem could already be reproduced in a balanced 190-tip E+P model with all cross-effects fixed to zero and correlated drift disabled.

The corresponding univariate processes sampled well independently, whereas the mixed E+P model did not. The main failure was concentrated in the Bernoulli diffusion scale Q_sigma[P].

Minimal diagnostic

For the same balanced 190-tip dataset:

• P only: Q_sigma[P] R-hat = 1.001, bulk ESS = 1,359
• centred E+P: R-hat = 1.499, bulk ESS = 7.7, tail ESS = 18.4, E-BFMI = 0.064–0.190

With cross-effects and correlated drift disabled, we verified numerically that the E+P target factorises into the corresponding univariate targets up to a constant.

We then noted that the P-only model represents the terminal innovation on a standard-normal scale, whereas the mixed model uses the realised-scale Bernoulli terminal drift.

Conditional non-centering

We constructed a target-equivalent diagnostic version in which the Bernoulli terminal innovation was conditionally non-centred while retaining the Gaussian dimensions and the Cholesky covariance structure.

Target-density and gradient equivalence were checked before sampling.

For the same E+P model:

• conditionally non-centred E+P: Q_sigma[P] R-hat = 1.002, bulk ESS = 1,281, tail ESS = 1,010, E-BFMI = 0.839–1.024

There were no divergences or maximum-treedepth hits, and all monitored population/process parameters passed our convergence criteria.

We also applied the corresponding conditional transformation to the full three-trait D0 model. This substantially improved Q_sigma[P], E-BFMI, and the divergence count, but the full model still did not pass the convergence gate; additional geometry remained in several cross-trait/drift parameters.

Possible package-level direction

A possible package-level change would be to allow the Bernoulli terminal innovation to use a conditional non-centred representation in mixed Gaussian–Bernoulli models while retaining the full covariance structure.

As Erik noted in our email discussion, this uses the same conditional-Gaussian algebra already present in the generated-quantities calculation, applied in the reverse conditional direction.

The compact 190-tip diagnostic package has already been shared in our email thread. I can add a minimal public reproducer here if useful for implementation or testing.

Thanks again for taking a look at this.

[ScottClaessens]

@yuhangxoox Thanks very much for all your work so far on this! It would be great to improve models with mixtures of continuous and non-continuous variables for all users.

I think that a useful starting point, as you say, would be to create a fresh reprex, ideally with simulated data, to concisely demonstrate the problem and the proposed solution. Then we can work through implementing it in the package.