Generalized ELPD (Beta-Divergence) for robust LOO cross-validation

[DJLacombeTTU]

Feature Request: Generalized ELPD (Beta-Divergence) for robust LOO cross-validation

Is your feature request related to a problem? Please describe.

Standard PSIS-LOO based on the logarithmic scoring rule is highly sensitive to influential observations and outliers. When dealing with heavy-tailed data or misspecified models, the standard expected log predictive density (ELPD) can be overly penalized by a few extreme points, leading to unstable model comparisons.

Describe the solution you'd like

I propose adding support for Generalized ELPD using beta-divergence (a type of Bregman divergence) to az.loo. By allowing a tunable beta parameter (typically between 1.01 and 1.05), we can smoothly downweight the influence of severe outliers, providing a more robust measure of predictive accuracy.

This mathematical framework is formally detailed in:

Robust Bayesian Predictive Model Selection using Bregman Divergence (arXiv:2606.10409)

Describe alternatives you've considered

Currently, researchers handling heavy outliers must either manually drop data points, switch to k-fold cross-validation (which is computationally expensive for large Bayesian models), or manually calculate the integral penalty outside of ArviZ.

Additional context

I have already implemented and tested this feature locally on my fork. The implementation:

1. Adds score_type="beta" and a beta kwarg to the loo functions.
2. Extracts the posterior_predictive group to compute the necessary integral penalty using xarray.integrate.
3. Passes the modified weights through the existing psislw architecture.

While running the test suite for this, I also caught and patched a TypeError (unsupported operand type for int and NoneType) in diagnostics.py that gets triggered when _get_r_eff returns None on synthetic mock data.

If the maintainers are open to this API approach, I have a clean, pre-commit-formatted branch with passing tests ready to submit as a Pull Request.

[aloctavodia]

I haven't had time to read the article yet, but it might be an interesting addition.