Fix code and math defects in SUG cross-validation and mixture sections - #983

Submission Checklist

• Builds locally (quarto render of cross-validation.qmd and finite-mixtures.qmd to HTML, Quarto 1.8.26)
• New functions marked with <<{ since VERSION }>> (no new functions)
• Declare copyright holder and open-source license: see below

Summary

I found these defects (with help from Claude) while I worked on #962 and #963 (I still need to think more about those). This PR fixes only clear errors; it does not rename variables or add content.

src/stan-users-guide/cross-validation.qmd:

• The generated quantities block of the random-folds example did not compile. I fixed the code and the related equations.
• permutation_rng exchanged each element with an element drawn uniformly from all $N$ positions. This does not give a uniform permutation for $N \ge 3$: there are $N^N$ equally likely paths, and $N!$ does not divide $N^N$. A simulation with $N = 3$ gives probabilities of 4/27 and 5/27 instead of 1/6. The function now does a Fisher-Yates shuffle with discrete_range_rng(n, N). I removed the footnote that said Stan has no uniform discrete RNG.
• In the Rao-Blackwellization derivation, I fixed an equation. The text now gives the correct reason that the expectation equals the linear predictor and the expectation of a normal distribution is its location.
• The loo package citation rendered as "Visualization in Bayesian workflow". See the all.bib item.

src/stan-users-guide/finite-mixtures.qmd:

• A comma was missing between the two arguments of log_mix, so the code did not compile.
• The text said to subtract log(N_new) after log_sum_exp. The correct value is log(M), the number of draws, for each test point.

src/bibtex/all.bib:

• The key GabryEtAl:2019 was defined two times: for the loo package and for "Visualization in Bayesian workflow". The loo package entry now has the key VehtariEtAl:2026. The two GabryEtAl:2019 citations in posterior-predictive-checks.qmd refer to the visualization paper.

Copyright and Licensing

Please list the copyright holder for the work you are submitting (this will be you or your assignee, such as a university or company):

Aalto University

By submitting this pull request, the copyright holder is agreeing to license the submitted work under the following licenses:

• Code: BSD 3-clause (https://opensource.org/licenses/BSD-3-Clause)
• Documentation: CC BY-ND 4.0 (https://creativecommons.org/licenses/by-nd/4.0/)