Implement unconstraining transform for LKJCorr

[johncant]

I've ported this bijector from tensorflow and added to LKJCorr. This ensures that initial samples drawn from LKJCorr are positive definite, which fixes #7101 . Sampling now completes successfully with no divergences.

There are several parts I'm not comfortable with:

• https://github.com/johncant/pymc/blob/fix_lkjcorr_positive_definiteness/pymc/distributions/multivariate.py#L1583 - it seems like a bad idea to run a pytensor graph here. Is there any way to get the LKJCorr n parameter from op or rv without evaling any pytensors?
• https://github.com/johncant/pymc/blob/fix_lkjcorr_positive_definiteness/pymc/distributions/transforms.py#L176-L177 - not sure whether or not this is the right way to create a constant pytensor tensor here.

@fonnesbeck @twiecki @jessegrabowski @velochy - please could you take a look? I would like to make sure that this fix makes sense before adding tests and making the linters pass.

Notes:

• Tests not yet written, linters not yet ran
• The original tensorflow bijector is defined in the opposite sense to pymc transforms, i.e. forward in tensorflow_probability is backward in pymc
• The original tensorflow bijector produces cholesky factors, not actual correlation matrices, so in this implementation, we have to do a cholesky decomposition in the forward transform.
• In the tensorflow bijector, the triagonal elements of a matrix are filled in a clockwise spiral, as opposed to numpy which defines indices in a row-major order.

Description

Backward method

1. Start with identity matrix and fill lower triangular elements with unconstrained real numbers.
2. Normalize each row so the L-2 norm is 1
3. This is now a Cholesky factor that will always result in positive definite correlation matrices

Forward method

1. Reconstruct the correlation matrix from its upper triangular elements
2. Perform cholesky decomposition to obtain L
3. The diagonal elements of L are multipliers we used to normalize the other elements.
4. Extract those diagonal elements and divide to undo the backward method

log_jac_det

This was quite complicated to implement, so I used the symbolic jacobian.

Related Issue

• [ ] Closes #7101

Checklist

• [ ] Checked that the pre-commit linting/style checks pass
• [ ] Included tests that prove the fix is effective or that the new feature works
• [ ] Added necessary documentation (docstrings and/or example notebooks)
• [ ] If you are a pro: each commit corresponds to a relevant logical change

Type of change

• [ ] New feature / enhancement
• [x] Bug fix
• [ ] Documentation
• [ ] Maintenance
• [ ] Other (please specify):

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📚 Documentation preview 📚: https://pymc--7380.org.readthedocs.build/en/7380/

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[johncant]

Hi, It's unlikely I'm going to have any time to work on this for the next 6 months. The hardest part is coming up with a closed form solution for log_det_jac, which I don't think I'm very close to doing.

[twiecki]

Thanks for the update @johncant and for pushing this as far as you did.