Bivariate von Mises Distribution

[OlaRonning]

This PR introduces the Sine Bivariate von Mises (BvM) distribution which is a distribution on the torus described by Singh et al.. The sampling method works for the univariate case and follows the process developed by Kent et al..

In bioinformatics, BvM is well suited for molding torsion angles for peptide bonds. I'll submit an example using this distribution together with the Sine Skewed distribution in a separate PR.

Sampling may fail if the distribution is bimodal; I've added a raise Exception if this occurs. However, I'm not sure of the preferred behavior in this case.

[fritzo]

Hi @OlaRonning, nice contribution. I'm curious what inference algorithms you'd like to support with this distribution. As I understand, this will support MCMC because the .log_prob() is differentiable, but SVI would be trickier because .sample() is not reparametrizable. Often when creating new distributions I mention in the docstring which inference algorithms are supported and what parameters can be learned.

[OlaRonning]

Hi @fritzo,

Thanks for the review! I had not considered the issue with reparameterizing and intended to use SVI; however, HMC should still work. I'll update the docstring as you suggest and look into the possibility of a transform RxR -> S^1xS^1 (not for this PR :)).

[fritzo]

Hi @OlaRonning, for SVI you might also consider a ProjectedNormal prior together with a ProjectedNormalReparam and an AutoMultivariateNormal guide. The ProjectedNormal distribution and reparametrizer aims to support the reparametrization trick while working around the torus topology by working in a higher dimensional euclidean space and projecting down to spherical (or toroidal) space. This combination supports learning of a bimodal posterior.

[OlaRonning]

That's neat, I'll definitely try this out! Thanks.

[OlaRonning]

Using infer.Predictive, I could not use autodiff in BvM._bfind; I resolved this by manually computing the first and second-order derivates.

[fritzo]

Looks good!

I have checked coding idioms but I have not checked math.

1. Can you please confirm that the the examples params in tests/distributions/conftest.py exercise a sufficiently diverse range of parameter space, such that test_gof can be trusted to verify agreement between .log_prob() and .sample()? I see that this test passes

   pytest -vx tests/distributions/test_distributions.py::test_gof -k SineBiv

   but I want to make sure you're testing with some symmetry breaking parameters, e.g. randomly sampled from their supports, or otherwise asymmetric so as to detect math errors like writing sin instead of cos.
2. Is there any math you'd like checked in particular?

[OlaRonning]

Thanks for the second review, @fritzo. I've added some more fixtures as you requested, and I believe these now sufficient exercise critical parameterizations (i.e., loc close to pi, high correlation, low correlation, and low concentration).

It would be nice to avoid the transcendental functions when sampling the second angle; however, this would involve reimplementing the von Mises distribution. Kent et al. provide an R implementation under supplementary material, together with their article, I believe the maths is reasonably straight forward.

The main difference between the R implementation and this is I log transform the terms in the bound and batch sampling, so those would be the critical points.

[OlaRonning]

With the bug fix in 1789fb0, my mixture model is behaving as expected.

[fritzo]

Hi @OlaRonning, could you try merging dev and pushing? I'm unsure why doc tests are failing.

[OlaRonning]

Hi @fritzo, been slightly distracted by deadlines. I'll work out the problem with docs on Thursday if #2838 doesn't resolve it. #2838 solved it!

[fritzo]

LGTM. I haven't thoroughly checked math and algorithms, but the interfaces look good and I trust the thorough tests.