Implement BivariateNormal distribution

[fritzo]

I think it would help to have a BivariateNormal distribution. The MultivariateNormal #52 is currently blocked in lack of batched/differentiable linear algebra operations, but for bivariate normals, it's simple to write the cholesky factorizations and inverses out by hand. This would also serve as a great reference implementation as we test the MultivariateNormal once that is implemented. Here is an example implementation I wrote in Pyro (but the batching and shaping are different, so this will take some reworking for use in PyTorch): https://github.com/uber/pyro/blob/8f2f858/pyro/distributions/testing/bivariate_normal.py#L14

[fritzo]

👍 Thanks @tbrx for starting this. In Pyro, I found bugs in our MultivariateNormal gradients only by hand-implementing a BivariateNormal.

[tbrx]

Is the plan to send this upstream (i.e. is a BivariateNormal independently of interest even if we were to have a fully functional MultivariateNormal)? Or is this intended primarily as (a) stopgap and (b) test target?

[fritzo]

I'm a bit worried that PyTorch 0.4 might miss some features of MultivariateNormal (batching, gradients, cuda support for trtrs). If there's a risk of these features not making it into release, I think it's a good idea to send this upstream. Certainly it will be valuable in testing, so @apaszke may approve it simply for testing purposes (it will help test PyTorch linear algebra, incidentally).

[apaszke]

Maybe we could implement MultivariateNormal that fails if it's not equivalent to BivariateNormal, and later remove this restriction?