Implement recognition of affine transforms

[fritzo]

To perform full exact inference of Gaussian distributions, we will need some ability to recognize affine transforms. This might either rely on autograd, or could simply pattern match +, *, and einsum patterns.

Tasks

• Handle Binary and Variable patterns Add an Affine term to represent multilinear functions of real Variables #116
• Rewrite Normal(Affine, ...) to an appropriate Gaussian Add rule for creating Gaussians from Affine inputs #119
• Handle Contract and einsum Add a single normal form for sum-product contractions #157
• Recognize affine funsors including Contraction and Einsum Implement affine funsor approximation #249
• Recognize GetitemOp and ReshapeOp as a linear transformations Implement affine funsor approximation #249
• Rewrite MultivariateNormal(Affine, ...) to an appropriate Gaussian Add pattern for MultivariateNormal(affine) #245
• Fix funsor.affine.is_affine() to be sound but incomplete Implement sound is_affine() test #282
• Rewrite Gaussian(x=affine) as a Gaussian. In three steps: Refactor towards Gaussian._eager_subs_affine() #284, Implement affine pattern recognition for Gaussian substitution #285, Affine gaussian 3 #286

[fehiepsi]

Thanks for keeping tracks of those tasks! It is convenient for me to understand what is going on.

[fritzo]

@fehiepsi Do you want to try working on the task

Rewrite Gaussian(x=affine) as a Gaussian

following the ideas in #245 ? If not, I'm happy to do this part to unblock our first release. If so, it might be a nice intro to Funsor and give us insight into how to incorporate structured Gamma priors in Funsor #190, generalizing your Pyro PR pyro-ppl/pyro#2025 .

[fehiepsi]

Hi @fritzo, could you do it instead? It is better for me to follow-up (though I want to try), given my lacking knowledge of Funsor. I am happy to review and learn from you.

[fritzo]

@fehiepsi sounds good, I'll add you as reviewer.