In the Wikepedia Exponential Family page, natural parameter of Wishart distribution is noted as: -1/2 * V^-1 and (n-p-1)/2 based on most widely used representation of WIshart distribution.

For optimization, I tried to transform (n-p-1)/2 to n/2. Mechanism of exponential familiy entropy calculation implemented at torch.distributions.exp_family.ExponentialFamily calculates entropy by the dot product of natural parameters and the derivative of log probability with respect to natural parameters. It does not yield correct entropy when natural parameters are transformed. Thus, I will revert the changes for now.

@neerajprad @vadimkantorov @fritzo I think I am not allowed to reproduce the macos errors. Does the waiting for the further maintenance of CI will solve the issue?

test (default, 1, 2, macos-11)

botocore.exceptions.ClientError: An error occurred (AccessDeniedException) when calling the Invoke operation: User: arn:aws:iam::308535385114:user/pytorch-pytorch-ossci-s3-read-write is not authorized to perform: lambda:InvokeFunction on resource: arn:aws:lambda:us-east-1:308535385114:function:rds-proxy because no identity-based policy allows the lambda:InvokeFunction action
Error: Process completed with exit code 1.

I think it is ready to be merged.

Added trivial change: fix wrong parameter input (df with ndim - 1 < df < ndim) in Scipy Wishart class.
macos test failure was due to my mistake.

@pytorchbot ciflow rerun

This command didn't do anything.
You don't need to manually issue ciflow rerun commands anymore. Just adding a ciflow/ label will trigger the workflow.

LGTM

@neerajprad has imported this pull request. If you are a Meta employee, you can view this diff on Phabricator.

Thank you for the richful feedbacks and reviews. I appreciated your kindness. I am considering implementing Multivariate T distribution for PyTorch if there is no one trying after I finish Inverse Wishart #70275. It's a shame that I am delaying it to this far.

I guess some people are trying to use T Process similary to Gaussian Process these days. It seems that probabilistic ML libraries are actively implementing it. It might be great if PyTorch offers official implementation for Multivariate T used for T process.

@nonconvexopt it's a pleasure to work with you. Yes Multivariate Student't t might be useful for T processes, but I'm not sure: IIUC it is just a Gamma-MultivariateNormal chain where the gamma rv can be analytically integrated out, and I believe in T processes there are local gammas, but in mutlivariate T there is a single gamma? Let's move this discussion to a new issue. Also cc @fehiepsi who worked out a lot of that math in Pyro.

There is MVT distribution in Pyro, but it only supports scale_tril input. If we port/reimplement in PyTorch, I think we need to support covariance_matrix, precision_matrix inputs too.