WishartTriL does not support df \in (k - 1, k)

My understanding is that the Wishart distribution is well defined so long as df > k - 1. k is the dimension parameter, so the Wishart distribution has support over k x k matrices. df is the "degrees of freedom" parameter. However, the docs for WishartTriL indicate that it only supports df >= k. And when I try constructing a WishartTriL with k > df > k - 1, I get an error.

The fact that Wishart is supported for any df > k - 1 seems to be indicated in the distribution overview section of the wikipedia for InverseWishart, and in the wikipedia for the Wishart distribution, in the last paragraph of the "Spectral Density" section.

I have in mind that it would be useful to use a Wishart with df < k in order to express a very uncertain prior over the covariance of a multivariate normal distribution.

The scipy docs suggest that scipy's Wishart distribution supports construction with k - 1 < df < k. They do note, however, that the behavior of their sampling algorithm can create invalid samples sometimes when df < k, and advise users to check for invalid sample generation.

Here is a code example showing the problem:

import tensorflow_probability.substrates.jax.distributions as tfd
import jax.numpy as jnp

nu = 0.01
Lambda = jnp.array([[1.0]])
X = jnp.array([[0.5]])

scale_tril = jnp.linalg.cholesky(Lambda)
tfd.WishartTriL(nu, scale_tril)

Constructing this distribution causes an error.

This is despite the fact that it is possible to construct and evaluate logpdfs of the following Gamma distribution, even though (if I understand correctly) this 1D Gamma is equivalent to the above Wishart distribution:

tfd.Gamma(nu / 2, 1 / (2 * Lambda[0, 0]))

(You can check this equivalence by changing nu to something greater than 1, so you can construct the Wishart distribution, and test that the logpdf values for the two distributions are the same.)

I am using tensorflow_probability version 0.23.0.

Thank you for the help!

Start with the tfd.WishartTriL construction shown in the issue and inspect how it validates df relative to the scale_tril dimension. Compare its behavior and logpdf with the provided Gamma example for k - 1 < df < k. Done means valid construction and correct logpdf behavior in that range, with the documented constraint updated accordingly.