Extend sum of mul rewrite for multiple axis #484
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twiecki
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Rewrite from prod of mul was not correct when only some axes were reduced by prod
Also: * Separates the sum of negation rewrite * Fixes bug in partial prod reduction
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Extend the Sum of Mul rewrite beyond the scalar case. We extract variables that have only degenerate dims along the reduced dimensions. In the case of a total sum only scalars can be pulled out, which was exactly the case supported by the old rewrite.
I am not so hot on the Prod of Mul rewrite, but since it was there already, we keep it. This actually fixes a bug when the prod only does a partial reduction, in which case the old logic of raising the extracted values to the power of the whole inner size was wrong. I added a test that failed in main in the first commit. The expanded rewrite comes up in the second commit, and fixes the failing test.
I also separated the rewrite for Sum(-mul) since it shares almost no logic with the larger rewrite. The usefulness of generalizing this rewrite showed up when exploring the grad of the batched MvNormal logp/dlogp in #482 .
And finally, I merged this with the nearly identical rewrite for sum of div, that already handled multiple axes