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Raise in kl_divergence of TransformedDistributions with a non-injective bijector - #2037
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…ve bijector KL(g(P) || g(Q)) equals KL(P || Q) only for injective g. For a shared non-injective bijector such as AbsoluteValue, the base KL overstates the divergence: KL(|N(-1, 1)| || |N(1, 1)|) was returned as 2.0 although the two distributions are identical. Raise NotImplementedError instead, as TransformedDistribution._entropy already does for such bijectors. Fixes tensorflow#2036
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Fixes #2036
What
For two
TransformedDistributions with equal bijectors,kl_divergencereturns the KL of the base distributions. The identityKL(g(P) || g(Q)) = KL(P || Q)holds only for injectiveg. For a non-injective bijector such astfb.AbsoluteValue, the KL of the pushforwards can be strictly smaller, because information is lost: by the chain rule of KL, the difference isE_{Y ~ g(P)} KL(P(. | g = Y) || Q(. | g = Y)) >= 0. For example,KL(|N(-1, 1)| || |N(1, 1)|)is returned as2.0although the two distributions are identical.TransformedDistribution._entropyalready raisesNotImplementedErrorfor non-injective bijectors. This PR makes_kl_transformed_transformedraise as well, with a message naming the bijector. Injective bijectors are unchanged.Test
testTransformedKLNonInjectiveBijectorFailsintransformed_distribution_test.py. It fails on currentmain(in all three execution regimes) and passes with this change.transformed_distribution_test.pypasses: 144 passed, 11 skipped (TensorFlow 2.21, CPU, Python 3.13).