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The derivation assumes constant base measure. For distributions that do not have constant base measure the expressions used are not correct as the entropy contribution is not as has been derived. There should be gradient of log base measure terms. In the case of Chisq and Poisson because we are projecting from Chi to Chi they cancel out and a happy coincidence gives us the correct result
We should subtract the gradient of log base measure and it should work
Opening this for historical reasons and so we can close the issue when fixed
From slack @ismailsenoz @Nimrais
related to ReactiveBayes/ExponentialFamily.jl#199
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