## sufficient statistics for machine learning

By chance, I came across this ICML¹⁹ paper of Milan Cvitkovic and nther Koliander, Minimal Achievable Sufficient Statistic Learning on a form of sufficiency for machine learning. The paper starts with “our” standard notion of sufficiency albeit in a predictive sense, namely that Z=T(X) is sufficient for predicting Y if the conditional distribution of Y given Z is the same as the conditional distribution of Y given X. It also acknowledges that minimal sufficiency may be out of reach. However, and without pursuing this question into the depths of said paper, I am surprised that any type of sufficiency can be achieved there since the model stands outside exponential families… In accordance with the Darmois-Pitman-Koopman lemma. Obviously, this is not a sufficiency notion in the statistical sense, since there is no likelihood (albeit there are parameters involved in the deep learning network). And Y is a discrete variate, which means that

$\mathbb P(Y=1|x),\ \mathbb P(Y=2|x),\ldots$

is a sufficient “statistic” for a fixed conditional, but I am lost at how the solution proposed in the paper, could be minimal when the dimension and structure of T(x) are chosen from the start. A very different notion, for sure!

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