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\end{definition}  Distributional \emph{decision} problems are the special case when the range of $f$ is $\Bool$. In this special case, the outputs of an optimal predictors can be thought of as probabilities\footnote{However, probabilities\footnote{That said,  $P^{kj}(x,y)=1$ doesn't imply $f(x) = 1$ and $P^{kj}(x,y)=0$ doesn't imply $f(x)=0$. We can try to fix this using a logarithmic error function instead of the squared norm, however this creates other difficulties and is outside the scope of the present work.}.