deletions | additions       diff --git a/We_are_now_ready_to__.tex b/We_are_now_ready_to__.tex  index 06ce279..27036dc 100644  --- a/We_are_now_ready_to__.tex  +++ b/We_are_now_ready_to__.tex 
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\E_{\mu^k \times U^{\R_P(k,j)}}[(P^{kj} - f)^2] \leq \E_{\mu^k \times U^{\R_Q(k,j)}}[(Q^{kj} - f)^2] + \varepsilon(k,j)  \end{equation}  \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\footnotes{However, $P^{kj}(x,y)=1$ 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 is outside the scope of this paper.}.