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Intuitively I can see why mode of predictions is the main prediction of a zero-one loss function, but mathematically I am not sure how it is derived?

Main prediction $= argmin_{y'}E_D[L(\widehat y, y')] \\$

$L_z(y,\widehat y) = \begin{align*} &\ \Big\{ \begin{aligned} 0 \quad if \quad \widehat y = y \\ 1 \quad otherwise \\ \end{aligned} \end{align*} $

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