= Minimum sum of errors in a simple hypothesis test
{title2=$C=\{f_1>f_0\},\quad\min(\alpha+\beta)=\int\min(f_0,f_1)$}
For two <simple hypotheses> with <probability densities> $f_0,f_1$, a <critical region> $C$ has sum of <Type I error> and <Type II error> probabilities $\int_Cf_0+\int_{C^c}f_1$. Pointwise minimization chooses $C=\{f_1>f_0\}$, with either decision allowed on the equality set. Thus the optimal <likelihood ratio> threshold is $1$, and the minimum is $\int\min(f_0,f_1)=1-\tfrac12\int|f_1-f_0|$. This criterion weights the two error probabilities equally; unequal costs or prior probabilities change the threshold.
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