 Base-rate dependence of autonomic emotion classification

Fix a true-positive probability $s>0$ and false-positive probability $f>0$. The positive predictive value at base rate $p$ is $v(p)=sp/(sp+f(1-p))$. Differentiation gives $v^{\prime }(p)=sf/[f+(s-f)p{]}^2>0$, while $v(p)\to 0$ as $p\to 0$. Thus a classifier can retain the same conditional performance yet become unreliable among its positive classifications when the target emotion is rare. This is a mathematical implication of the specified binary model, not an empirical performance estimate.