When priors and costs exactly cancel

Question

A screen reads X. Under "clean" (H0), X \sim 	ext{Normal}(0, 1) with prior \pi0 = 0.8. Under "flagged" (H1), X \sim 	ext{Normal}(2, 1) with prior \pi1 = 0.2. A false alarm costs 1 unit; a miss costs 4 units. You declare "flagged" when X c. Find the cost-minimizing cutoff c.

Accepted Answer

The optimal threshold. For two equal-variance normals, c = \frac{\mu0 + \mu1}{2} + \frac{\sigma^2}{\mu1 - \mu0}\,\ln\!\left(\frac{\pi0\, cI}{\pi1\, c{II}}ight). Plug in. (\mu0+\mu1)/2 = 1, \sigma^2/(\mu1-\mu0) = 1/2, and \frac{\pi0 cI}{\pi1 c{II}} = \frac{0.8 	imes 1}{0.2 	imes 4} = \frac{0.8}{0.8} = 1. Since \ln(1) = 0, the shift vanishes: c = 1 + 	frac{1}{2}	imes 0 = 1.0. Read it. The prior favors "clean" 4-to-1, which alone would push the cutoff up, but a miss costs 4 times a false alarm, which alone would pull it down. Here the two tilts are exactly equal and opposite, so the optima…

When priors and costs exactly cancel

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