Equal priors, a costly miss

Question

A classifier reads X. Under "negative" (H0), X \sim 	ext{Normal}(0, 1); under "positive" (H1), X \sim 	ext{Normal}(3, 1). The two states are equally likely, \pi0 = \pi1 = 0.5. A false alarm costs 1 unit; a miss costs 9 units. You declare "positive" when X c. Find the cost-minimizing cutoff c.

Accepted Answer

The optimal threshold. c = \frac{\mu0 + \mu1}{2} + \frac{\sigma^2}{\mu1 - \mu0}\,\ln\!\left(\frac{\pi0\, cI}{\pi1\, c{II}}ight). Plug in. (\mu0+\mu1)/2 = 1.5, \sigma^2/(\mu1-\mu0) = 1/3, and with equal priors the ratio reduces to the cost ratio: \frac{\pi0 cI}{\pi1 c{II}} = \frac{0.5 	imes 1}{0.5 	imes 9} = \frac{1}{9}. c = 1.5 + 	frac{1}{3}\ln\!\left(	frac{1}{9}ight) = 1.5 + 	frac{1}{3}	imes(-2.197) = 1.5 - 0.732 \approx 0.77. Read it. With equal priors, only the costs tilt the cutoff. Because a miss is 9 times as costly as a false alarm, the optimal cutoff drops well below the mid…

Equal priors, a costly miss

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