Consider a binary hypothesis test in which there is a cost associated with each type of decision In addition to the cost C10 for a false alarm and C01 for a miss, we also have the costs C00 for correctly deciding hypothesis H0 and the C11 for correctly deciding hypothesis H1 Based on the observatio...

The key to this problem is to observe that P A 0 H 0 1 P A 1 H 0 PA 1 H 1 1 ...

Consider a binary hypothesis test in which there is a cost associated with each type of decision.

Question:

Consider a binary hypothesis test in which there is a cost associated with each type of decision. In addition to the cost Cʹ10 for a false alarm and Cʹ01 for a miss, we also have the costs Cʹ00 for correctly deciding hypothesis H0 and the Cʹ11 for correctly deciding hypothesis H1. Based on the observation of a continuous random vector X, design the hypothesis test that minimizes the total expected cost
E[Cʹ] = P[A1|H0]P[H0]Cʹ10
+ P[A0|H0]P[H0]Cʹ00
+ P[A0|H1)P[H1]Cʹ01
+ P[A1|H1)P[H1]Cʹ11.
Show that the decision rule that minimizes total cost is the same as decision rule of the minimum cost test in Theorem 11.3, with the costs C01 and C10 replaced by the differential costs Cʹ01 - Cʹ11 and Cʹ10 - Cʹ00.