Binary classification: error probability minimization

[Bipolarity]

Typically in problems involving binary classification (i.e. radar detection, medical testing), one will try to find a binary classification scheme that minimizes the total probability of error.

For example, consider a radar detection system where a signal is corrupted with noise, so that if the signal is present and has value A, the radar detects Y = A + X where X is noise, and if the signal is not present, the radar detects Y = X.

Given the observation Y, one wishes to find a decision rule regarding whether or not the signal was present that will minimize the probability of error. Error occurs either as false positives (type I) or false negatives (type II).

If you know that the noise X is Gaussian with zero-mean and unit variance, one can (with some calculations) show that a good decision rule is to see whether Y<A/2 or Y>A/2 to decide whether or not the signal is present. I think most would agree that this minimizes the total probability of error. However, how would one prove this? There are, after all, an infinite set of possibilities for the decision rule. One could have some weird decision rule like:
A > |Y| > A/2 --> signal is present, otherwise signal is absent, but these would be suboptimal. How one would PROVE that the rule Y>A/2 is optimal in the sense that it minimizes error?

Thanks!

BiP

 

[Stephen Tashi]

I think most would agree that this minimizes the total probability of error.

How do you define the "total probability of error"?

Binary detectors are often analyzed by looking at their "receiver operating characteristic" (ROC) curve.

 

[Bipolarity]

The total probability of error is the sum of probabilities of type 1 and type 2 errors respectively. I am aware of the ROC curves, but that does not answer my question.