Editing Detection theory (section)

===Bayes criterion===

In some cases, it is far more important to respond appropriately to ''H1'' than it is to respond appropriately to ''H2''.  For example, if an alarm goes off, indicating H1 (an incoming bomber is carrying a [[nuclear weapon]]), it is much more important to shoot down the bomber if H1 = TRUE, than it is to avoid sending a fighter squadron to inspect a [[false alarm]] (i.e., H1 = FALSE, H2 = TRUE) (assuming a large supply of fighter squadrons).  The [[Thomas Bayes|Bayes]] criterion is an approach suitable for such cases.<ref name=Schonhoff/>

Here a [[utility]] is associated with each of four situations:
* <math>U_{11}</math>:  One responds with behavior appropriate to H1 and H1 is true: fighters destroy bomber, incurring fuel, maintenance, and weapons costs, take risk of some being shot down;
* <math>U_{12}</math>:  One responds with behavior appropriate to H1 and H2 is true: fighters sent out, incurring fuel and maintenance costs, bomber location remains unknown;
* <math>U_{21}</math>:  One responds with behavior appropriate to H2 and H1 is true: city destroyed;
* <math>U_{22}</math>:  One responds with behavior appropriate to H2 and H2 is true: fighters stay home, bomber location remains unknown;

As is shown below, what is important are the differences, <math>U_{11} - U_{21}</math> and <math>U_{22} - U_{12}</math>.

Similarly, there are four probabilities, <math>P_{11}</math>, <math>P_{12}</math>, etc., for each of the cases (which are dependent on one's decision strategy).

The Bayes criterion approach is to maximize the expected utility:

<math> E\{U\} = P_{11} \cdot U_{11} + P_{21} \cdot U_{21} + P_{12} \cdot U_{12} + P_{22} \cdot U_{22} </math>

<math> E\{U\} = P_{11} \cdot U_{11} + (1-P_{11}) \cdot U_{21} + P_{12} \cdot U_{12} + (1-P_{12}) \cdot U_{22} </math>

<math> E\{U\} = U_{21} + U_{22} + P_{11} \cdot (U_{11} - U_{21}) - P_{12} \cdot (U_{22} - U_{12}) </math>

Effectively, one may maximize the sum,

<math>U' = P_{11} \cdot (U_{11} - U_{21}) - P_{12} \cdot (U_{22} - U_{12}) </math>,

and make the following substitutions:

<math>P_{11} = \pi_1 \cdot \int_{R_1}p(y|H1)\, dy </math>

<math>P_{12} = \pi_2 \cdot \int_{R_1}p(y|H2)\, dy </math>

where <math>\pi_1</math> and <math>\pi_2</math> are the ''a priori'' probabilities, <math>P(H1)</math> and <math>P(H2)</math>, and <math>R_1</math> is the region of observation events, ''y'', that are responded to as though ''H1'' is true.

<math> \Rightarrow U' = \int_{R_1} \left \{ \pi_1 \cdot (U_{11} - U_{21}) \cdot p(y|H1) - \pi_2 \cdot (U_{22} - U_{12}) \cdot p(y|H2) \right \} \, dy </math>

<math>U'</math> and thus <math>U</math> are maximized by extending <math>R_1</math> over the region where

<math>\pi_1 \cdot (U_{11} - U_{21}) \cdot p(y|H1) - \pi_2 \cdot (U_{22} - U_{12}) \cdot p(y|H2) > 0 </math>

This is accomplished by deciding H2 in case

<math>\pi_2 \cdot (U_{22} - U_{12}) \cdot p(y|H2) \ge \pi_1 \cdot (U_{11} - U_{21}) \cdot p(y|H1) </math>

<math> \Rightarrow L(y) \equiv \frac{p(y|H2)}{p(y|H1)} \ge \frac{\pi_1 \cdot (U_{11} - U_{21})}{\pi_2 \cdot (U_{22} - U_{12})} \equiv \tau_B </math>

and H1 otherwise, where ''L(y)'' is the so-defined ''[[Likelihood function|likelihood ratio]]''.