Editing Detection theory (section)

==Mathematics==

=== P(H1|y) > P(H2|y) / MAP testing ===
In the case of making a decision between two [[Hypothesis|hypotheses]], ''H1'', absent, and ''H2'', present, in the event of a particular [[observation]], ''y'', a classical approach is to choose ''H1'' when ''p(H1|y) > p(H2|y)'' and ''H2'' in the reverse case.<ref name=Schonhoff>Schonhoff, T.A. and Giordano, A.A. (2006) ''Detection and Estimation Theory and Its Applications''. New Jersey: Pearson Education ({{ISBN|0-13-089499-0}})</ref> In the event that the two ''[[a posteriori]]'' [[probability|probabilities]] are equal, one might choose to default to a single choice (either always choose ''H1'' or always choose ''H2''), or might randomly select either ''H1'' or ''H2''. The ''[[A priori and a posteriori|a priori]]'' probabilities of ''H1'' and ''H2'' can guide this choice, e.g. by always choosing the hypothesis with the higher ''a priori'' probability.

When taking this approach, usually what one knows are the conditional probabilities, ''p(y|H1)'' and ''p(y|H2)'', and the ''[[A priori and a posteriori|a priori]]'' probabilities <math>p(H1) = \pi_1</math> and <math>p(H2) = \pi_2</math>.  In this case,

<math>p(H1|y) = \frac{p(y|H1) \cdot \pi_1}{p(y)} </math>,

<math>p(H2|y) = \frac{p(y|H2) \cdot \pi_2}{p(y)} </math>

where ''p(y)'' is the total probability of event ''y'',

<math> p(y|H1) \cdot \pi_1 + p(y|H2) \cdot \pi_2 </math>.

''H2'' is chosen in case

<math> \frac{p(y|H2) \cdot \pi_2}{p(y|H1) \cdot \pi_1 + p(y|H2) \cdot \pi_2} \ge \frac{p(y|H1) \cdot \pi_1}{p(y|H1) \cdot \pi_1 + p(y|H2) \cdot \pi_2} </math>

<math> \Rightarrow \frac{p(y|H2)}{p(y|H1)} \ge \frac{\pi_1}{\pi_2}</math>

and ''H1'' otherwise.

Often, the ratio <math>\frac{\pi_1}{\pi_2}</math> is called <math>\tau_{MAP}</math> and <math>\frac{p(y|H2)}{p(y|H1)}</math> is called <math>L(y)</math>, the ''[[Likelihood function|likelihood ratio]]''.

Using this terminology, ''H2'' is chosen in case <math>L(y) \ge \tau_{MAP}</math>.  This is called MAP testing, where MAP stands for "maximum ''a posteriori''").

Taking this approach minimizes the expected number of errors one will make.

===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]]''.

===Normal distribution models===

Das and Geisler <ref name="Das">{{cite journal |last1=Das|first1=Abhranil|last2=Geisler|first2=Wilson|arxiv=2012.14331|title=A method to integrate and classify normal distributions|journal=Journal of Vision |year=2021 |volume=21 |issue=10 |page=1 |doi=10.1167/jov.21.10.1 |pmid=34468706 |pmc=8419883 }}</ref> extended the results of signal detection theory for normally distributed stimuli, and derived methods of computing the error rate and [[confusion matrix]] for [[Ideal observer analysis|ideal observers]] and non-ideal observers for detecting and categorizing univariate and multivariate normal signals from two or more categories.