Editing Detection theory (section)

=== 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.