Editing Bayes' theorem (section)

=== Bayes' rule in odds form ===
Bayes' theorem in [[odds|odds form]] is:{{cn|date=April 2025}}

:<math>O(A_1:A_2\vert B) = O(A_1:A_2) \cdot \Lambda(A_1:A_2\vert B) </math>

where

:<math>\Lambda(A_1:A_2\vert B) = \frac{P(B\vert A_1)}{P(B\vert A_2)}</math>

is called the [[Bayes factor]] or [[likelihood ratio]]. The odds between two events is simply the ratio of the probabilities of the two events. Thus:

:<math>O(A_1:A_2) = \frac{P(A_1)}{P(A_2)},</math>

:<math>O(A_1:A_2\vert  B) = \frac{P(A_1\vert  B)}{P(A_2\vert  B)},</math>

Thus the rule says that the posterior odds are the prior odds times the [[Bayes factor]]; in other words, the posterior is proportional to the prior times the likelihood.

In the special case that <math>A_1 = A</math> and <math>A_2 = \neg A</math>, one writes <math>O(A)=O(A:\neg A) =P(A)/(1-P(A))</math>, and uses a similar abbreviation for the Bayes factor and for the conditional odds. The odds on <math>A</math> is by definition the odds for and against <math>A</math>. Bayes' rule can then be written in the abbreviated form

:<math>O(A\vert B) = O(A)  \cdot \Lambda(A\vert B) ,</math>

or, in words, the posterior odds on <math>A</math> equals the prior odds on <math>A</math> times the likelihood ratio for <math>A</math> given information <math>B</math>. In short, '''posterior odds equals prior odds times likelihood ratio'''.

For example, if a medical test has a [[Sensitivity and specificity|sensitivity]] of 90% and a [[Sensitivity and specificity|specificity]] of 91%, then the positive Bayes factor is <math>\Lambda_+ = P(\text{True Positive})/P(\text{False Positive}) = 90\%/(100\%-91\%)=10</math>. Now, if the [[prevalence]] of this disease is 9.09%, and if we take that as the prior probability, then the prior odds is about 1:10. So after receiving a positive test result, the posterior odds of having the disease becomes 1:1, which means that the posterior probability of having the disease is 50%. If a second test is performed in serial testing, and that also turns out to be positive, then the posterior odds of having the disease becomes 10:1, which means a posterior probability of about 90.91%. The negative Bayes factor can be calculated to be 91%/(100%-90%)=9.1, so if the second test turns out to be negative, then the posterior odds of having the disease is 1:9.1, which means a posterior probability of about 9.9%.

The example above can also be understood with more solid numbers: assume the patient taking the test is from a group of 1,000 people, 91 of whom have the disease (prevalence of 9.1%). If all 1,000 take the test, 82 of those with the disease will get a true positive result (sensitivity of 90.1%), 9 of those with the disease will get a false negative result ([[False positives and false negatives|false negative rate]] of 9.9%), 827 of those without the disease will get a true negative result (specificity of 91.0%), and 82 of those without the disease will get a false positive result (false positive rate of 9.0%). Before taking any test, the patient's odds for having the disease is 91:909. After receiving a positive result, the patient's odds for having the disease is

:<math>\frac{91}{909}\times\frac{90.1\%}{9.0\%}=\frac{91\times90.1\%}{909\times9.0\%}=1:1</math>
which is consistent with the fact that there are 82 true positives and 82 false positives in the group of 1,000.