Changes

In other words, there is only a 16% chance that a person testing positive actually has the disease.

In other words, there is only a 16% chance that a person testing positive actually has the disease.

An extended form of Bayes's theorem is obtained by noting that it applies to [[probability distributions]] as well as to events. Let ''y'' be a (vector valued) observable quantity that we want to use to estimate some unknown, unobservable (vector valued) quantity <math>\theta</math>. Prior to seeing the data ''y'', we summarise our knowledge about <math>\theta</math> by a probability distribution <math>p(\theta)</math>. Assume that we have a model of the relationship between ''y'' and <math>theta</math>. Call this <math>p(y|\theta)</math>. We can use Bayes' theorem to update our knowledge of <math>\theta</math> by incorporating the information contained in the observed data ''y''.

An extended form of Bayes's theorem is obtained by noting that it applies to [[probability distribution]]s as well as to events. Let ''y'' be a (vector valued) observable quantity that we want to use to estimate some unknown, unobservable (vector valued) quantity <math>\theta</math>. Prior to seeing the data ''y'', we summarise our knowledge about <math>\theta</math> by a probability distribution <math>p(\theta)</math>. Assume that we have a model of the relationship between ''y'' and <math>\theta</math>. Call this <math>p(y|\theta)</math>. We can use Bayes' theorem to update our knowledge of <math>\theta</math> by incorporating the information contained in the observed data ''y''.

We have                         

We have                         

               <math>p(\theta|y) = p(\theta)p(y|\theta)/p(y)</math>

               <math>p(\theta|y) = p(\theta)p(y|\theta)/p(y)</math>