I'm currently writing a blog post that uses Bayes' Law but don't want to muddy the post with a review in layman's terms. So I have something to link, here is a short description and a chance to flex my teaching muscles before the school year starts.
For those who aren't sure, Bayes' Law tells us that the probability event occurs given we know that event has occurred can easily be computed. It is written as vertical bar is meant like the word "given", in other words, the event is distinct from the event i.e. given . Bayes' law, states that
This effectively is a re-scaling of the events by the total probability of the given event: .
For example, if is the event that a is rolled on a fair die and is the event that the roll is odd. We know of course that since half of the rolls are odd. The event in this case is the same as since the roll can only be if the roll is already odd. Thus
and we can compute the conditional probability
As we expect, one out of every three odd rolls is a .
Bayes' Law Extended Form
Instead of considering a single event we can consider a range of possible events occur. We require that one of these -events must occur and that they cover all possible events that could occur. For example is the event that H2O is vapor, is the event that H2O is water and is the event that H2O is ice.
In such a case we know that since the -events are distinct
Using Bayes' law, we can reinterpret as
and the above becomes
The same is true if we replace with an arbitrary number of events .