5 – SL NB 04 Bayes Theorem V1 V2

So, let’s look at a formal version of Bayes Theorem. Initially, we start with an event, and this event could be A or B. The probabilities for each are here, P of A, and P of B. Now, we observe a third event, and that event can either happen or not happen both for A and for B. R is going to help us find more exact probabilities for A and B in the following way. Let’s say we can calculate the probability of R given A, and also, of R complement which is node R given A. And similarly for R given B, and R complement given B. Now, our set of scenarios are these four, R n A, R complement n A, R n B, and R complement n B. But since we know R occurred, then we know that the second and the fourth events are not possible. So, our new universe consists of the two events, R n A and R n B. We calculate the probability for A n R or equivalently A intersection R, and by the Law of Conditional Probability, this is P of A times B of R given A. Similarly, for B intersection R. Now, since these probabilities do not add to one, we just divide them both by their sum so that the new normalized probabilities now do add to one. Thus, we get the following formulas for P of A given R, and P of B given R. These are our new and improved probabilities for A and B after we know that R occurred. Again, P of A and P of B are called the prior probabilities which is, what we knew before we knew that R occurred. P of A given R and P of B given R, are posterior probabilities which is, what we inferred after we knew that R occurred. And here it is, the formula for Bayes Theorem..

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