Establish the prior probability of an event. Let Event A be the event that a stock's price rises, and the probability of Event A, or P(A) in Bayes' Theorem, be 50 percent. This is the prior probability of Event A -- the stock's price has proven to rise in 50 percent of the cases studied.
Introduce Event B. Let Event B be the event that interest rates rise, and establish its probability, say 50 percent. Call this P(B) -- it is the probability of Event B happening, and is independent of Event A.
Connect Event A and Event B. Calculate the likelihood, based on new data, that interest rates will rise (Event B) when stock prices rise (Event A). Say that the likelihood -- this is called P(B/A) -- is 20 percent.
Calculate the posterior probability using Bayes' Theorem. The theorem says that the posterior probability P(A/B) equals the likelihood (20 percent) times the prior probability of Event A (50 percent), divided by the prior probability of Event B (50 percent). The posterior probability of stock prices rising when interest rates rise is then (.20 x .50) divided by .50, or 5 percent.