Bayes Theorem: In Probability Theory And Statistics

Abstract: In probability theory and statistics, Bayes’ theorem

(alternatively Bayes’ law or Bayes' rule) describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For example , “if cancer is related to age, then, using Bayes’ theorem, a person’s age can be used to more accurately assess the probability that they have cancer, compared to the assessment of the probability of cancer made without knowledge of the person's age.”

One of the many applications of Bayes’ theorem is Bayesian inference, a particular approach to statistical inference. When applied, the probabilities involved in Bayes’ theorem may have different probability interpretations. With the Bayesian probability interpretation

Jeffreys wrote that Bayes’ theorem “is to the theory of probability what the Pythagorean theorem is to geometry.”

Bayes' Theorem

The particular formula from Bayesian probability we are going to use is called Bayes' Theorem, sometimes called Bayes' formula or Bayes' rule. This particular rule is most often used to calculate what is called the posterior probability. The posterior probability is the conditional probability of a future uncertain event that is based upon relevant evidence relating to it historically. In other words, if you gain new information or evidence and you need to update the probability of an event occurring, you can use Baye's Theorem to estimate this new probability.

The formula is: P(A) is the probability of A occurring, and is called the prior probability.

P(A|B) is the conditional probability of A given that B occurs. This is the posterior probability due to its variable dependency on B. This assumes that the A is not independent of

This example can be extrapolated to individual companies given changes within their own balance sheets, bonds given changes in credit rating, and many other examples. (Learn how to analyze the balance sheet in our article, Breaking Down The Balance Sheet.)

So what if one does not know the exact probabilities but has only estimates? This is where the subjectivists' view comes strongly into play. Many people put a lot of faith into the estimates and simplified probabilities given by experts in their field; this also gives us the great ability to confidently produce new estimates for new and more complicated questions introduced by those inevitable roadblocks in financial forecasting. Instead of guessing or using simple probability trees to overcome these road blocks, we can now use Bayes' Theorem if we possess the right information with which to start. (See Analyst Forecasts Spell Disaster For Some Stocks to read about the effects of a bad