Bayes' Theorem

Bayes’ Theorem describes a mathematical procedure which is used to find out or specify the possibility of one event happening in relation to another event also taking place. In other words, this theorem is applied when it’s necessary to discover conditional probability, when the probability of a certain result is estimated in relation to additional data being obtained. That additional data might be another unrelated event happening, or a new circumstance being considered, or outcomes of an experiment, and so on. With the use of Bayes Theorem, it’s possible to adjust the possibility of a certain outcome after receiving new information. The theorem might also be alternatively called Bayes’ Law, or Bayes Rule in some cases.

Bayes’ Theorem history

The theorem in question was formulated in the 18th century by English philosopher and statistician Thomas Bayes, after whom it was named. As the theorem requires complex computation, it wasn’t generally popular for the following centuries after its creation, with Boolean calculations being used more widely. But with the development and spread of informational technologies and modern ways of calculating, Bayes’ Theorem has been highly appreciated by theorists and statisticians, and now it’s extensively used in various spheres, including medicine, finance, recreational mathematics, production statistics, etc. For Bayes’ Theorem calculations in those spheres, test results are often used and reevaluated with involvement of new significant data, that might affect the probability.

Among other applications, Bayes’ Theorem is also frequently employed when using a statistical technique called Bayes’ inference, which is one of the forms of statistical inference in general. 

Bayes’ Theorem essentials

For better understanding of Bayes’ Theorem, it’s important to be familiar with several related concepts:

• prior probability, which is the probability stated before receiving new information;
• posterior probability, which refers to the refined probability recalculated after receiving new data;
• conditional probability, which depends on additional received information.

So, Bayes’ Theorem provides a possibility to utilize prior probability distribution to get posterior probability containing new information on conditional probability of two events happening in relation to each other. It’s also might be exploited to study possible effects of a circumstance occurring during an event, and the probability of certain outcomes if that circumstance actually takes place.

In financial sphere, Bayes Theorem is often applied when evaluating risks of loan issuing to a certain client, with new facts about that client being gathered and added to the assessment. It’s also used for analysis and risk management, particularly to update statistical distributions after gaining new data on the situation.

Bayes Theorem was also found to be useful in an area of machine learning, due to its unique approach to the relationship between certain data chunks and the probability of that data to be true. Networks based on Bayes principles allows creating complex pictures of many possible outcomes and combinations of several factors happening or not happening. Although in some cases such representations might get too complicated, statisticians and developers still frequently use Bayes Law in their work, as manual application of Bayes Theorem is often more time-consuming, and machine calculations allow reaching the same goals quicker.

Bayes’ Theorem formula

Considering all the abovementioned information, it’s possible to derive a formula for Bayes’ Theorem. Let’s assume there are events A and B, and probabilities P(A) and P(B) of these events, and it’s required to find out the possibility of A taking place given B also happens.

It’s worth noting, though, that A and B are independent events, with one not being the cause of another. So, the Bayes’s theorem is expressed by the following equation:

In this equation, P(A) and P(B) are the already known prior probabilities, which are used for calculation of the conditional posterior probability.