## What Is Bayes’ Theorem?

Bayes’ Theorem, named after 18th-century British mathematician Thomas Bayes, is a mathematical formula for determining conditional probability. Conditional probability is the likelihood of an outcome occurring, based on a previous outcome having occurred in similar circumstances. Bayes’ theorem provides a way to revise existing predictions or theories (update probabilities) given new or additional evidence.

In finance, Bayes’ Theorem can be used to rate the risk of lending money to potential borrowers. The theorem is also called Bayes’ Rule or Bayes’ Law and is the foundation of the field of Bayesian statistics.

### Key Takeaways

- Bayes’ Theorem allows you to update the predicted probabilities of an event by incorporating new information.
- Bayes’ Theorem was named after 18th-century mathematician Thomas Bayes.
- It is often employed in finance in calculating or updating risk evaluation.
- The theorem has become a useful element in the implementation of machine learning.
- The theorem was unused for two centuries because of the high volume of calculation capacity required to execute its transactions.

## Understanding Bayes’ Theorem

Applications of Bayes’ Theorem are widespread and not limited to the financial realm. For example, Bayes’ theorem can be used to determine the accuracy of medical test results by taking into consideration how likely any given person is to have a disease and the general accuracy of the test. Bayes’ theorem relies on incorporating prior probability distributions in order to generate posterior probabilities.

Prior probability, in Bayesian statistical inference, is the probability of an event occurring before new data is collected. In other words, it represents the best rational assessment of the probability of a particular outcome based on current knowledge before an experiment is performed.

Posterior probability is the revised probability of an event occurring after taking into consideration the new information. Posterior probability is calculated by updating the prior probability using Bayes’ theorem. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred.

## Special Considerations

Bayes’ Theorem thus gives the probability of an event based on new information that is, or may be, related to that event. The formula can also be used to determine how the probability of an event occurring may be affected by hypothetical new information, supposing the new information will turn out to be true.

For instance, consider drawing a single card from a complete deck of 52 cards.

The probability that the card is a king is four divided by 52, which equals 1/13 or approximately 7.69%. Remember that there are four kings in the deck. Now, suppose it is revealed that the selected card is a face card. The probability the selected card is a king, given it is a face card, is four divided by 12, or approximately 33.3%, as there are 12 face cards in a deck.