Bayes' Theorem is a mathematical formula for determining conditional probability. In finance, Bayes' theorem can be used to assess the risk of lending money to potential borrowers. The applications of Bayes' theorem are widespread and are not limited to the financial field. Let's see then what Bayes' Theorem is, how it is calculated and how to apply it through a couple of examples.
What is Bayes' Theorem
Bayes' Theorem, named after the 18th century British mathematician Thomas Bayes, is a mathematical formula for determining conditional probability. Conditional probability is the probability that an outcome will occur, given whether a previous outcome has occurred under similar circumstances. Bayes' theorem allows existing predictions or theories to be revised (updating probabilities) based on new or additional evidence. In finance, Bayes' theorem can be used to assess 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.
What is Bayes' Theorem for?
The applications of Bayes' theorem are widespread and are not limited to the financial field. For example, Bayes' theorem can be used to determine the accuracy of medical test results by taking into account the probability that a given person has a disease and the overall accuracy of the test. Bayes' theorem is based on the incorporation of prior probability distributions to generate posterior probabilities. In Bayesian statistical inference, the prior probability is the probability that an event will occur before new data are collected. In other words, it represents the best rational assessment of the probability of a particular outcome based on current knowledge before performing an experiment.
What is the formula for Bayes' Theorem?
Posterior probability is the revised probability that an event will occur after taking into account new information. The posterior probability is calculated by updating the prior probability using Bayes' theorem. In statistical terms, the posterior probability is the probability that event A will occur if event B has occurred. Thus, Bayes' Theorem gives the probability that an event will occur based on new information that is or may be available. 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, assuming the new information turns out to be true.
Explanation of the formula of Bayes' Theorem.
Examples of use of Bayes' Theorem
Below I am going to present two examples of Bayes' Theorem in which the first example applies Bayes' theorem to breathalyzer tests. The second example shows how the formula can be derived in a stock investing example using Nvidia (NVDA).
Numerical example of Bayes' Theorem
As a numerical example, let's imagine that there is a breathalyzer test that is 98% accurate, meaning that 98% of the time, it shows a true positive result for someone who has drunk alcohol, and 98% of the time, shows a true negative result for non-alcohol consumers. Next, let's assume that 0,5% of people consume alcohol. If a randomly selected person tests positive for alcohol, the following calculation can be made to determine the probability that the person is actually an alcohol consumer.
(0,98 x 0,005) / [(0,98 x 0,005) + ((1 – 0,98) x (1 – 0,005))] = 0,0049 / (0,0049 + 0,0199) = 19,76, XNUMX%.
Bayes' Theorem shows that even if a person tested positive in this scenario, there is approximately an 80% chance that they will not consume alcohol.
Deduce the formula of Bayes' theorem
Bayes' Theorem follows simply from the axioms of conditional probability, which is the probability of an event given that another event has occurred. For example, a simple probability question might be: "What is the probability that Nvidia's stock price will fall?" Conditional probability takes this question one step further: “What is the probability that the NVDA stock price will fall given that the Nasdaq Index (NDAQ) fell earlier?” The conditional probability of A given that B has happened can be expressed as: If A is: “The price of NVDA falls”, then P(NVDA) is the probability that NVDA falls; and B is: “the NDAQ has already gone down”, and P(NDAQ) is the probability that the NDAQ has gone down; then the conditional probability expression is read as “the probability that NVDA will fall given a decline in the NDAQ is equal to the probability that the price of NVDA will fall and the NDAQ will fall over the probability of a decline in the NDAQ index.”
P(NVDA|NDAQ) = P(NVDA and NDAQ) / P(NDAQ) P(NVDA and NDAQ) is the probability that both A and B occur. It is also the same as the probability of A occurring multiplied by the probability of B occurring given that A occurs, expressed as P(NVDA) x P(NDAQ|NVDA). The fact that these two expressions are equal leads to Bayes' theorem, which is written as
yes, P(NVDA and NDAQ) = P(NVDA) x P(NDAQ|NVDA) = P(NDAQ) x P(NVDA|NDAQ)
then, P(NVDA|NDAQ) = [P(NVDA) x P(NDAQ|NVDA)] / P(NDAQ). Where P(NVDA) and P(NDAQ) are the probabilities that Nvidia and the Nasdaq will fall, without taking into account each other. The formula explains the relationship between the probability of the hypothesis before seeing the evidence that P(NVDA), and the probability of the hypothesis after obtaining the evidence P(NVDA|NDAQ), given a hypothesis for Nvidia given the evidence in the Nasdaq.