Bayes theorem

Bayes' Theorem

Bayes' Theorem is a fundamental concept in probability theory and statistics, providing a mathematical framework for updating the probability of a hypothesis based on new evidence. Named after the Reverend Thomas Bayes, an 18th-century statistician and theologian, the theorem has wide applications in various fields, including medicine, finance, and machine learning.

Definition

Bayes' Theorem describes the probability of an event, based on prior knowledge of conditions that might be related to the event. The theorem is expressed mathematically as:

\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]

Where:

• \(P(A|B)\) is the posterior probability, the probability of event \(A\) occurring given that \(B\) is true.
• \(P(B|A)\) is the likelihood, the probability of event \(B\) occurring given that \(A\) is true.
• \(P(A)\) is the prior probability, the initial probability of event \(A\).
• \(P(B)\) is the marginal likelihood, the total probability of event \(B\).

Applications in Medicine

In the medical field, Bayes' Theorem is particularly useful for diagnostic testing. It allows clinicians to update the probability of a disease as new test results become available. For example, if a patient tests positive for a disease, Bayes' Theorem can be used to calculate the probability that the patient actually has the disease, taking into account the test's sensitivity and specificity.

Example

Consider a disease that affects 1% of the population. A test for the disease has a sensitivity of 99% and a specificity of 95%. If a patient tests positive, Bayes' Theorem can be used to calculate the probability that the patient actually has the disease.

\[ P(Disease|Positive) = \frac{P(Positive|Disease) \cdot P(Disease)}{P(Positive)} \]

Where:

• \(P(Positive|Disease) = 0.99\)
• \(P(Disease) = 0.01\)
• \(P(Positive) = P(Positive|Disease) \cdot P(Disease) + P(Positive|No\ Disease) \cdot P(No\ Disease)\)

Historical Background

Thomas Bayes was an English statistician and minister who formulated the theorem that bears his name. His work was published posthumously in 1763 by his friend Richard Price. The theorem was initially met with skepticism but gained prominence in the 20th century with the advent of computational methods that allowed for its practical application.

Mathematical Derivation

Bayes' Theorem can be derived from the definition of conditional probability. Given two events \(A\) and \(B\), the conditional probability of \(A\) given \(B\) is defined as:

\[ P(A|B) = \frac{P(A \cap B)}{P(B)} \]

Similarly, the conditional probability of \(B\) given \(A\) is:

\[ P(B|A) = \frac{P(A \cap B)}{P(A)} \]

Rearranging the second equation gives:

\[ P(A \cap B) = P(B|A) \cdot P(A) \]

Substituting this into the first equation yields Bayes' Theorem:

\[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \]

Also see

• Probability theory
• Conditional probability
• Statistical inference
• Thomas Bayes
• Likelihood function