Bay’s Theorem

Bayes' Rule

Definition: 

Bayes' theorem, named after Thomas Bayes, is a mathematical formula used to calculate conditional probability. It determines the probability of an event occurring given the probability of related events.

Conditional Probability:

It addresses the probability of an event A occurring given that event B has already occurred.

Formula: P(A∣B) = $\frac{P(B\mid A)\cdot P\left(A\right)}{P\left(B\right)}$

• Interpretation:
  • P(A∣B): Probability of event A given event B.
  • P(B∣A): Probability of event B given event A.
  • P(A) and P(B): Probabilities of events A and B independently.

Example:

• Scenario: Determining the probability of a patient having liver disease given that they are an alcoholic.
• Given:
  • P(A): Probability of liver disease in the population (10%).
  • P(B): Probability of being an alcoholic (5%).
  • P(B∣A): Probability that an alcoholic has liver disease (7%).
• Calculation:

• P(A∣B) = $\frac{P(B\mid A)\cdot P\left(A\right)}{P\left(B\right)}$

• P(A∣B) = $\frac{0.07\cdot 0.1}{0.05}=0.14 OR 14\%$

• Implication: Even though being an alcoholic increases the probability of having liver disease to 14% (from the base rate of 10%), it remains unlikely for any specific patient.

Practical Use:

• Medical Diagnostics: Assessing disease probabilities based on test results.
• Statistical Inference: Incorporating new information to update prior beliefs.

Considerations:

• False Positives: Tests with high false positive rates can skew probabilities.
• Complexity: Requires careful definition of events and tests for accurate application.

Bayes' theorem is fundamental in various fields, including medicine, economics, and artificial intelligence, providing a structured approach to probabilistic reasoning and decision-making.