Bayes Theorem

Bayes' Theorem is a fundamental principle of probability and statistics that explains how the probability of an event or hypothesis changes when new evidence becomes available. Rather than relying only on initial assumptions, it provides a systematic method for updating beliefs based on observed data. The theorem is mathematically expressed as P(A|B) = [P(B|A) × P(A)] / P(B), where P(A|B) is the posterior probability, representing the likelihood of event A after observing event B. P(B|A) is the likelihood, or the probability of observing B if A is true. P(A) is the prior probability, reflecting the initial belief about A before considering new evidence, while P(B) is the overall probability of observing B. By combining prior knowledge with new information, Bayes' Theorem produces more accurate probability estimates. It is extensively used in machine learning, artificial intelligence, medical diagnosis, fraud detection, spam email filtering, weather forecasting, financial risk analysis, and decision-making systems, where probabilities are continuously refined as additional data becomes available.