Bayesian theory is a statistical framework that treats probability as a degree of belief or confidence. Unlike traditional statistics, Bayesian theory allows us to update our beliefs as new evidence becomes available. This makes it particularly powerful for making decisions under uncertainty.
The mathematical foundation of Bayesian theory is Bayes theorem. The formula shows how to calculate the posterior probability P of H given E, which equals the likelihood P of E given H, times the prior probability P of H, divided by the evidence P of E. This elegant formula allows us to update our beliefs systematically.
Let's consider a medical diagnosis example. Suppose a disease affects only one percent of the population, and we have a test that is ninety five percent accurate. If someone tests positive, what's the probability they actually have the disease? This is a classic example where Bayesian thinking reveals surprising results.
Let's calculate the answer using Bayes theorem. First, we calculate P of Positive, which is zero point nine five times zero point zero one plus zero point zero five times zero point nine nine, equals zero point zero five nine. Then applying Bayes formula, P of Disease given Positive equals zero point nine five times zero point zero one divided by zero point zero five nine, which equals zero point one six one. Surprisingly, even with a positive test, there's only a sixteen point one percent chance of actually having the disease!
To summarize what we've learned: Bayesian theory provides a mathematical framework for updating beliefs with new evidence. Prior knowledge significantly affects our conclusions. Counterintuitive results like our medical example reveal the importance of considering base rates. This theory has wide applications in medicine, artificial intelligence, and decision making under uncertainty.
The mathematical foundation of Bayesian theory is Bayes theorem. The formula shows how to calculate the posterior probability P of H given E, which equals the likelihood P of E given H, times the prior probability P of H, divided by the evidence P of E. This elegant formula allows us to update our beliefs systematically.
Let's consider a medical diagnosis example. Suppose a disease affects only one percent of the population, and we have a test that is ninety five percent accurate. If someone tests positive, what's the probability they actually have the disease? This is a classic example where Bayesian thinking reveals surprising results.
Let's calculate the answer using Bayes theorem. First, we calculate P of Positive, which is zero point nine five times zero point zero one plus zero point zero five times zero point nine nine, equals zero point zero five nine. Then applying Bayes formula, P of Disease given Positive equals zero point nine five times zero point zero one divided by zero point zero five nine, which equals zero point one six one. Surprisingly, even with a positive test, there's only a sixteen point one percent chance of actually having the disease!
To summarize what we've learned: Bayesian theory provides a mathematical framework for updating beliefs with new evidence. Prior knowledge significantly affects our conclusions. Counterintuitive results like our medical example reveal the importance of considering base rates. This theory has wide applications in medicine, artificial intelligence, and decision making under uncertainty.