Bayesian Probability Calculator
Define Probabilities
Bayesian Calculation Steps & Result
Input probabilities and click "Calculate" to see results.
Input Probabilities:
P(H) - Prior probability of Hypothesis:N/A
P(E|H) - Prob. of Evidence given Hypothesis is true:N/A
P(E|~H) - Prob. of Evidence given Hypothesis is false:N/A
Intermediate Calculations:
P(~H) - Prior probability of Hypothesis being false (1 - P(H)):N/A
P(E) - Overall probability of observing Evidence:N/A
P(E) = P(E|H) * P(H) + P(E|~H) * P(~H)
Posterior Probability P(H|E) - Updated probability of Hypothesis given Evidence
N/A
Understanding Bayes' Theorem & Export
Bayes' Theorem Explained
Bayes' Theorem is a fundamental concept in probability theory and statistics that describes how to update the probability of a hypothesis based on new evidence. It's a way to formally incorporate new information into your existing beliefs.
The theorem is stated mathematically as:
P(H|E) = [P(E|H) * P(H)] / P(E)
Where:- P(H|E) is the Posterior Probability: The probability of hypothesis H being true *after* observing evidence E. This is what you calculate.
- P(E|H) is the Likelihood: The probability of observing evidence E *if* hypothesis H is true.
- P(H) is the Prior Probability: Your initial belief in the probability of hypothesis H being true *before* observing evidence E.
- P(E) is the Probability of the Evidence (also known as marginal likelihood or normalizing constant): The overall probability of observing evidence E, whether H is true or false. It's calculated using the law of total probability:
P(E) = P(E|H) * P(H) + P(E|~H) * P(~H)
(where P(~H) is the probability of H being false, i.e., 1 - P(H)).
Applications: Medical diagnosis (e.g., probability of disease given test results), spam filtering, A/B testing analysis, updating beliefs in scientific research, and many areas of machine learning and AI.