Bayes Theorem Calculator
Calculate posterior probability from prior, likelihood, and evidence using Bayes' theorem.
Inputs
Posterior P(H|E)
0.180000
Step by step
Numerator: P(E|H) × P(H)
0.9 × 0.01
= 0.009000
Posterior: numerator ÷ P(E)
0.009000 ÷ 0.05
= 0.180000
How it works
Bayes' theorem calculates the probability of a hypothesis given observed evidence: P(H|E) = P(E|H) × P(H) / P(E). It is the foundation of Bayesian inference, spam filters, medical diagnosis models, and any system that updates beliefs based on new data.
Formula
Bayes' Theorem
P(H|E) = P(E|H) * P(H) / P(E)
- P(H)
- Prior probability of hypothesis
- P(E|H)
- Likelihood of evidence given hypothesis
- P(E)
- Total probability of evidence
Frequently Asked Questions
What is the difference between prior and posterior?
The prior is your initial belief about a hypothesis before seeing evidence. The posterior is the updated belief after incorporating the evidence through Bayes' theorem.
How is Bayes' theorem used in ML?
It underpins Naive Bayes classifiers, Bayesian neural networks, Bayesian optimization for hyperparameter tuning, and probabilistic programming frameworks like PyMC and Stan.
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