Posterior Probability Calculator
Calculate the posterior probability given a prior distribution and observed evidence.
Inputs
Posterior P(H|E)
0.784615
Evidence P(E)
0.325000
Step by step
P(E) = P(E|H)P(H) + P(E|¬H)P(¬H)
0.85×0.3 + 0.1×0.70
= 0.325000
P(H|E) = P(E|H)×P(H) / P(E)
0.85×0.3 / 0.325000
= 0.784615
How it works
The posterior probability updates our belief in a hypothesis after observing evidence. It uses the total probability rule to compute P(E) from both the true and false hypotheses, then applies Bayes' theorem. This is the fundamental operation in Bayesian inference and sequential updating.
Formula
Posterior via Bayes
P(H|E) = P(E|H)*P(H) / [P(E|H)*P(H) + P(E|~H)*P(~H)]
- P(H)
- Prior belief in hypothesis
- P(E|H)
- Likelihood of evidence if hypothesis true
Frequently Asked Questions
Can the posterior become a prior for the next update?
Yes — this is sequential Bayesian updating. After observing new evidence, today's posterior becomes tomorrow's prior, allowing beliefs to be continuously refined as data arrives.
How does this relate to Bayesian neural networks?
Bayesian NNs maintain a posterior distribution over weights rather than point estimates, enabling uncertainty quantification. The posterior is updated via approximate methods like variational inference.