Posterior Probability Calculator
Calculate the posterior probability given a prior distribution and observed evidence.
About Posterior Probability Calculator
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
Worked example
With the default values below, Posterior Probability Calculator returns Posterior P(H|E) of 0.784615. Change any input above and every figure here is recalculated in your browser.
Inputs used
- Prior P(H)accepts 0–1
- 0.3
- P(Evidence | H true)accepts 0–1
- 0.85
- P(Evidence | H false)accepts 0–1
- 0.1
Results
- Posterior P(H|E)
- 0.784615
- Evidence P(E)
- 0.325000
Step by step
- 1P(E) = P(E|H)P(H) + P(E|¬H)P(¬H)= 0.325000
- 2P(H|E) = P(E|H)×P(H) / P(E)= 0.784615
Assumptions and limits
- The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
- In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
- Figures are rounded for display to 6 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.
How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.
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.