Bayesian Inference Calculator

A probability can never exceed 100% by definition, and Bayes' theorem enforces that automatically as long as the three inputs describe a consistent probability model. Specifically, Evidence Probability P(E) can never be smaller than the joint probability of the hypothesis and evidence both occurring (Likelihood × Prior Probability) -- that joint event is a subset of the evidence occurring at all. If you enter an Evidence Probability below that joint value, this calculator treats it as at least the joint probability so the result stays a valid probability rather than an impossible number above 100%.

Evidence Probability P(E) is how likely you'd see that evidence under ANY circumstance, not just when the hypothesis is true. If the evidence would have been common regardless of whether the hypothesis holds, observing it tells you comparatively little -- so a higher Evidence Probability weakens how much the evidence should shift your belief, which is exactly what dividing by a larger P(E) does to the posterior. That relationship only kicks in once Evidence Probability rises above the joint probability of the hypothesis and evidence both occurring (Likelihood × Prior Probability): below or exactly at that joint value, this calculator holds Evidence Probability at the joint itself (see "Why can't I get a Posterior Probability above 100%?"), so every raw Evidence Probability you enter in that lower range produces the identical Posterior Probability -- a flat plateau at the maximum-possible value, not a smoothly changing one -- until your entered value actually exceeds the joint.

Bayes Factor is the ratio of your Posterior Odds to your Prior Odds -- how much the evidence multiplied your confidence in the hypothesis. A Bayes Factor above 1 means the evidence favored the hypothesis (your odds improved); a value below 1 means the evidence actually weakened the case for the hypothesis, even if the hypothesis still remains more likely than not overall.

For this two-hypothesis setup (H vs. ¬H), Bayes Factor and Likelihood Ratio are the same number: both equal P(E|H) ÷ P(E|¬H), the ratio of how probable the evidence is under the hypothesis versus under its complement. Bayes Factor is conceptually framed as the overall change in odds (Posterior Odds ÷ Prior Odds), while Likelihood Ratio isolates just the evidence's own diagnostic strength -- but by Bayes' theorem applied to both H and ¬H, those two framings algebraically reduce to the identical value, which is why this calculator computes Bayes Factor directly from Likelihood Ratio rather than re-deriving it through the odds ratio (a redundant path that broke down with a division-by-zero at Prior Probability's own 0% and 100% boundary values).

No -- Prior Probability only represents your belief before seeing the evidence. A high Posterior Probability can come from a moderate prior combined with strong evidence (a high Likelihood relative to Evidence Probability), just as a low posterior can result from weak evidence even with a favorable prior. Look at Posterior Probability and Bayes Factor together to judge both where you ended up and how much the evidence itself actually moved the needle.