The curve, the so-called posterior distribution of β, shows which effect sizes are still plausible after taking prior knowledge (prior) and data into account. The blue area to the right of the chosen threshold shows how likely an effect above this boundary is.

Drag the threshold with the mouse and think about: from which effect size would the difference be substantively or clinically relevant to me? For example: should the effect be larger than 2, 2.5, or 5 BDI points?

The percentage directly answers the question: how likely is it that the true effect is larger than my chosen threshold? Example: P(β > 3) = 80% means: after taking the prior and the data into account, the probability that the effect is larger than 3 BDI points is 80%.

A classical 95% confidence interval or a p-value does not answer this question directly. A p-value can, for example, show whether a result would be surprising under a specific null hypothesis. But it does not say: "How probable is it that the true effect is larger than 3?" With the posterior distribution, that probability can be read directly off the area under the curve.

Three curves: Prior (green, dashed — knowledge before the data), Likelihood (orange, dotted — what the data alone say) and Posterior (blue, filled — combined result). Below: a width comparison of 95% intervals.

Switch between non-informative and informative prior. The likelihood (data) stays identical — only the prior changes, and therefore the posterior.

With an informative prior (10 prior studies: μ = 5, σ = 0.6) the posterior narrows considerably. Same data, more precise posterior — prior knowledge formally reduces uncertainty.

Frequentism structurally ignores prior knowledge. Bayes formalises cumulative learning: today's posterior is tomorrow's prior. Scientific progress as an accumulated state of knowledge.