The posterior distribution of β — the plausibility of all effect sizes given prior knowledge and data. The blue area to the right of the threshold shows how much probability mass lies in that region.

Drag the threshold with the mouse and ask yourself: above which β value is the effect clinically meaningful? E.g. 2.2 or 5.0 BDI points — the threshold is freely adjustable.

The percentage = P(β > threshold | data, prior) — a direct probability statement about the parameter. Trying multiple thresholds is no problem: the posterior does not change, only the query does.

The 95% CI is not a probability statement about β. P(β > 3) cannot be read off directly. Testing the same dataset for β > 3 and β > 2.2 constitutes multiple testing with an inflated error rate. Bayesian: no problem — the posterior is fixed.

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.

Kruschke (2014) — HDI/ROPE decision logic
McElreath (2020) — Bayesian workflow, prior as cumulative knowledge