The canonical cultivation-theory design: heavy television viewers are hypothesized to overestimate the prevalence of crime more than light viewers. There is no closed form for "how likely is the difference, δ = θ_A − θ_B, to exceed zero?" so this page draws thousands of posterior samples and compares them directly: a Monte Carlo posterior comparison.
Both groups share a mildly informative Beta(2,2) prior, genuinely uncertain, only ruling out the extremes. This is the default weakly informative prior recommended for binomial proportions (see Kruschke, 2015; Gelman et al., 2013).
The sentence a p-value cannot write: "Given this data and prior, there is roughly an – chance that heavy viewers overestimate crime more than light viewers." This is a direct probability statement about the effect, not a long-run frequency about an estimator.
Conventional benchmarks (Cohen, 1988): |h| ≈ 0.2 small, ≈ 0.5 medium, ≈ 0.8 large. The arcsine transform equalizes variance across the 0–1 range, so h behaves far better than a raw percentage-point gap, especially near the boundaries, which is why it's the standard effect size for comparing proportions.