Posterior Landscape (Arthur's Seat)
Show terrain
Posterior Density (Emerging from Exploration)
Visits are accumulated from iteration 0 with no burn-in discarded, so early transient positions are included in this map.
The Profound Point
With the terrain hidden, you see how MCMC discovers the posterior without knowing the landscape. The chains only know local information (can I go higher from here?) yet collectively map out the full distribution. This is how Bayesian inference works on problems we can't visualise.
Prior Distribution
Click on the terrain to position the prior ellipse
Prior Shape
MCMC Settings
Convergence Diagnostics
What is R-hat?
R-hat (Gelman-Rubin statistic) compares variance within chains to variance between chains. When chains converge to the same distribution:
- R-hat < 1.1 — Chains have converged
- 1.1 ≤ R-hat < 1.2 — Nearly converged
- R-hat ≥ 1.2 — Chains haven't converged yet
Here R-hat is computed on the log-elevation trace only, so it tracks agreement in "height" rather than in position.
Traceplots (Log-elevation)
Watch for burn-in (initial climb) then stable mixing around high-probability regions. The actual log-posterior is 5 × this trace, so the shape is identical up to scale.
The Bayesian Perspective
In Bayesian inference, we don't just find the best parameter values — we explore the full posterior distribution. The posterior tells us which parameter values are plausible given our data.
We use 5 × log(elevation) as the log-posterior density — equivalent to raising the terrain to the 5th power. Sharpening the peaks like this makes the chains' preference for high ground easier to see, and creates more realistic traceplot behaviour: watch for the initial "burn-in" climb as chains find high-probability regions, then stable mixing once they've converged.
One subtlety: the prior ellipse only sets where the chains start. It doesn't enter the target distribution, so the long-run distribution the chains settle into is unaffected by where you place it.
Further reading: the marble-vs-jumping-bean analogy in Statistical Simulation: A Complete Example on JonStats, or the blog version: Part Thirteen — On Marbles and Jumping Beans.