NUTS Sampler: 2D Contour & Phase-Space View

No-U-Turn Sampler (2D)

Step size ε 0.12

Max tree depth 6

Drag the main contour map to pan, scroll/pinch to zoom. The two lower panels track momentum phase-space and Hamiltonian energy for the trajectory currently being grown.

Live Stats

Samples drawn

Tree depth (last)

–

Leapfrog steps (last)

–

Avg depth

–

Energy drift (last)

–

Zoom

1.0×

How it works

NUTS targets a posterior p(θ) by simulating a physical particle on the energy surface U(θ) = −log p(θ), with an auxiliary momentum r and total energy H(θ,r) = U(θ) + ½|r|². Each proposal resamples r ~ N(0,I), then integrates Hamilton's equations with the leapfrog scheme:

r(½) = r − (ε/2)∇U(θ)
θ'   = θ + ε·r(½)
r'   = r(½) − (ε/2)∇U(θ')

NUTS removes the fixed-trajectory-length tuning knob of plain HMC: it builds a balanced binary tree of leapfrog steps, doubling in a random direction (forward or backward) at each level, until the endpoints start turning back toward each other:

stop when (θ⁺ − θ⁻)·r⁻ < 0  or  (θ⁺ − θ⁻)·r⁺ < 0

That is the "no U-turn" criterion. A sample is then drawn from the states visited along the tree, weighted so the draw is still a valid sample from p(θ) — no whole-trajectory Metropolis rejection needed.

Main contour map — a top-down view of p(θ₁,θ₂) with the leapfrog tree path (cyan) and accumulated draws (violet). Thin strips along the bottom/right edges are live marginal histograms of θ₁ and θ₂ built from the accumulated samples.
Phase-space panel — plots the auxiliary momentum (r₁,r₂) visited along the current tree; a well-behaved leapfrog trajectory traces a smooth, roughly closed loop.
Energy panel — plots H(θ,r) at each leapfrog step of the current tree against the starting energy (dashed line); good step sizes keep the drift small, this is exactly the energy diagnostic used to detect divergences in Stan/PyMC.
ε (step size) — smaller steps track curvature more faithfully but need more leapfrog evaluations; larger steps risk visible energy drift (numerical divergence).
Max tree depth — a safety cap on doublings (2^depth leapfrog steps worst case) if no U-turn ever triggers.

This is the exact algorithm (Hoffman & Gelman, 2014) that powers Stan, PyMC and NumPyro. The bent "banana" ridge is a classic torture-test posterior whose curving valley defeats naive random-walk Metropolis but NUTS glides along with ease.

Drawn samples Leapfrog tree path Current chain position

Momentum phase space (r₁, r₂)

Energy H(θ,r) vs leapfrog step

No-U-Turn Sampler: 2D Contour, Phase-Space and Energy View