See sampled points build a probability distribution, compare samplers using modern diagnostics, and inspect normalized neural energy models. Failed mixing and finite-step bias remain visible alongside stronger baselines.
Repository · Equations and contracts · Measured comparisonsCovariance has unit diagonal and correlation ρ. ULA uses x′ = x − ηΣ⁻¹x + √(2η)z. A stable chain can still have the wrong stationary covariance.
Equal axes; one-standard-deviation covariance ellipses. Teal: exact target. Gold: ULA stationary law when stable. This is an analytic calculation, not an estimated posterior region.
Standard Gaussian position target, unit mass, initial position 0.8 and momentum 0.3. Leapfrog conserves energy approximately; the acceptance step corrects its integration error.
Owned HMC experiments also randomize trajectory length. Adaptive HMC tunes its step size and diagonal geometry in warmup; NUTS uses the maintained Pyro implementation. Warmup states never enter reported diagnostics.
Gray contours: target density. Colored dots: retained samples. Colors identify independent chains; the larger dots mark their current positions.
Colored bars: accumulated sampled marginal density. Black curve: independent exact-draw reference when available. Histogram area uses all displayed draws, including out-of-window states.
Metrics and protocol · Numeric chain/model evidence
Rhat below 1.01, bulk and tail ESS above 400, and zero divergences define the displayed diagnostic screen. It is neither a proof of convergence nor a correction for ULA/SGLD/SGHMC bias. Repeated states are retained; no thinning is used for inference.