Hamiltonian Monte Carlo and diagnostics

For a positive differentiable target on unconstrained coordinates $q$, define potential $U(q)=-\log\widetilde\pi(q)$ and introduce Gaussian momentum with positive-definite mass matrix $M$. The joint density is proportional to $\exp[-H(q,p)]$, where kinetic energy is quadratic in momentum. Integrating momentum leaves the desired position marginal because its Gaussian normalizer is constant in position. Constraints require valid coordinate transforms and the associated Jacobian terms; gradients of an incorrectly transformed density faithfully explore the wrong target.

For a twice continuously differentiable potential, exact Hamiltonian flow conserves energy and phase-space volume along trajectories that exist within its domain. It moves along an energy surface, so flow alone does not sample the entire augmented distribution. Momentum refreshment and a valid trajectory proposal rule are essential parts of a sampler. A nontrivial mass matrix changes movement through differently scaled directions without changing the intended position marginal when incorporated consistently. Multimodality, narrow necks, and strongly varying curvature can remain difficult even with exact gradients.

Leapfrog alternates half a momentum update, a full position update, and another half momentum update. For differentiable potential and constant mass, the exact-arithmetic map is reversible after momentum reversal and preserves volume. It does not conserve the Hamiltonian exactly. A standard fixed-length HMC proposal therefore receives a Metropolis acceptance probability based on the change in Hamiltonian. Ignoring this correction generally leaves discretization bias. Floating-point computation also calls for numerical checks rather than interpreting the exact-arithmetic derivation as an implementation guarantee.

Large step sizes can cause inaccurate trajectories or numerical divergence, especially where curvature changes rapidly. Very small step sizes cost more gradient evaluations for the same travel time. Integration length also matters: repeatedly returning near a starting point wastes proposals. The No-U-Turn Sampler adapts trajectory construction using a valid sampling procedure; it is not equivalent to stopping an arbitrary trajectory whenever a picture looks close to a turn. Warmup adaptation learns scales and step sizes, after which retained sampling follows the specified algorithm.

Divergent transitions warn that the numerical trajectory encountered problematic geometry and can indicate biased exploration; inspecting where they occur can suggest reparameterization or stronger modeling constraints. Maximum tree-depth warnings primarily indicate potentially inefficient trajectory truncation and need separate interpretation. Rank-normalized split R-hat compares multiple chains and parts of chains; values near one support agreement for examined quantities but cannot reveal every shared failure, such as all chains remaining in one missed-mode basin.

Bulk and tail effective sample sizes address different summaries, and Monte Carlo standard error should be small enough for the intended decision. A posterior standard deviation measures substantive uncertainty, not simulation accuracy. Inspect parameter combinations and scientifically relevant predictions in addition to individual coordinates, since a hard-to-explore direction can hide in a projection. Diagnostics should be interpreted with traces, initialization, model structure, and numerical warnings. Finite diagnostic success is evidence about the run, not a mathematical proof that the chain converged or that the model is calibrated for reality.