What does Hamiltonian Monte Carlo's momentum variable buy over random-walk Metropolis proposals?

## The problem with random-walk proposals A random-walk proposal has no idea where it is. It offers `theta + noise` in a direction chosen without reference to the posterior's shape, so to be accepted at a reasonable rate the step must be small relative to the posterior's tightest direction. The chain then diffuses: distance covered grows like the square root of the number of iterations, and in high dimensions almost every proposed direction points out of the thin shell where the posterior mass actually lives. Cranking the step size up only converts crawling into rejection. Hamiltonian Monte Carlo attacks this by giving the proposal a sense of direction. ## The mechanical picture Define a potential energy as the negative log unnormalised posterior, `U(theta) = -log p_unnorm(theta)`. High posterior density is low altitude; the posterior's bulk is a valley, and the tails are the slopes rising away from it. Each iteration: 1. Draw a fresh momentum vector `m`, usually from a multivariate normal, independently of the current position. 2. Simulate the motion of a frictionless particle starting at `theta` with momentum `m`, moving over the surface `U` for some fixed simulated time. 3. Take the end point of that trajectory as the proposed `theta'`, and accept or reject it with a Metropolis test. The intuition is the puck. Shove it and it slides down into the valley, gaining speed, then climbs the far slope, decelerates, and turns back — always tracing a path that follows the terrain instead of ignoring it. Total energy, potential plus kinetic, is conserved along the exact path, which is precisely the statement that the proposal ends somewhere with comparable posterior density to where it started, however far away that is. This is why long moves survive the acceptance test: distance and acceptance stop being in tension, which is exactly the tradeoff that traps a random walk. The momentum being redrawn each iteration is what makes the chain explore rather than orbit forever on one energy level. ## Why there is still an accept-reject step The dynamics cannot be simulated exactly. In practice the trajectory is approximated with a leapfrog integrator that alternates half-steps of momentum update and full steps of position update, governed by a step size and a number of steps. The integrator conserves energy well but not perfectly, so the end point carries a small energy error. The final Metropolis test uses that error: the acceptance probability is one when energy is exactly conserved and drops as the error grows. This is what makes HMC exact despite being built on an approximate simulation, and it is why an implementation with no acceptance test is simply wrong. The two knobs interact. Too large a step size makes the integrator inaccurate, energy error blows up, and acceptance collapses. Too small a step size makes each trajectory expensive in gradient evaluations. Too few steps and the proposal barely leaves the starting point, reducing HMC to an expensive random walk; too many and the trajectory can curve back towards where it started, spending gradients to go nowhere. Adaptive variants exist that pick trajectory length automatically rather than requiring a hand-set number of steps. ## The requirements it imposes HMC needs the gradient of the log posterior with respect to every parameter. Three consequences follow. First, the parameter space must be continuous and the log posterior differentiable. Discrete parameters cannot be moved by these dynamics at all; the standard treatment is to marginalise them out of the model analytically and recover them afterwards if needed, or to update them with a different kind of step. Second, constrained parameters — a variance that must be positive, a probability in the unit interval, a simplex — are handled by transforming to an unconstrained space and carrying the Jacobian term, so the dynamics run on a space with no walls to hit. Third, cost per proposal is high. One HMC proposal costs as many gradient evaluations as it has leapfrog steps, so it may be tens or hundreds of times more expensive than one random-walk proposal. The bargain only pays when those expensive proposals move much further and are accepted, which is generally true in moderate to high dimensions and may not be true for a small, well-behaved, cheap-to-evaluate model. ## When it is not the answer HMC is not universally superior. For a low-dimensional model with conditionally conjugate structure, Gibbs sampling may be simpler, tuning-free and entirely sufficient. For a model with discrete latent structure that cannot be marginalised, HMC does not apply directly. And difficult posterior geometry — regions where curvature changes sharply with position — remains difficult for HMC too; momentum makes exploration efficient where the geometry is reasonable, it does not make pathological geometry disappear. An interviewer asking this question usually wants the physical intuition stated clearly, the reason distant proposals still get accepted, and honest acknowledgment of the gradient requirement and the cost per proposal. Reciting the leapfrog recursion without the intuition, or claiming HMC needs no acceptance step, both read as memorised rather than understood.