Direct likelihood correction of Picard HMC

Determine whether Metropolis–Hastings or first-order rejection-sampling correction mechanisms can be applied directly to Picard Hamiltonian Monte Carlo trajectories despite the nonlinearity of the Picard map and the resulting complicated likelihood ratio.

Background

The paper explains that standard correction mechanisms require evaluating the likelihood ratio between a target transition and a proposal, or constructing an unbiased estimator of that ratio. For Picard HMC, the nonlinear Picard map makes this likelihood ratio difficult to express and evaluate, unlike the symplectic leapfrog integrator used in conventional Metropolized HMC.

The Gaussian cloud sampler avoids this unresolved direct-correction issue by tracking Gaussian distributions centered at the hidden ideal Picard iterates and correcting the resulting cloud laws instead. Thus, the question of whether the correction mechanisms themselves can be applied directly to the original Picard HMC transition remains unresolved in the stated discussion.