Generalized Differentiable Hamiltonian Dynamic Nested Sampling

Nested sampling reduces the evidence integral to a one-dimensional quadrature over prior volume, at the cost of a complex inner problem: at every iteration, a new sample must be drawn from the prior restricted to a likelihood level set. These constrained draws dominate the running time and can bias the estimator. We present Generalized Differentiable Hamiltonian Dynamic Nested Sampling (GDHDNS), built around three complementary ideas to address this bottleneck. First, the live-point covariance provides a natural local preconditioner: its inverse defines a mass matrix that contracts with the constrained region, costs $\mathcal{O}(N{d}^2)$, and requires no Hessian. Second, a reflected Hamiltonian flow is integrable inside a convex level set, so its endpoint remains correlated with the point seeding it no matter how long the trajectory; this biases the evidence upward as a result. We break that correlation with a preconditioned slice refresh while retaining the flow's gradient-guided efficiency. Third, we show that batched removal has the same volume schedule as sequential removal, permitting unbiased vectorization, and that a run returns $\partial \log \mathcal{Z}/\partial \varphi$ with the evidence. We evaluate GDHDNS on sweeps up to $d=128$, a strong-lensing reconstruction problem, an interatomic potential, and an SDE mixed-effects model. Across the dimension sweeps it is the only method with unbiased evidence at every tested dimension, while running $5$–$24\times$ faster than `dynesty` and up to four orders of magnitude faster than `PolyChord`.