The optimal refreshment rate of the Bouncy Particle Sampler: uniqueness to within 10⁻⁷, a certified critical point at 1.423266366804207…, and the event ratio 0.7811

Abstract

For the Bouncy Particle Sampler on a standard Gaussian target in ℝᵈ with velocity refreshment rate ρ, Bierkens, Kamatani and Roberts proved that the log-density process, sped up by d, converges to an Ornstein–Uhlenbeck process whose diffusion coefficient σ²(ρ) is the Laplace transform at ρ of the autocovariance of an explicit one-dimensional piecewise deterministic process. They located the maximiser of σ² by Monte Carlo at "around 1.424" and derived from it the tuning rule that refreshments should make up 0.7812 of all events. Shiba and Kamatani have since obtained σ² in closed form, in terms of the Mills-ratio function M(a)=e^(a²/2)∫ₐ^∞ e^(-t²/2) dt, and located the maximiser numerically at 1.423…. Starting from that closed form we prove that σ² is strictly increasing on [1/2,1.4232663] and strictly decreasing on [1.4232664,2], that σ²(ρ)<σ²(ρ^*) for ρ ∈ [frac110,1/2] and for ρ ≥ 2, and hence that σ² attains its maximum on [frac110,∞) and every maximiser lies in a window of width 10⁻⁷; that (σ²)' changes sign inside (1.423266366804207, 1.423266366804208); that 1.8383133642103960<σ²(1.423266366804207)<1.8383133642103961; and that the ratio of refreshment events to all events at every point of that bracket lies in (0.781066629766458, 0.781066629766459). So the printed constants 1.424 and 0.7812 are both off in their fourth significant figure — to four figures the values are 1.423 and 0.7811 — and the tuning rule, named "the 78.12 version and targeted verbatim by an adaptive sampler of Bertazzi and Bierkens, should read 0.78107 rather than 0.7812. Literal uniqueness of the maximiser inside the 10⁻⁷ window and the range (0,frac110) are left open. Every statement is verified in Lean 4 against Mathlib: the numerical steps are exact rational interval enclosures (16,576 derivative boxes and 8,256 value boxes) checked by compiled evaluation, and the passage from an enclosure to a real inequality is a kernel proof.

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