There is renewed interest in formulating integration as an inference problem,
motivated by obtaining a full distribution over numerical error that can be
propagated through subsequent computation. Current methods, such as Bayesian
Quadrature, demonstrate impressive empirical performance but lack theoretical
analysis. An important challenge is to reconcile these probabilistic
integrators with rigorous convergence guarantees. In this paper, we present the
first probabilistic integrator that admits such theoretical treatment, called
Frank-Wolfe Bayesian Quadrature (FWBQ). Under FWBQ, convergence to the true
value of the integral is shown to be exponential and posterior contraction
rates are proven to be superexponential. In simulations, FWBQ is competitive
with state-of-the-art methods and out-performs alternatives based on
Frank-Wolfe optimisation. Our approach is applied to successfully quantify
numerical error in the solution to a challenging model choice problem in
cellular biology.

本文介绍了第一个能够满足严格收敛保证的概率积分器 ——Frank-Wolfe Bayesian Quadrature（FWBQ），证明了在 FWBQ 下，到达积分的真实值是指数级的，后验收缩速率被证明为超指数级。在仿真中，FWBQ 竞争性地与最先进的现有方法相比，并优于基于 Frank-Wolfe 优化的替代方法。本文的方法成功地应用于量化细胞生物学中一个具有挑战性的模型选择问题中的数值误差。