by the Bayes rule. If the prior on $\mathcal{C}[0,1]$ is that of a Brownian Motion (i.e. $f(t)=B_t$ where $B_t$ is a Brownian motion and $B_0$ is normal), then $\E\big[f(x)\big|f(x_1),\ldots,f(x_n)\big]$ is the piecewise linear interpolation of $f$ between the points $x_1,\ldots,x_n$ and one re-discovers the trapezoidal quadrature rule. If the prior on $\mathcal{C}[0,1]$ is that of the first integral of a Brownian Motion (i.e. $f(t)\sim \int_0^t B_s\,ds$) then the posterior $\E\big[f(x)\big|f(x_1),\ldots,f(x_n)\big]$ is the cubic spline interpolant and integrating $k$ times yields splines of order $2k+1$.

Subsequent to Poincar[é]{}’s early discovery [@Poincare:1896], Sul’din [@Suldin1959] and (in particular) Larkin [@Larkin1972] initiated the systematic investigation of the correspondence between conditioning Gaussian measures/processes and numerical approximation. As noted by Larkin [@Larkin1972], despite Sard’s introduction of probabilistic concepts in the theory of linear approximation [@Sard1963], and Kimeldorf and Wahba’s exposition [@Kimeldorf70] of the correspondence between Bayesian estimation and spline smoothing/interpolation, the application of probabilistic concepts and techniques to numerical integration/approximation “attracted little attention among numerical analysts” (perhaps due to the counterintuitive nature of the process of randomizing a known function). However, a natural framework for understanding this process of randomization can be found in the pioneering works of Wo[ź]{}niakowski [@Woniakowski1986], Packel [@Packel1987], and Traub, Wasilkowski, and 