Active Bayesian Quadrature and Bayesian Monte Carlo

Summary

Because a Gaussian BQ model has a posterior variance that is independent of the observed values, its optimal design is fixed a priori and no genuine adaptivity is possible. Adaptive Bayesian quadrature schemes therefore adopt non-Gaussian, warped models — modelling the log-integrand (BBQ) or the square-root of the integrand (WSABI) — so that non-negative integrands (likelihoods, model evidence) are respected and evaluations can inform where to look next. These schemes use uncertainty sampling on the integrand variance to place nodes actively, turning the solver into a true learning agent; empirically WSABI beats Monte Carlo and annealed importance sampling in wall-clock time. Bayesian Monte Carlo (O’Hagan) is the ancestral GP-based integrator this all specialises.

Overview

This note collects the active / adaptive branch of Bayesian quadrature and its relation to Bayesian Monte Carlo. The motivation is twofold. (1) Non-negative integrands. Many important integrals — probabilistic model evidence $p(\mathcal{D})=\int p(\mathcal{D}\mid x)p(x)\mathrm{d}x$, marginal likelihoods, partition functions — have non-negative integrands, often spanning many orders of magnitude. A plain GP on $f$ ignores non-negativity and the large dynamic range. (2) Genuine adaptivity. From Bayesian Quadrature, a Gaussian model’s variance ignores $Y$, so node selection is open-loop; to let collected values steer future evaluations we must leave the Gaussian family. Both motivations lead to warped GP models. The agent then chooses evaluations to reduce uncertainty (see The Numerical Agent) — the posterior variance drives active node selection, the third pillar of the recurring PN thesis.

Main Content

Bayesian Monte Carlo (O’Hagan 1991)

Bayesian Monte Carlo (BMC) is the original GP-based integrator: put a GP prior on $f$, condition on evaluations, and report the induced Gaussian on $F=\int f\mathrm{d}\nu$ — exactly the construction of Bayesian Quadrature with $\nu$ a probability measure (so $F={\mathbb{E}}_{\nu }[f]$). BMC replaces the Monte Carlo average with the BQ posterior mean ${\ell }_X^{\top }{k}_{XX}^{-1}Y$, retaining a calibrated error bar. It is the point of departure for all schemes below; “Bayesian quadrature” and “Bayesian Monte Carlo” are often used interchangeably for the non-adaptive GP integrator.

Model evidence: the canonical non-negative integrand

Model evidence / marginal likelihood as a quadrature problem

$$\underset{F}{\underbrace{p(\mathcal{D})}}=\int \underset{f(x)}{\underbrace{p(\mathcal{D}\mid x)}}\underset{\mathrm{d}\nu (x)}{\underbrace{p(x)\mathrm{d}x}}.\text{\hspace{1em}(10.11)}$$

The integrand $f(x)=p(\mathcal{D}\mid x)\ge 0$ (a likelihood) is non-negative with large dynamic range; $\nu =p$ is the prior. Solving such integrals is a key step in Bayesian inference and, speculatively, toward AI.

To date, all adaptive BQ schemes target this non-negative setting.

Warped models for non-negativity

BBQ — doubly-Bayesian quadrature (Osborne et al. 2012)

Model the logarithm of the integrand with a GP, (approximately) enforcing $f>0$ and accommodating the large dynamic range: $\log f\sim \mathcal{G}\mathcal{P}$. Uses the integral variance (Eq. 10.4) as its loss — arguably the most desirable model (log-GP) with the most desirable loss (integral square-error). But it requires a first-order (linearised) approximation of $\exp$ and maintaining candidate points $x_c$; it is computationally demanding and expresses the dynamic-range prior only weakly.

WSABI — warped sequential active Bayesian integration (Gunter et al. 2014)

Model the square-root of the integrand (minus a constant $\alpha \in \mathbb{R}$) with a GP: given data $\mathcal{D}$,

$$f(x)=\alpha +\frac{1}{2}\tilde{f}(x{)}^2,p(\tilde{f}\mid \mathcal{D})=\mathcal{G}\mathcal{P}(\tilde{f};\tilde{m},\tilde{\mathbb{V}}),$$

with $\tilde{m},\tilde{\mathbb{V}}$ the usual GP posterior mean/covariance. Squaring guarantees $f\ge \alpha$. A squared-GP has smaller dynamic range than an exponentiated GP (a step back from BBQ in expressiveness), but the required approximations are far cheaper. Two implementations of the square transform:

• Linearised (WSABI-L):

$$m^{\mathcal{L}}(x)=\alpha +\frac{1}{2}\tilde{m}(x{)}^2,{\mathbb{V}}^{\mathcal{L}}(x,x^{\prime })=\tilde{m}(x)\tilde{\mathbb{V}}(x,x^{\prime })\tilde{m}(x^{\prime }).\text{\hspace{1em}(10.12)}$$

• Moment-matched (WSABI-M):

$$m^{\mathcal{M}}(x)=\alpha +\frac{1}{2}(\tilde{m}(x{)}^2+\tilde{\mathbb{V}}(x,x)),{\mathbb{V}}^{\mathcal{M}}(x,x^{\prime })=\frac{1}{2}\tilde{\mathbb{V}}(x,x^{\prime }{)}^2+\tilde{m}(x)\tilde{\mathbb{V}}(x,x^{\prime })\tilde{m}(x^{\prime }).\text{\hspace{1em}(10.13)}$$

In either case the resulting posterior over the integrand is again a GP, so the standard GP quadrature equations apply. A closely related scheme, MMLT (moment-matched log transformation, Chai & Garnett 2019), moment-matches the exponential transform, cheaper than BBQ.

Uncertainty sampling: the active design rule

Both WSABI and MMLT use the pointwise variance in the integrand (not the integral) as loss, sampling where the function is most uncertain — uncertainty sampling:

$$x_{i+1}=\arg \underset{x}{\max }\mathbb{V}(x,x),$$

with $\mathbb{V}(x,x)={\mathbb{V}}^{\mathcal{L}}(x,x)$ (linearised) or ${\mathbb{V}}^{\mathcal{M}}(x,x)$ (moment-matched). Reducing pointwise variance should also reduce integral variance in the limit — but the link to the ultimate goal (integral square-error) is indirect.

Weak adaptivity (Kanagawa & Hennig 2019)

A general adaptive BQ scheme with sequential design $x_{i+1}=\arg {\max }_x{a}_i(x)$, where

$$a_i(x)=T(q^2(x)\tilde{\mathbb{V}}(x,x))b_i(x),$$

($\tilde{\mathbb{V}}$ the GP posterior variance, $q$ a positive function e.g. the integration measure, $T$ the warping/transformation, $b_i$ an “adaptivity” function) is weakly adaptive — and hence consistent, subject to regularity — if $b_i$ is bounded away from $0$ and $\infty$. Moment-matched WSABI and MMLT satisfy this; linearised WSABI needs a minor modification. Weak adaptivity guarantees the scheme is not “too adaptive”: it can be “stuck for a while, but not forever”, giving a worst-case rate approaching that of non-adaptive BQ.

Weak adaptivity plays a role analogous to detailed balance / ergodicity for MCMC: a weak consistency guarantee that licenses a “zoo” of empirically-evaluated adaptive schemes. Empirically, adaptive schemes usually improve on non-adaptive BQ (stronger than the theory guarantees).

When uncertainty sampling helps — and when it is wasteful

Pathological case for integrand-variance loss (Garnett's example)

Let $F={\int }_0^{2\pi }\sin (x+\varphi )\mathrm{d}x$ where the phase $\varphi$ is unknown. The pointwise integrand variance $\mathrm{var}(f(x))$ depends on $p(\varphi )$ and is generally non-zero, so uncertainty sampling would take (possibly many) evaluations. But the integral is $F=0$ with certainty ($p(F)=\delta (F)$, $\mathrm{var}(F)=0$) regardless of $\varphi$! Targeting integrand variance is thus wasteful here; targeting integral variance takes zero evaluations. In practice, however, uncertainty sampling gives algorithms that are substantially faster (in wall-clock time) than integral-variance schemes like BBQ.

Speed: computation as investment

WSABI must maintain a full GP ($\mathcal{O}(N^3)$ in evaluations), manage GP hyperparameters, and solve a global optimisation problem to pick each node — substantial overhead versus Monte Carlo’s near-free PRNG draws. Yet:

WSABI beats Monte Carlo and AIS in wall-clock time

On an 8-dimensional GP-regression marginal-likelihood benchmark (yacht hydrodynamics) and a 4-dimensional GP-classification citation-network task, WSABI reaches a given estimation error in less wall-clock time than Monte Carlo and even annealed importance sampling (AIS). The heavy per-iteration overhead does not prevent faster overall convergence: the computation spent on Bayesian inference is an investment that pays dividends in fewer, better-placed evaluations — a central lesson for Lessons from Integration. This speed is because of, not despite, the Bayesian modelling.

Connections

• Extends the Gaussian Bayesian Quadrature to non-Gaussian warped models where the variance depends on data, enabling true adaptivity.
• Instantiates the agent view of The Numerical Agent: the posterior variance is a utility driving active evaluation.
• BMC is the non-adaptive GP integrator of Bayesian Quadrature applied to $F={\mathbb{E}}_{\nu }[f]$.
• Contrasts integrand-variance (uncertainty sampling; WSABI/MMLT) vs integral-variance (BBQ) losses; connects to calibration in Convergence and Priors in Bayesian Quadrature.
• Supplies the “computation as investment” and “randomness can be wasteful” lessons of Lessons from Integration.

See Also

• Bayesian Quadrature — the Gaussian base model these warp; why Gaussian BQ is open-loop.
• The Numerical Agent — the solver-as-agent framing of active node selection.
• Convergence and Priors in Bayesian Quadrature — consistency and rates for adaptive schemes.
• Lessons from Integration — WSABI as evidence that Bayesian overhead is an investment.