25 research outputs found
A modern retrospective on probabilistic numerics
This article attempts to place the emergence of probabilistic numerics as a mathematical–statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul′din and Larkin in the 1960s and how their pioneering early ideas have reached a degree of maturity in the intervening period, mediated by paradigms such as average-case analysis and information-based complexity. We provide a subjective assessment of the state of research in probabilistic numerics and highlight some difficulties to be addressed by future works
A Locally Adaptive Bayesian Cubature Method
Bayesian cubature (BC) is a popular inferential perspective on the cubature
of expensive integrands, wherein the integrand is emulated using a stochastic
process model. Several approaches have been put forward to encode sequential
adaptation (i.e. dependence on previous integrand evaluations) into this
framework. However, these proposals have been limited to either estimating the
parameters of a stationary covariance model or focusing computational resources
on regions where large values are taken by the integrand. In contrast, many
classical adaptive cubature methods focus computational resources on spatial
regions in which local error estimates are largest. The contributions of this
work are three-fold: First, we present a theoretical result that suggests there
does not exist a direct Bayesian analogue of the classical adaptive trapezoidal
method. Then we put forward a novel BC method that has empirically similar
behaviour to the adaptive trapezoidal method. Finally we present evidence that
the novel method provides improved cubature performance, relative to standard
BC, in a detailed empirical assessment
Semi-Exact Control Functionals From Sard's Method
A novel control variate technique is proposed for post-processing of Markov chain Monte Carlo output, based both on Stein's method and an approach to numerical integration due to Sard. The resulting estimators of posterior expected quantities of interest are proven to be polynomially exact in the Gaussian context, while empirical results suggest the estimators approximate a Gaussian cubature method near the Bernstein-von-Mises limit. The main theoretical result establishes a bias-correction property in settings where the Markov chain does not leave the posterior invariant. Empirical results are presented across a selection of Bayesian inference tasks. All methods used in this paper are available in the R package ZVCV
Optimality Criteria for Probabilistic Numerical Methods
It is well understood that Bayesian decision theory and average case analysis
are essentially identical. However, if one is interested in performing
uncertainty quantification for a numerical task, it can be argued that standard
approaches from the decision-theoretic framework are neither appropriate nor
sufficient. Instead, we consider a particular optimality criterion from
Bayesian experimental design and study its implied optimal information in the
numerical context. This information is demonstrated to differ, in general, from
the information that would be used in an average-case-optimal numerical method.
The explicit connection to Bayesian experimental design suggests several
distinct regimes in which optimal probabilistic numerical methods can be
developed.Comment: Prepared for the proceedings of the RICAM workshop on Multivariate
Algorithms and Information-Based Complexity, November 201
A Riemannian-Stein Kernel Method
This paper presents a theoretical analysis of numerical integration based on
interpolation with a Stein kernel. In particular, the case of integrals with
respect to a posterior distribution supported on a general Riemannian manifold
is considered and the asymptotic convergence of the estimator in this context
is established. Our results are considerably stronger than those previously
reported, in that the optimal rate of convergence is established under a basic
Sobolev-type assumption on the integrand. The theoretical results are
empirically verified on
Black box probabilistic numerics
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