### Preprint: Cameron-Martin theorems for Cauchy-distributed random sequences

Han Cheng Lie and I have just uploaded a preprint of our latest paper, on Cameron–Martin-type theorems for sequences of Cauchy-distributed random variables, to the arXiv.
Inspired by questions of prior robustness left unanswered in this earlier paper on *α*-stable Banach space priors, this paper addresses the basic probabilistic question:
when is an infinite-dimensional Cauchy distribution, e.g. on sequence space, mutually absolutely continuous with its image under a translation?
In the Gaussian case, the celebrated Cameron–Martin theorem says that this equivalence of measures holds if a weighted \(\ell^{2}\) norm (the Cameron–Martin norm) of the translation vector is finite.
We show that, in the Cauchy case, the same weighted version of the translation vector needs to lie in the sequence space \(\ell^{1} \cap \ell \log \ell\).
More precisely, if the Cauchy distribution on the *n*^{th} term of the sequence has width parameter \(\gamma_{n} > 0\), and the translation vector is the sequence \(\varepsilon = (\varepsilon_n)_{n = 1}^{\infty}\), then a sufficient condition for mutual absolute continuity is that

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| < \infty \)

and, with the usual convention that \(0 \log 0 = 0\),

\( \displaystyle \sum_{n = 1}^{\infty} \left| \frac{\varepsilon_{n}}{\gamma_{n}} \log \left| \frac{\varepsilon_{n}}{\gamma_{n}} \right| \right| < \infty . \)

We also discuss similar results for dilation of the scale parameters, i.e. \(\gamma_{n} \mapsto \sigma_{n} \gamma_{n}\) for some real sequence \(\sigma = (\sigma_n)_{n = 1}^{\infty}\).

Published on Monday 15 August 2016 at 10:00 UTC #publication #preprint

### Probabilistic Numerics at MCQMC

There will be a workshop on Probabilistic Numerics at this year's MCQMC conference at Stanford University. The workshop will be held on Thursday, 18 August 2016, 15:50–17:50, at the Li Ka Shing Center on the Stanford University campus. Speakers include:

- Mark Girolami (University of Warwick & Alan Turing Institute) — Probabilistic Numerical Computation: A New Concept?
- François-Xavier Briol (University of Warwick & University of Oxford) — Probabilistic Integration: A Role for Statisticians in Numerical Analysis?
- Chris Oates (University of Technology Sydney) — Probabilistic Integration for Intractable Distributions
- Jon Cockayne (University of Warwick) — Probabilistic meshless methods for partial differential equations and Bayesian inverse problems

**Update, 19 August 2016.** The slides from the talks can be found here, on Chris Oates' website.

Published on Sunday 31 July 2016 at 14:00 UTC #prob-num #event

### Turing Lecture by Mark Girolami on Probabilistic Numerics

Last Friday 8 July 2016, as part of this year's Turing Lectures on *The Intersection of Mathematics, Statistics and Computation*, Mark Girolami gave a lecture on “Probabilistic Numerical Computation: A New Concept?”, containing some of our joint work on probabilistic numerics.
The video can be found on the Alan Turing Institute's Youtube channel.

Published on Tuesday 12 July 2016 at 12:00 UTC #video #talk #prob-num

### UQ Talks: Hans Kersting

Next week Hans Kersting (MPI Tübingen) will give a talk in the UQ research seminar about “UQ in probabilistic ODE solvers”.

**Time and Place.** Tuesday 14 June 2016, 12:15–13:15, ZIB Seminar Room 2006, Zuse Institute Berlin, Takustraße 7, 14195 Berlin

**Abstract.** In an ongoing push to construct probabilistic extensions of classic ODE solvers for application in statistics and machine learning, two recent papers have provided distinct methods that return probability measures instead of point estimates, based on sampling and filtering respectively.
While both approaches leverage classical numerical analysis, by building on well-studied solutions of existing seminal solvers, the different constructions of probability measures strike a divergent balance between a formal quantification of epistemic uncertainty and a low computational overhead.

On the one hand, Conrad et al. proposed to randomise existing non-probabilistic one-step solvers by adding suitably scaled Gaussian noise after every step and thereby inducing a probability measure over the solution space of the ODE which contracts to a Dirac measure on the true unknown solution in the order of convergence of the underlying classic numerical method. But the computational cost of these methods is significantly above that of classic solvers.

On the other hand, Schober et al. recast the estimation of the solution as state estimation by a Gaussian (Kalman) filter and proved that employing a integrated Wiener process prior returns a posterior Gaussian process whose maximum likelihood (ML) estimate matches the solution of classic Runge–Kutta methods. In an attempt to amend this method's rough uncertainty calibration while sustaining its negligible cost overhead, we propose a novel way to quantify uncertainty in this filtering framework by probing the gradient using Bayesian quadrature.

Published on Monday 6 June 2016 at 10:00 UTC #event #uq-talk #prob-num

### Preprint: Probabilistic meshless methods for PDEs and BIPs

Jon Cockayne, Chris Oates, Mark Girolami and I have just uploaded a preprint of our latest paper, “Probabilistic meshless methods for partial differential equations and Bayesian inverse problems” to the arXiv. This paper forms part of the push for probabilistic numerics in scientific computing.

**Abstract.** This paper develops a class of meshless methods that are well-suited to statistical inverse problems involving partial differential equations (PDEs). The methods discussed in this paper view the forcing term in the PDE as a random field that induces a probability distribution over the residual error of a symmetric collocation method. This construction enables the solution of challenging inverse problems while accounting, in a rigorous way, for the impact of the discretisation of the forward problem. In particular, this confers robustness to failure of meshless methods, with statistical inferences driven to be more conservative in the presence of significant solver error. In addition, (i) a principled learning-theoretic approach to minimise the impact of solver error is developed, and (ii) the challenging setting of inverse problems with a non-linear forward model is considered. The method is applied to parameter inference problems in which non-negligible solver error must be accounted for in order to draw valid statistical conclusions.

Published on Thursday 26 May 2016 at 09:00 UTC #publication #preprint #prob-num