Ejemplo de análisis de peligrosidad del proceso

Let (Ω,F,P) be a probability space and (X,B(X)) be a measurable space with

X ⊂ _Rd _{for d ∈}

N and let B(X) be a Borel σ-algebra. Let f : X → R be some

Recall that a quadrature rule describes any functional of the form of a linear combination of function values: ˆΠ[f] = Pn

i=1wif(xi) for some states (or samples) {xi}ni=1 ⊂ X and weights {wi}ni=1 ⊂ R. The notation ˆΠ[f] is motivated by the

fact that this expression can be re-written as the integral of f with respect to an empirical measure ˆΠ = Pn

i=1wiδ(xi), where δ(xi) is a Dirac measure (i.e. for all

A ∈ B(X), δxi(A) = 1 if xi ∈ A, δxi(A) = 0 if xi ∈/ A). The weights wi can be

negative and need not satisfyPn

i=1wi = 1.

BQ begins by defining a stochastic process g : X ×Ω → R formally seen

as a prior model for the integrand f. The most popular choice, originally made by Larkin [1972], is to consider a GP, but others could also be used. Recall from Chapter 2 that a GP can be characterised by its mean function and its covariance function: m(x) =EP[g(x, ω)] andc(x,x

0_{) =}

EP[(g(x, ω)−m(x))(g(x

0_{, ω)−m(x}0_))].

From now on, we assume without loss of generality thatm ≡0. Conditioning the GP at quadrature pointsX={xi}ni=1⊂ X gives a new GP denotedgn:X ×Ω→R.

This GP has mean mn(x) = m(x) +c(x,X)C−1(f −m) and covariance function

cn(x,x0) = c(x,x0)−c(x,X)C−1c(X,x0). For simplicity we will assume there is

no measurement error. After obtaining the conditioned GP gn, the final step is

to produce a distribution on the value of the integral Π[gn] by considering the

pushforward of the process gn through the integration operator. A sketch of the

procedure is presented in Figure 3.1 and the relevant formulae are now provided.

Proposition 1(BQ posterior distribution on the solution of the integral).

The distribution of Π[gn]is Gaussian with mean and variance1

E[Π[gn]] = Π[c(·,X)]C−1f, (3.1)

V[Π[gn]] = ΠΠ[c(·,·)]−Π[c(·,X)]C−1Π[c(X,·)]. (3.2)

All of the proofs in this thesis can be found in Appendix B, ordered by Chapter and in the order in which they appear in the main text. In particular, see B.1 for all the proofs in this chapter.

Here, ΠΠ[c(·,·)] denotes the integral of c with respect to each argument. It can be seen that the computational cost of obtaining this full posterior (in the worst-case O(n3)) is much higher than that of obtaining a point estimate for the integral using MC methods. However, many methods for scaling GPs (discussed in the previous chapter) can be used to speed this up. Karvonen and S¨arkk¨a [2018] also proposed a novel scalable method specifically targeted to scaling BQ.

1_{The mean and variance are taken with respect to}

P, but we do not repeatedly specify this to

n=0 n=3 n=8

x

Integr

and

Solution of the integral

P

oster

ior distr

ib

ution

Figure 3.1: Sketch of Bayesian quadrature. The top row shows the approximation of the integrandf (red) by the posterior mean mn (blue) as the numbernof function

evaluations is increased. The dashed lines represent point-wise 95% posterior credi- ble intervals. The bottom row shows the Gaussian distribution with meanE[Π[gn]]

and variance V[Π[gn]] and the dashed black line gives the true value of the integral

Π[f].

Since BQ formally associates g with a prior on f, Π[gn] in turn provides

a posterior distribution over the value of the integral Π[f] representing our epis- temic uncertainty. An interesting remark is that Equation 3.1 takes the form of a quadrature rule: E[Π[gn]] = ˆΠBQ[f] := n X i=1 wBQ_i f(xi), (3.3)

with weight vector given bywBQ := (Π[c(X,·)]C−1)>. Furthermore, the posterior variance in Equation 3.2 does not depend on function values {f(xi)}ni=1, but only

on the location of the states{xi}ni=1 and the choice of covariance functionc. This

is useful as it allows state locations and weights to be precomputed and reused. However, it also means that the variance is completely driven by the choice of prior. A valid quantification of uncertainty thus relies on a well-specified prior; we consider this issue further in Section 3.42.

The BQ mean (Equation 3.1) coincides with classical quadrature rules for specific choices of covariance functionc. For example, in one dimension a Brownian covariance function c(x, x0) = min(x, x0) leads to a posterior mean mn that is a

2

Note that other choices of priors for f will give posteriors which do not necessarily have this property.

piecewise linear interpolant off between the states {xi}ni=1, i.e. the trapezium rule

[Suldin, 1959]. Similarly, S¨arkk¨a et al. [2016] constructed a covariance function c

for which Gauss-Hermite quadrature is recovered, and Karvonen and S¨arkk¨a [2017] showed how other polynomial-based quadrature rules can be recovered. In another research direction, Karvonen et al. [2018] showed how it is possible to design a BQ rule whose mean corresponds to the point estimate of any cubature rule.

Clearly the point estimator in Equation 3.3 is a natural object; it has also received attention in both the kernel quadrature literature [Sommariva and Vianello, 2006] and empirical interpolation literature [Kristoffersen, 2013]. In those contexts, the point estimator is derived from different assumptions on the integrand: namely, that it is an element of a RKHS with kernelc, rather than a draw from a GP with covariancec.

Although other stochastic processes could of course be used as priors [Cock- ayne et al., 2017], GPs are popular due to their conjugacy properties, and the terminology Bayesian quadrature usually refers to this case. Note that other names for BQ with GP priors include Gaussian-process quadrature or kernel quadrature. Alternative prior which are conjugate include Student-t process, and these could afford heavier tails for values assumed by the integrand.

There has been a wide range of applications of BQ, including to other numer- ical methods in optimisation, linear algebra and functional approximation [Kersting and Hennig, 2016; Fitzsimons et al., 2017], inference in complex computer models [Oates et al., 2017d], and problems in econometrics [Oettershagen, 2017] and com- puter graphics [Brouillat et al., 2009; Marques et al., 2013; Briol et al., 2015b; Xi et al., 2018].