Lineamientos para la prevención de conflictos

1.7

Background reading

This section gathers some background material in the areas studied in this thesis. The focus is on survey works.

Frequentist analysis of Bayesian procedures Ghosal & van der Vaart [34], Giné &

Nickl [38, chapter 8]. For diffusions in particular: van Zanten [87]

Stochastic diffusions Durrett [26], Rogers & Williams [72,73], Bass [7], Bhattacharya

& Waymire [9]

Bayesian computation Cotter, Roberts, Stuart & White [22], Stuart [77]

PDE inverse problems Katchalov–Kury–Lassas [47], Isakov [44]

The Calderón problem/EIT Uhlmann [81], Salo [75]

Chapter 2

Contraction rates for scalar

diffusions with high-frequency data

Notation

Most of the notation to be used in this chapter is informally gathered here.

X: A solution to dXt = b(Xt) dt + σ(Xt) dWt.

˙

X: The periodised diffusion ˙X = X mod 1. b, σ: Drift function, diffusion coefficient.

µ= µb; πb: Invariant distribution/density of ˙X.

P_b(x): Law of X on C([0, ∞]) (on C([0, ∆]) in Section 2.4) for initial condition X0 = x.

Eb; Pb; Varb: Expectation/probablity/variance according to the law of X started from

µb.

Eµ; Varµ, and similar: Expectation/variance according to the subscripted measure.

W(x)σ : Notation for P

(x)

b when b = 0.

pb(t, x, y), ˙pb(t, x, y): Transition densities of X, ˙X (with respect to Lebesgue measure).

˜pb: Density (with respect to W(x)σ ) of P

(x) b on C([0, ∆]). Ib(x) = Rx 0(2b/σ2(y)) dy. X(n) = (X 0, . . . , Xn∆); x(n) = (x0, . . . , xn∆); p (n)

b (x(n)) = πb(x0)Qni=1pb(∆, x(i−1)∆, xi∆).

b0: The true parameter generating the data.

µ0, π0, p0 etc.: Shorthand for µb0, πb0, pb0 etc.

L0: A constant such that n∆2log(1/∆) ≤ L0 for all n.

C_per1 ([0, 1]): The space of continuously differentiable functions f : R → R satisfying f(x + 1) = f(x) for x ∈ R.

Θ = Θ(K0): The maximal paramater space: Θ = {f ∈ Cper1 ([0, 1]) s.t. ∥f∥C1

per ≤ K0}.

Θs(A0) = {f ∈ Θ : ∥f∥_Bs

2,∞ ≤ A0}, for B

s

2,∞ a (real scalar, 1–periodic) Besov space.

I = {K0, σL, σU}.

Sm: Wavelet approximation space of resolution m, generated by periodised Meyer-type

wavelets: Sm = span{ψlk : −1 ≤ l < m, 0 ≤ k < 2l}, where ψ−1,0 is used as

notation for the constant function 1.

Dm = dim(Sm) = 2m; πm =(L2–)orthogonal projection onto Sm.

wm(δ) = δ1/2(log(δ−1)1/2+ log(m)1/2) if m ≥ 1, wm:= w1 if m < 1.

1A: Indicator of the set (or event) A.

K(p, q): Kullback–Leibler divergence between densities p, q: K(p, q) = Ep[log(p/q)].

KL(b0, b) = Eb0log(p0/pb). B_KL(n)(ε) =  b ∈Θ s.t. K(p(n)₀ , p(n)_b ) ≤ (n∆ + 1)ε2_,Var b0  log p(n)₀ /p(n)_b   ≤(n∆ + 1)ε2  . Bε= {b ∈ Θ s.t. K(π0, πb) ≤ ε2, Varb0(log π0 πb) ≤ ε 2_{, KL(b} 0, b) ≤ ∆ε2, Varb0(log p0 pb) ≤ ∆ε2_}.

Π: The prior distribution.

Π(· | X(n)_{): The posterior distribution given data X}(n)_.

⟨·, ·⟩₂: the L2_{([0, 1]) inner product, ⟨f, g⟩} 2 =

R1

0 f(x)g(x) dx (this chapter uses real-valued

functions, so no need for the complex conjugate of g). ∥·∥₂: The L2_{([0, 1])–norm, ∥f∥}2 2 = R1 0 f(x)2dx. ∥·∥_µ: The L2(µ)–norm, ∥f∥2 µ= R1 0 f(x)2µ(dx) = R1 0 f(x)2πb(x) dx.

∥·∥_∞: The L∞_{– (supremum) norm.}

∥∥_C1 per: The C 1 per–norm, ∥f∥C1 per = ∥f∥∞+ ∥f ′_∥ ∞.

∥·∥_n: The empirical L2–norm ∥f∥2

n=

Pn

2.0 Introduction 39

2.0

Introduction

Let’s reprise the stochastic diffusion model described in Section 1.1, and expand on its key features. Consider a scalar diffusion process (Xt)t≥0 starting at some X0 and evolving

according to the stochastic differential equation

dXt= b(Xt) dt + σ(Xt) dWt, (2.1)

where Wt is a standard Brownian motion. It is of considerable interest to estimate the

parameters b and σ, which are arbitrary functions (until we place further assumptions on their form), so that the model is naturally nonparametric. The problems of estimating σ and b can essentially be decoupled in the setting to be considered here (see Section2.1), so in this chapter we consider estimation of the drift function b when the diffusion coefficient

σ is assumed to be given.

It is realistic to assume that we do not observe the full trajectory (Xt)t≤T but rather

the process sampled at discrete time intervals (Xk∆)k≤n. The estimation problem for

b and σ has been studied extensively and minimax rates have been attained in two

sampling frameworks: low-frequency, where ∆ is fixed and asymptotics are taken as

n → ∞ (see Gobet–Hoffmann–Reiss [39]), and high-frequency, where asymptotics are

taken as n → ∞ and ∆ = ∆n →0, typically assuming also that n∆2 →0 and n∆ → ∞

(see Hoffmann [43], Comte et al. [21]). See also e.g. [23], [40], [68], [82] for more papers addressing nonparametric estimation for diffusions.

For typical frequentist methods, one must know from which sampling regime the data is drawn. In particular, the low-frequency estimator of [39] is consistent in the high-frequency setting but numerical simulations suggest it does not attain the minimax rate (see the discussion in Chorowski [18]), while the high-frequency estimators of [43] and [21] are not even consistent with low-frequency data. Often real data arrives all at once so that the appropriate asymptotic regime is not clear, hence it is desirable to use estimators which perform well independent of the regime. The only previous result known to the author in this direction in the nonparametric setting considered here is found in [18], where Chorowski is able to estimate the diffusion coefficient σ but not the drift, and obtains the minimax rate when σ has 1 derivative but not for smoother diffusion coefficients. See also Chorowski & Trabs [19], wherein an estimator adapting to the sampling regime is given but only for the sampling period bounded below, and Coca [20] for an estimator adapting to the sampling regime in a Lévy process setting.

For this chapter we consider estimation of the parameters in a diffusion model from a nonparametric Bayesian perspective. An attraction of Bayesian methods for the discretely

sampled diffusion model is that the statistician need only specify a prior, and the prior need not reference the sampling regime, so Bayesian methodology provides a natural candidate for a unified approach to the high- and low-frequency settings. Note also that, building on ideas outlined in Section 1.5, Bayesian methods for diffusion estimation can be implemented in practice (e.g. see Papaspiliopoulos et al. [65]). Under the frequentist assumption of a fixed true parameter, the results of this chapter imply that Bayesian methods can adapt both to the sampling regime and also to unknown smoothness of the drift function (see the remarks after Propositions2.4 and 2.2 respectively for details).

It has previously been shown that in the low-frequency setting we have a posterior

contraction rate, guaranteeing that posteriors corresponding to reasonable priors concen-

trate their mass on neighbourhoods of the true parameter shrinking at the fastest possible rate (up to log factors) – see Nickl & Söhl [62]. To complete a proof that such posteriors contract at a rate adapting to the sampling regime, it remains to prove a corresponding contraction rate in the high-frequency setting. This forms the key contribution of the current chapter: we prove that a large class of “reasonable” priors will exhibit posterior contraction at the optimal rate (up to log factors) in L2_{–distance. This in turn guarantees}

that point estimators based on the posterior will achieve the frequentist minimax optimal rate (see the remark after Theorem 2.1) in both high- and low-frequency regimes.

The broad structure of the proof, inspired by that in [62], is as described Section 1.4.1. The main ingredients are:

• A concentration inequality for a (frequentist) estimator, from which we construct tests of the true b0 against a set of suitable (sufficiently separated) alternatives.

See Section2.3.

• A small ball result, to relate the L2–distance to the information-theoretic Kullback–

Leibler “distance”. See Section2.4.

Although the main proof structure reflects that used in [62] for the low-frequency case, the details are very different. Estimators for the low-frequency setting are typically based on the mixing properties of (Xk∆) viewed as a Markov chain and the spectral structure of

its transition matrix (see Gobet–Hoffmann–Reiss [39]) and fail to take full advantage of the local information one sees when ∆ → 0. Here we instead use an estimator introduced in Comte et al. [21], which uses the assumption ∆ → 0 to view estimation of b as a regression problem. To prove this estimator concentrates depends on a key insight of this chapter: the Markov chain concentration results used in the low-frequency setting (which give worse bounds as ∆ → 0) must be supplemented by Hölder type continuity results,