Posterior consistency via precision operators for Bayesian nonparametric drift estimation in SDEs

Abstract

We study a Bayesian approach to nonparametric estimation of the periodic drift function of a one-dimensional diffusion from continuoustime data. Rewriting the likelihood in terms of local time of the process, and specifying a Gaussian prior with precision operator of differential form, we show that the posterior is also Gaussian with precision operator also of differential form. The resulting expressions are explicit and lead to algorithms which are readily implementable. Using new functional limit theorems for the local time of diffusions on the circle, we bound the rate at which the posterior contracts around the true drift function.

Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

“…Therefore there exists a subsequence which satisfies (after labelling by n again) G(u n ) → G(u † ) almost surely. (43) Now let {ψ ℓ } ∞ ℓ=1 be an r-regular orthonormal wavelet basis for L 2 (R d ) with r ≥ s. We then, by (42), have E| u n , ψ ℓ | ≤ C ℓ E u n B s 1 ≤ C ℓ ( u † B s 1 + K), for all ℓ ∈ N. For ℓ = 1, the above bound implies the existence of {u n 1 (k) } k∈N ⊂ {u n } n∈N and η 1 ∈ R such that E| u n 1 (k) , ψ 1 | → η 1 as k → ∞. Considering ℓ = 2, 3, .…”

Gaussian process methods for one-dimensional diffusions: Optimal rates and adaptation

“…It can be proved that this defines a valid prior onL 2 (T), cf. Pokern et al (2013), Section 2.2. The main convergence result proved in Pokern et al (2013) asserts that if in this setup the true drift b 0 generating the data has (Sobolev) regularity α + 1/2, then the corresponding posterior distribution of b contracts around b 0 at the rate T −α/(1+2α) as T → ∞, with respect to the L 2 -norm.…”

“…Pokern et al (2013), Section 2.2. The main convergence result proved in Pokern et al (2013) asserts that if in this setup the true drift b 0 generating the data has (Sobolev) regularity α + 1/2, then the corresponding posterior distribution of b contracts around b 0 at the rate T −α/(1+2α) as T → ∞, with respect to the L 2 -norm. In the concluding section of Pokern et al (2013) it was already conjectured that this result is not completely sharp.…”

“…We are able to obtain the improved result by following a different mathematical route than in Pokern et al (2013). The latter paper uses more or less explicit representations of the posterior mean and covariance in terms of weak solutions of certain differential equations to study the asymptotic behaviour of the posterior using techniques from PDE theory.…”

Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions

“…Alternative methodology for nonparametric inference in diffusion models has been put forward recently, notably of a Bayesian flavour, see Roberts and Stramer [24], Papaspiliopoulos et al [20], Pokern et al [22], van der Meulen et al [30], van Waaij and van Zanten [34] and references therein. While such Bayesian methods are attractive in applications [15], [27], [35], particularly since they provide associated uncertainty quantification procedures ('credible regions'), our understanding of their frequentist sampling performance is extremely limited.…”