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  • JOURNAL ARTICLE
    Crisan D, Míguez J, 2018,

    Nested particle filters for online parameter estimation in discrete-time state-space Markov models

    , Bernoulli, Vol: 24, Pages: 2429-2460, ISSN: 1350-7265

    © 2018 ISI/BS. We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent “sequential Monte Carlo square” (SMC2) algorithm. However, unlike the SMC2scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in Lp(p ≥ 1) with convergence rate proportional to1N+1M, where N is the number of Monte Carlo samples in the parameter space and N × M is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC2algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.

  • CONFERENCE PAPER
    Dutordoir V, Salimbeni H, Deisenroth M, Hensman Jet al., 2018,

    Gaussian Process Conditional Density Estimation

    Conditional Density Estimation (CDE) models deal with estimating conditionaldistributions. The conditions imposed on the distribution are the inputs of themodel. CDE is a challenging task as there is a fundamental trade-off betweenmodel complexity, representational capacity and overfitting. In this work, wepropose to extend the model's input with latent variables and use Gaussianprocesses (GP) to map this augmented input onto samples from the conditionaldistribution. Our Bayesian approach allows for the modeling of small datasets,but we also provide the machinery for it to be applied to big data usingstochastic variational inference. Our approach can be used to model densitieseven in sparse data regions, and allows for sharing learned structure betweenconditions. We illustrate the effectiveness and wide-reaching applicability ofour model on a variety of real-world problems, such as spatio-temporal densityestimation of taxi drop-offs, non-Gaussian noise modeling, and few-shotlearning on omniglot images.

  • JOURNAL ARTICLE
    Noven RC, Veraart AED, Gandy A, 2018,

    A latent trawl process model for extreme values

    , JOURNAL OF ENERGY MARKETS, Vol: 11, Pages: 1-24, ISSN: 1756-3607
  • JOURNAL ARTICLE
    Gulisashvili A, Horvath B, Jacquier A, 2018,

    Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

    , Quantitative Finance, Vol: 18, Pages: 1753-1765, ISSN: 1469-7688

    We study the mass at the origin in the uncorrelated SABR stochasticvolatility model, and derive several tractable expressions, in particular whentime becomes small or large. As an application--in fact the original motivationfor this paper--we derive small-strike expansions for the implied volatilitywhen the maturity becomes short or large. These formulae, by definitionarbitrage free, allow us to quantify the impact of the mass at zero on existingimplied volatility approximations, and in particular how correct/erroneousthese approximations become.

  • JOURNAL ARTICLE
    Crisan D, McMurray E, 2018,

    Smoothing properties of McKean–Vlasov SDEs

    , Probability Theory and Related Fields, Vol: 171, Pages: 97-148, ISSN: 0178-8051

    © 2017, The Author(s). In this article, we develop integration by parts formulae on Wiener space for solutions of SDEs with general McKean–Vlasov interaction and uniformly elliptic coefficients. These integration by parts formulae hold both for derivatives with respect to a real variable and derivatives with respect to a measure understood in the sense of Lions. They allows us to prove the existence of a classical solution to a related PDE with irregular terminal condition. We also develop bounds for the derivatives of the density of the solutions of McKean–Vlasov SDEs.

  • JOURNAL ARTICLE
    Davis M, Obloj J, Siorpaes P, 2018,

    Pathwise stochastic calculus with local times

    , ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, Vol: 54, Pages: 1-21, ISSN: 0246-0203
  • JOURNAL ARTICLE
    Olofsson S, Deisenroth MP, Misener R, 2018,

    Design of Experiments for Model Discrimination using Gaussian Process Surrogate Models

    , Vol: 44, Pages: 847-852, ISSN: 1570-7946

    © 2018 Elsevier B.V. Given rival mathematical models and an initial experimental data set, optimal design of experiments for model discrimination discards inaccurate models. Model discrimination is fundamentally about finding out how systems work. Not knowing how a particular system works, or having several rivalling models to predict the behaviour of the system, makes controlling and optimising the system more difficult. The most common way to perform model discrimination is by maximising the pairwise squared difference between model predictions, weighted by measurement noise and model uncertainty resulting from uncertainty in the fitted model parameters. The model uncertainty for analytical model functions is computed using gradient information. We develop a novel method where we replace the black-box models with Gaussian process surrogate models. Using the surrogate models, we are able to approximately marginalise out the model parameters, yielding the model uncertainty. Results show the surrogate model method working for model discrimination for classical test instances.

  • JOURNAL ARTICLE
    Crisan D, Flandoli F, Holm DD, 2018,

    Solution Properties of a 3D Stochastic Euler Fluid Equation

    , Journal of Nonlinear Science, ISSN: 0938-8974

    © 2018, The Author(s). We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.

  • CONFERENCE PAPER
    Salimbeni H, Cheng C-A, Boots B, Deisenroth MPet al., 2018,

    Orthogonally Decoupled Variational Gaussian Processes.

  • CONFERENCE PAPER
    Salimbeni H, Deisenroth M, 2017,

    Doubly stochastic variational inference for deep Gaussian processes

    , NIPS 2017, Publisher: Advances in Neural Information Processing Systems (NIPS), Pages: 4589-4600, ISSN: 1049-5258

    Gaussian processes (GPs) are a good choice for function approximation as theyare flexible, robust to over-fitting, and provide well-calibrated predictiveuncertainty. Deep Gaussian processes (DGPs) are multi-layer generalisations ofGPs, but inference in these models has proved challenging. Existing approachesto inference in DGP models assume approximate posteriors that forceindependence between the layers, and do not work well in practice. We present adoubly stochastic variational inference algorithm, which does not forceindependence between layers. With our method of inference we demonstrate that aDGP model can be used effectively on data ranging in size from hundreds to abillion points. We provide strong empirical evidence that our inference schemefor DGPs works well in practice in both classification and regression.

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