760 research outputs found
MCMC methods for functions modifying old algorithms to make\ud them faster
Many problems arising in applications result in the need\ud
to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the unknown function. We describe an approach to modifying a whole range of MCMC methods which ensures that their speed of convergence is robust under mesh refinement. In the applications of interest the data is often sparse and the prior specification is an essential part of the overall modeling strategy. The algorithmic approach that we describe is applicable whenever the desired probability measure has density with respect to a Gaussian process or Gaussian random field prior, and to some useful non-Gaussian priors constructed through random truncation. Applications are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method for functions. This leads to algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems
Analysis of the Gibbs sampler for hierarchical inverse problems
Many inverse problems arising in applications come from continuum models
where the unknown parameter is a field. In practice the unknown field is
discretized resulting in a problem in , with an understanding
that refining the discretization, that is increasing , will often be
desirable. In the context of Bayesian inversion this situation suggests the
importance of two issues: (i) defining hyper-parameters in such a way that they
are interpretable in the continuum limit and so that their
values may be compared between different discretization levels; (ii)
understanding the efficiency of algorithms for probing the posterior
distribution, as a function of large Here we address these two issues in
the context of linear inverse problems subject to additive Gaussian noise
within a hierarchical modelling framework based on a Gaussian prior for the
unknown field and an inverse-gamma prior for a hyper-parameter, namely the
amplitude of the prior variance. The structure of the model is such that the
Gibbs sampler can be easily implemented for probing the posterior distribution.
Subscribing to the dogma that one should think infinite-dimensionally before
implementing in finite dimensions, we present function space intuition and
provide rigorous theory showing that as increases, the component of the
Gibbs sampler for sampling the amplitude of the prior variance becomes
increasingly slower. We discuss a reparametrization of the prior variance that
is robust with respect to the increase in dimension; we give numerical
experiments which exhibit that our reparametrization prevents the slowing down.
Our intuition on the behaviour of the prior hyper-parameter, with and without
reparametrization, is sufficiently general to include a broad class of
nonlinear inverse problems as well as other families of hyper-priors.Comment: to appear, SIAM/ASA Journal on Uncertainty Quantificatio
Statistical Inference for Complex Time Series Data
During recent years the focus of scientific interest has turned from low dimensional stationary time series to nonstationary time series and high dimensional time series. In addition new methodological challenges are coming from high frequency finance where data are recorded and analyzed on a millisecond basis. The three topics ânonstationarityâ, âhigh dimensionalityâ and âhigh frequencyâ are on the forefront of present research in time series analysis. The topics also have some overlap in that there already exists work on the intersection of these three topics, e.g. on locally stationary diffusion models, on high dimensional covariance matrices for high frequency data, or on multivariate dynamic factor models for nonstationary processes. The aim of the workshop was to bring together researchers from time series analysis, nonparametric statistics, econometrics and empirical finance to work on these topics. This aim was successfully achieved and the workshops was very well attended
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Asymptotic theory for Bayesian nonparametric inference in statistical models arising from partial differential equations
Partial differential equations (PDEs) are primary mathematical tools to model the behaviour of complex real-world systems. PDEs generally include a collection of parameters in their formulation, which are often unknown in applications and need to be estimated from the data. In the present thesis, we investigate the theoretical performance of nonparametric Bayesian procedures in such parameter identification problems in PDEs. In particular, inverse regression models for elliptic equations and stochastic diffusion
models are considered.
In Chapter 2, we study the statistical inverse problem of recovering an unknown function from a linear indirect measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernsteinâvon Mises theorem for a large collection of linear functionals of the unknown, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The general result is applied to three concrete examples that cover both the mildly and severely ill-posed cases: specifically, elliptic inverse problems, an elliptic boundary value problem, and the recovery of the initial condition of the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regulariser is an efficient estimator, and we derive frequentist guarantees for certain credible balls centred around it.
Chapter 3 is concerned with statistical nonlinear inverse problems. We focus on the prototypical example of recovering the unknown conductivity function in an elliptic PDE in divergence form from discrete noisy point evaluations of the PDE solution. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate, algebraic in inverse sample size, for the estimation error of the associated posterior means.
Finally, in Chapter 4 we extend the posterior consistency analysis to dynamical models based on stochastic differential equations. We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and p-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension.
Chapter 1 is dedicated to introducing the statistical models considered in Chapters 2 - 4, and to providing an overview of the theoretical results derived therein. The main theorems of Chapter 2 and Chapter 3 are illustrated via the results of simulations, and detailed comments are provided on the implementation.Richard Nicklâs ERC grant No. 647812; EPSRC grant EP/L016516/1 for the
Cambridge Centre for Analysi
Lack-of-fit tests in semiparametric mixed models.
In this paper we obtain the asymptotic distribution of restricted likelihood ratio tests in mixed linear models with a fixed and finite number of random effects. We explain why for such models the often quoted 50:50 mixture of a chi-s quared random variable with one degree of freedom and a point mass at zero does not hold. Our motivation is a study of the use of wavelets for lack-of-fit testing within a mixed model framework. Even though wavelet shave received a lot of attention in the last say 15 years for the estimation of piecewise smooth functions, much less is known about their ability to check the adequacy of a parametric model when fitting the observed data. In particular we study the testing power of wavelets for testing a hypothesized parametric model within a mixed model framework. Experimental results show that in several situations the wavelet-based test significantly outperforms the com-petitor based on penalized regression splines. The obtained results are also applicable for testing in mixed models in general, and shed some new insight into previous results.Lack-off-fittest; Likelihood ratio test; Mixed models; One-sided test; Penalization; Restricted maximum likelihood; Variance components; Wavel; Asymptotic distribution; Distribution; Likelihood; Tests; Models; Model; Random effects; Effects; Studies; Lack-of-fit; Mixed model; Framework; Functions; Data; Power; Regression;
Fourier analysis of stationary time series in function space
We develop the basic building blocks of a frequency domain framework for
drawing statistical inferences on the second-order structure of a stationary
sequence of functional data. The key element in such a context is the spectral
density operator, which generalises the notion of a spectral density matrix to
the functional setting, and characterises the second-order dynamics of the
process. Our main tool is the functional Discrete Fourier Transform (fDFT). We
derive an asymptotic Gaussian representation of the fDFT, thus allowing the
transformation of the original collection of dependent random functions into a
collection of approximately independent complex-valued Gaussian random
functions. Our results are then employed in order to construct estimators of
the spectral density operator based on smoothed versions of the periodogram
kernel, the functional generalisation of the periodogram matrix. The
consistency and asymptotic law of these estimators are studied in detail. As
immediate consequences, we obtain central limit theorems for the mean and the
long-run covariance operator of a stationary functional time series. Our
results do not depend on structural modelling assumptions, but only functional
versions of classical cumulant mixing conditions, and are shown to be stable
under discrete observation of the individual curves.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1086 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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