234 research outputs found
Reversible jump MCMC for nonparametric drift estimation for diffusion processes
In the context of nonparametric Bayesian estimation a Markov chain Monte
Carlo algorithm is devised and implemented to sample from the posterior
distribution of the drift function of a continuously or discretely observed
one-dimensional diffusion. The drift is modeled by a scaled linear combination
of basis functions with a Gaussian prior on the coefficients. The scaling
parameter is equipped with a partially conjugate prior. The number of basis
function in the drift is equipped with a prior distribution as well. For
continuous data, a reversible jump Markov chain algorithm enables the
exploration of the posterior over models of varying dimension. Subsequently, it
is explained how data-augmentation can be used to extend the algorithm to deal
with diffusions observed discretely in time. Some examples illustrate that the
method can give satisfactory results. In these examples a comparison is made
with another existing method as well
Nonparametric Bayesian methods for one-dimensional diffusion models
In this paper we review recently developed methods for nonparametric Bayesian
inference for one-dimensional diffusion models. We discuss different possible
prior distributions, computational issues, and asymptotic results
Consistency of Bayesian nonparametric inference for discretely observed jump diffusions
We introduce verifiable criteria for weak posterior consistency of
identifiable Bayesian nonparametric inference for jump diffusions with unit
diffusion coefficient and uniformly Lipschitz drift and jump coefficients in
arbitrary dimension. The criteria are expressed in terms of coefficients of the
SDEs describing the process, and do not depend on intractable quantities such
as transition densities. We also show that products of discrete net and
Dirichlet mixture model priors satisfy our conditions, again under an
identifiability assumption. This generalises known results by incorporating
jumps into previous work on unit diffusions with uniformly Lipschitz drift
coefficients.Comment: 20 page
Full adaptation to smoothness using randomly truncated series priors with Gaussian coefficients and inverse gamma scaling
We study random series priors for estimating a functional parameter (f\in
L^2[0,1]). We show that with a series prior with random truncation, Gaussian
coefficients, and inverse gamma multiplicative scaling, it is possible to
achieve posterior contraction at optimal rates and adaptation to arbitrary
degrees of smoothness. We present general results that can be combined with
existing rate of contraction results for various nonparametric estimation
problems. We give concrete examples for signal estimation in white noise and
drift estimation for a one-dimensional SDE
Bayesian estimation of discretely observed multi-dimensional diffusion processes using guided proposals
Estimation of parameters of a diffusion based on discrete time observations
poses a difficult problem due to the lack of a closed form expression for the
likelihood. From a Bayesian computational perspective it can be casted as a
missing data problem where the diffusion bridges in between discrete-time
observations are missing. The computational problem can then be dealt with
using a Markov-chain Monte-Carlo method known as data-augmentation. If unknown
parameters appear in the diffusion coefficient, direct implementation of
data-augmentation results in a Markov chain that is reducible. Furthermore,
data-augmentation requires efficient sampling of diffusion bridges, which can
be difficult, especially in the multidimensional case.
We present a general framework to deal with with these problems that does not
rely on discretisation. The construction generalises previous approaches and
sheds light on the assumptions necessary to make these approaches work. We
define a random-walk type Metropolis-Hastings sampler for updating diffusion
bridges. Our methods are illustrated using guided proposals for sampling
diffusion bridges. These are Markov processes obtained by adding a guiding term
to the drift of the diffusion. We give general guidelines on the construction
of these proposals and introduce a time change and scaling of the guided
proposal that reduces discretisation error. Numerical examples demonstrate the
performance of our methods
Statistics of Stochastic Differential Equations on Manifolds and Stratified Spaces (hybrid meeting)
Statistics for stochastic differential equations (SDEs) attempts to use SDEs as statistical models for real-world phenomena. This involves an understanding of qualitative properties of this class of stochastic processes which includes Brownian motion as well as estimation of parameters in the SDE or a nonparametric estimation of drift and diffusivity fields from observations. Observations can be in continuous time, in high frequency discrete time considering the limit of small inter-observation times or in discrete time with constant inter-obseration times. Application areas of SDEs where state spaces are naturally viewed as manifolds or stratified spaces include multivariate stochastic volatility models, stochastic evolution of shapes (e.g. of biological cells), time-varying image deformations for video analysis and phylogenetic trees
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