248 research outputs found
Consistent nonparametric Bayesian inference for discretely observed scalar diffusions
We study Bayes procedures for the problem of nonparametric drift estimation
for one-dimensional, ergodic diffusion models from discrete-time, low-frequency
data. We give conditions for posterior consistency and verify these conditions
for concrete priors, including priors based on wavelet expansions.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ385 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
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
Introduction to Automatic Backward Filtering Forward Guiding
In this document I aim to give an informal treatment of automatic Backward
Filtering Forward Guiding, a general algorithm for conditional sampling from a
Markov process on a directed acyclic graph. I'll show that the underlying ideas
can be understood with a basic background in probability and statistics. The
more technical treatment is the paper Automatic backward filtering forward
guiding for Markov processes and graphical models (Van der Meulen and Schauer,
2021). I specifically assume some background knowledge on likelihood based
inference and Bayesian statistics. The final sections are more demanding and
assume familiarity with continuous-time stochastic processes constructed from
their infinitesimal generator.
Clearly, all work discussed here is the result of research carried out over
the past decade together with various collaborators, most importantly Moritz
Schauer (Chalmers University of Technology and University of Gothenburg,
Sweden). Section 8 is based on joint work with Marcin Mider (Trium Analysis
Online GmbH, Germany) and Frank Sch\"afer (University of Basel, Switzerland) as
well
A non-parametric Bayesian approach to decompounding from high frequency data
Given a sample from a discretely observed compound Poisson process, we
consider non-parametric estimation of the density of its jump sizes, as
well as of its intensity We take a Bayesian approach to the
problem and specify the prior on as the Dirichlet location mixture of
normal densities. An independent prior for is assumed to be
compactly supported and to possess a positive density with respect to the
Lebesgue measure. We show that under suitable assumptions the posterior
contracts around the pair at essentially (up to a logarithmic
factor) the -rate, where is the number of observations and
is the mesh size at which the process is sampled. The emphasis is on
high frequency data, , but the obtained results are also valid for
fixed . In either case we assume that . Our
main result implies existence of Bayesian point estimates converging (in the
frequentist sense, in probability) to at the same rate.
We also discuss a practical implementation of our approach. The computational
problem is dealt with by inclusion of auxiliary variables and we develop a
Markov Chain Monte Carlo algorithm that samples from the joint distribution of
the unknown parameters in the mixture density and the introduced auxiliary
variables. Numerical examples illustrate the feasibility of this approach
Simulation of elliptic and hypo-elliptic conditional diffusions
Suppose is a multidimensional diffusion process. Assume that at time zero
the state of is fully observed, but at time only linear combinations
of its components are observed. That is, one only observes the vector
for a given matrix . In this paper we show how samples from the conditioned
process can be generated. The main contribution of this paper is to prove that
guided proposals, introduced in Schauer et al. (2017), can be used in a unified
way for both uniformly and hypo-elliptic diffusions, also when is not the
identity matrix. This is illustrated by excellent performance in two
challenging cases: a partially observed twice integrated diffusion with
multiple wells and the partially observed FitzHugh-Nagumo model
Nonparametric Bayesian inference for multidimensional compound Poisson processes
Given a sample from a discretely observed multidimensional compound Poisson
process, we study the problem of nonparametric estimation of its jump size
density and intensity . We take a nonparametric Bayesian
approach to the problem and determine posterior contraction rates in this
context, which, under some assumptions, we argue to be optimal posterior
contraction rates. In particular, our results imply the existence of Bayesian
point estimates that converge to the true parameter pair at
these rates. To the best of our knowledge, construction of nonparametric
density estimators for inference in the class of discretely observed
multidimensional L\'{e}vy processes, and the study of their rates of
convergence is a new contribution to the literature.Comment: Published at http://dx.doi.org/10.15559/15-VMSTA20 in the Modern
Stochastics: Theory and Applications (https://www.i-journals.org/vtxpp/VMSTA)
by VTeX (http://www.vtex.lt/
Fast and scalable non-parametric Bayesian inference for Poisson point processes
We study the problem of non-parametric Bayesian estimation of the intensity
function of a Poisson point process. The observations are independent
realisations of a Poisson point process on the interval . We propose two
related approaches. In both approaches we model the intensity function as
piecewise constant on bins forming a partition of the interval . In
the first approach the coefficients of the intensity function are assigned
independent gamma priors, leading to a closed form posterior distribution. On
the theoretical side, we prove that as the posterior
asymptotically concentrates around the "true", data-generating intensity
function at an optimal rate for -H\"older regular intensity functions (). In the second approach we employ a gamma Markov chain prior on the
coefficients of the intensity function. The posterior distribution is no longer
available in closed form, but inference can be performed using a
straightforward version of the Gibbs sampler. Both approaches scale well with
sample size, but the second is much less sensitive to the choice of .
Practical performance of our methods is first demonstrated via synthetic data
examples. We compare our second method with other existing approaches on the UK
coal mining disasters data. Furthermore, we apply it to the US mass shootings
data and Donald Trump's Twitter data.Comment: 45 pages, 22 figure
Bayesian wavelet de-noising with the caravan prior
According to both domain expert knowledge and empirical evidence, wavelet
coefficients of real signals tend to exhibit clustering patterns, in that they
contain connected regions of coefficients of similar magnitude (large or
small). A wavelet de-noising approach that takes into account such a feature of
the signal may in practice outperform other, more vanilla methods, both in
terms of the estimation error and visual appearance of the estimates. Motivated
by this observation, we present a Bayesian approach to wavelet de-noising,
where dependencies between neighbouring wavelet coefficients are a priori
modelled via a Markov chain-based prior, that we term the caravan prior.
Posterior computations in our method are performed via the Gibbs sampler. Using
representative synthetic and real data examples, we conduct a detailed
comparison of our approach with a benchmark empirical Bayes de-noising method
(due to Johnstone and Silverman). We show that the caravan prior fares well and
is therefore a useful addition to the wavelet de-noising toolbox.Comment: 32 pages, 15 figures, 4 table
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