62 research outputs found
Bayesian Inference for partially observed SDEs Driven by Fractional Brownian Motion
We consider continuous-time diffusion models driven by fractional Brownian
motion. Observations are assumed to possess a non-trivial likelihood given the
latent path. Due to the non-Markovianity and high-dimensionality of the latent
paths, estimating posterior expectations is a computationally challenging
undertaking. We present a reparameterization framework based on the Davies and
Harte method for sampling stationary Gaussian processes and use this framework
to construct a Markov chain Monte Carlo algorithm that allows computationally
efficient Bayesian inference. The Markov chain Monte Carlo algorithm is based
on a version of hybrid Monte Carlo that delivers increased efficiency when
applied on the high-dimensional latent variables arising in this context. We
specify the methodology on a stochastic volatility model allowing for memory in
the volatility increments through a fractional specification. The methodology
is illustrated on simulated data and on the S&P500/VIX time series and is shown
to be effective. Contrary to a long range dependence attribute of such models
often assumed in the literature, with Hurst parameter larger than 1/2, the
posterior distribution favours values smaller than 1/2, pointing towards medium
range dependence
A 4D-Var Method with Flow-Dependent Background Covariances for the Shallow-Water Equations
The 4D-Var method for filtering partially observed nonlinear chaotic
dynamical systems consists of finding the maximum a-posteriori (MAP) estimator
of the initial condition of the system given observations over a time window,
and propagating it forward to the current time via the model dynamics. This
method forms the basis of most currently operational weather forecasting
systems. In practice the optimization becomes infeasible if the time window is
too long due to the non-convexity of the cost function, the effect of model
errors, and the limited precision of the ODE solvers. Hence the window has to
be kept sufficiently short, and the observations in the previous windows can be
taken into account via a Gaussian background (prior) distribution. The choice
of the background covariance matrix is an important question that has received
much attention in the literature. In this paper, we define the background
covariances in a principled manner, based on observations in the previous
assimilation windows, for a parameter . The method is at most times
more computationally expensive than using fixed background covariances,
requires little tuning, and greatly improves the accuracy of 4D-Var. As a
concrete example, we focus on the shallow-water equations. The proposed method
is compared against state-of-the-art approaches in data assimilation and is
shown to perform favourably on simulated data. We also illustrate our approach
on data from the recent tsunami of 2011 in Fukushima, Japan.Comment: 32 pages, 5 figure
Online Smoothing for Diffusion Processes Observed with Noise
We introduce a methodology for online estimation of smoothing expectations
for a class of additive functionals, in the context of a rich family of
diffusion processes (that may include jumps) -- observed at discrete-time
instances. We overcome the unavailability of the transition density of the
underlying SDE by working on the augmented pathspace. The new method can be
applied, for instance, to carry out online parameter inference for the
designated class of models. Algorithms defined on the infinite-dimensional
pathspace have been developed in the last years mainly in the context of MCMC
techniques. There, the main benefit is the achievement of mesh-free mixing
times for the practical time-discretised algorithm used on a PC. Our own
methodology sets up the framework for infinite-dimensional online filtering --
an important positive practical consequence is the construct of estimates with
the variance that does not increase with decreasing mesh-size. Besides
regularity conditions, our method is, in principle, applicable under the weak
assumption -- relatively to restrictive conditions often required in the MCMC
or filtering literature of methods defined on pathspace -- that the SDE
covariance matrix is invertible
Monte Carlo Co-Ordinate Ascent Variational Inference
In Variational Inference (VI), coordinate-ascent and gradient-based
approaches are two major types of algorithms for approximating
difficult-to-compute probability densities. In real-world implementations of
complex models, Monte Carlo methods are widely used to estimate expectations in
coordinate-ascent approaches and gradients in derivative-driven ones. We
discuss a Monte Carlo Co-ordinate Ascent VI (MC-CAVI) algorithm that makes use
of Markov chain Monte Carlo (MCMC) methods in the calculation of expectations
required within Co-ordinate Ascent VI (CAVI). We show that, under regularity
conditions, an MC-CAVI recursion will get arbitrarily close to a maximiser of
the evidence lower bound (ELBO) with any given high probability. In numerical
examples, the performance of MC-CAVI algorithm is compared with that of MCMC
and -- as a representative of derivative-based VI methods -- of Black Box VI
(BBVI). We discuss and demonstrate MC-CAVI's suitability for models with hard
constraints in simulated and real examples. We compare MC-CAVI's performance
with that of MCMC in an important complex model used in Nuclear Magnetic
Resonance (NMR) spectroscopy data analysis -- BBVI is nearly impossible to be
employed in this setting due to the hard constraints involved in the model
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