8,528 research outputs found
Variational Gaussian Process Diffusion Processes
Diffusion processes are a class of stochastic differential equations (SDEs)
providing a rich family of expressive models that arise naturally in dynamic
modelling tasks. Probabilistic inference and learning under generative models
with latent processes endowed with a non-linear diffusion process prior are
intractable problems. We build upon work within variational inference
approximating the posterior process as a linear diffusion process, point out
pathologies in the approach, and propose an alternative parameterization of the
Gaussian variational process using a continuous exponential family description.
This allows us to trade a slow inference algorithm with fixed-point iterations
for a fast algorithm for convex optimization akin to natural gradient descent,
which also provides a better objective for the learning of model parameters.Comment: 26 pages, 11 figure
A new variational radial basis function approximation for inference in multivariate diffusions
In this paper we present a radial basis function based extension to a recently proposed variational algorithm for approximate inference for diffusion processes. Inference, for state and in particular (hyper-) parameters, in diffusion processes is a challenging and crucial task. We show that the new radial basis function approximation based algorithm converges to the original algorithm and has beneficial characteristics when estimating (hyper-)parameters. We validate our new approach on a nonlinear double well potential dynamical system
Derivations of variational gaussian process approximation framework
Recently, within the VISDEM project (EPSRC funded EP/C005848/1), a novel variational approximation framework has been developed for inference in partially observed, continuous space-time, diffusion processes. In this technical report all the derivations of the variational framework, from the initial work, are provided in detail to help the reader better understand the framework and its assumptions
Approximate inference for state-space models
This thesis is concerned with state estimation in partially observed diffusion processes with
discrete time observations. This problem can be solved exactly in a Bayesian framework, up
to a set of generally intractable stochastic partial differential equations. Numerous approximate
inference methods exist to tackle the problem in a practical way. This thesis introduces a novel
deterministic approach that can capture non normal properties of the exact Bayesian solution.
The variational approach to approximate inference has a natural formulation for partially
observed diffusion processes. In the variational framework, the exact Bayesian solution is the
optimal variational solution and, as a consequence, all variational approximations have a universal
ordering in terms of optimality. The new approach generalises the current variational
Gaussian process approximation algorithm, and therefore provides a method for obtaining super
optimal algorithms in relation to the current state-of-the-art variational methods.
Every diffusion process is composed of a drift component and a diffusion component. To
obtain a variational formulation, the diffusion component must be fixed. Subsequently, the exact
Bayesian solution and all variational approximations are characterised by their drift component.
To use a particular class of drift, the variational formulation requires a closed form for the
family of marginal densities generated by diffusion processes with drift components from the
aforementioned class. This requirement in general cannot be met. In this thesis, it is shown how
this coupling can be weakened, allowing for more flexible relations between the variational drift
and the variational approximations of the marginal densities of the true posterior process. Based
on this revelation, a selection of novel variational drift components are proposed
A variational approach to path estimation and parameter inference of hidden diffusion processes
We consider a hidden Markov model, where the signal process, given by a
diffusion, is only indirectly observed through some noisy measurements. The
article develops a variational method for approximating the hidden states of
the signal process given the full set of observations. This, in particular,
leads to systematic approximations of the smoothing densities of the signal
process. The paper then demonstrates how an efficient inference scheme, based
on this variational approach to the approximation of the hidden states, can be
designed to estimate the unknown parameters of stochastic differential
equations. Two examples at the end illustrate the efficacy and the accuracy of
the presented method.Comment: 37 pages, 2 figures, revise
Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes
The application of Stochastic Differential Equations (SDEs) to the analysis
of temporal data has attracted increasing attention, due to their ability to
describe complex dynamics with physically interpretable equations. In this
paper, we introduce a non-parametric method for estimating the drift and
diffusion terms of SDEs from a densely observed discrete time series. The use
of Gaussian processes as priors permits working directly in a function-space
view and thus the inference takes place directly in this space. To cope with
the computational complexity that requires the use of Gaussian processes, a
sparse Gaussian process approximation is provided. This approximation permits
the efficient computation of predictions for the drift and diffusion terms by
using a distribution over a small subset of pseudo-samples. The proposed method
has been validated using both simulated data and real data from economy and
paleoclimatology. The application of the method to real data demonstrates its
ability to capture the behaviour of complex systems
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