2,846 research outputs found
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
An Optimal Control Derivation of Nonlinear Smoothing Equations
The purpose of this paper is to review and highlight some connections between
the problem of nonlinear smoothing and optimal control of the Liouville
equation. The latter has been an active area of recent research interest owing
to work in mean-field games and optimal transportation theory. The nonlinear
smoothing problem is considered here for continuous-time Markov processes. The
observation process is modeled as a nonlinear function of a hidden state with
an additive Gaussian measurement noise. A variational formulation is described
based upon the relative entropy formula introduced by Newton and Mitter. The
resulting optimal control problem is formulated on the space of probability
distributions. The Hamilton's equation of the optimal control are related to
the Zakai equation of nonlinear smoothing via the log transformation. The
overall procedure is shown to generalize the classical Mortensen's minimum
energy estimator for the linear Gaussian problem.Comment: 7 pages, 0 figures, under peer reviewin
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
Moment-Based Variational Inference for Markov Jump Processes
We propose moment-based variational inference as a flexible framework for
approximate smoothing of latent Markov jump processes. The main ingredient of
our approach is to partition the set of all transitions of the latent process
into classes. This allows to express the Kullback-Leibler divergence between
the approximate and the exact posterior process in terms of a set of moment
functions that arise naturally from the chosen partition. To illustrate
possible choices of the partition, we consider special classes of jump
processes that frequently occur in applications. We then extend the results to
parameter inference and demonstrate the method on several examples.Comment: Accepted by the 36th International Conference on Machine Learning
(ICML 2019
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