102 research outputs found
Bayesian Static Parameter Estimation for Partially Observed Diffusions via Multilevel Monte Carlo
In this article we consider static Bayesian parameter estimation for
partially observed diffusions that are discretely observed. We work under the
assumption that one must resort to discretizing the underlying diffusion
process, for instance using the Euler-Maruyama method. Given this assumption,
we show how one can use Markov chain Monte Carlo (MCMC) and particularly
particle MCMC [Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle
Markov chain Monte Carlo methods (with discussion). J. R. Statist. Soc. Ser. B,
72, 269--342] to implement a new approximation of the multilevel (ML) Monte
Carlo (MC) collapsing sum identity. Our approach comprises constructing an
approximate coupling of the posterior density of the joint distribution over
parameter and hidden variables at two different discretization levels and then
correcting by an importance sampling method. The variance of the weights are
independent of the length of the observed data set. The utility of such a
method is that, for a prescribed level of mean square error, the cost of this
MLMC method is provably less than i.i.d. sampling from the posterior associated
to the most precise discretization. However the method here comprises using
only known and efficient simulation methodologies. The theoretical results are
illustrated by inference of the parameters of two prototypical processes given
noisy partial observations of the process: the first is an Ornstein Uhlenbeck
process and the second is a more general Langevin equation
Bayesian Parameter Inference for Partially Observed Diffusions using Multilevel Stochastic Runge-Kutta Methods
We consider the problem of Bayesian estimation of static parameters
associated to a partially and discretely observed diffusion process. We assume
that the exact transition dynamics of the diffusion process are unavailable,
even up-to an unbiased estimator and that one must time-discretize the
diffusion process. In such scenarios it has been shown how one can introduce
the multilevel Monte Carlo method to reduce the cost to compute posterior
expected values of the parameters for a pre-specified mean square error (MSE).
These afore-mentioned methods rely on upon the Euler-Maruyama discretization
scheme which is well-known in numerical analysis to have slow convergence
properties. We adapt stochastic Runge-Kutta (SRK) methods for Bayesian
parameter estimation of static parameters for diffusions. This can be
implemented in high-dimensions of the diffusion and seemingly under-appreciated
in the uncertainty quantification and statistics fields. For a class of
diffusions and SRK methods, we consider the estimation of the posterior
expectation of the parameters. We prove that to achieve a MSE of
, for given, the associated work is
. Whilst the latter is achievable for the Milstein
scheme, this method is often not applicable for diffusions in dimension larger
than two. We also illustrate our methodology in several numerical examples
Multilevel Monte Carlo for a class of Partially Observed Processes in Neuroscience
In this paper we consider Bayesian parameter inference associated to a class
of partially observed stochastic differential equations (SDE) driven by jump
processes. Such type of models can be routinely found in applications, of which
we focus upon the case of neuroscience. The data are assumed to be observed
regularly in time and driven by the SDE model with unknown parameters. In
practice the SDE may not have an analytically tractable solution and this leads
naturally to a time-discretization. We adapt the multilevel Markov chain Monte
Carlo method of [11], which works with a hierarchy of time discretizations and
show empirically and theoretically that this is preferable to using one single
time discretization. The improvement is in terms of the computational cost
needed to obtain a pre-specified numerical error. Our approach is illustrated
on models that are found in neuroscience
A Multilevel Approach for Stochastic Nonlinear Optimal Control
We consider a class of finite time horizon nonlinear stochastic optimal
control problem, where the control acts additively on the dynamics and the
control cost is quadratic. This framework is flexible and has found
applications in many domains. Although the optimal control admits a path
integral representation for this class of control problems, efficient
computation of the associated path integrals remains a challenging Monte Carlo
task. The focus of this article is to propose a new Monte Carlo approach that
significantly improves upon existing methodology. Our proposed methodology
first tackles the issue of exponential growth in variance with the time horizon
by casting optimal control estimation as a smoothing problem for a state space
model associated with the control problem, and applying smoothing algorithms
based on particle Markov chain Monte Carlo. To further reduce computational
cost, we then develop a multilevel Monte Carlo method which allows us to obtain
an estimator of the optimal control with mean squared
error with a computational cost of
. In contrast, a computational cost
of is required for existing methodology to achieve
the same mean squared error. Our approach is illustrated on two numerical
examples, which validate our theory
Unbiased Parameter Estimation for Partially Observed Diffusions
In this article we consider the estimation of static parameters for partially
observed diffusion process with discrete-time observations over a fixed time
interval. In particular, we assume that one must time-discretize the partially
observed diffusion process and work with the model with bias and consider
maximizing the resulting log-likelihood. Using a novel double randomization
scheme, based upon Markovian stochastic approximation we develop a new method
to unbiasedly estimate the static parameters, that is, to obtain the maximum
likelihood estimator with no time discretization bias. Under assumptions we
prove that our estimator is unbiased and investigate the method in several
numerical examples, showing that it can empirically out-perform existing
unbiased methodology.Comment: 27 pages, 8 figure
Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art
Stochasticity is a key characteristic of intracellular processes such as gene
regulation and chemical signalling. Therefore, characterising stochastic
effects in biochemical systems is essential to understand the complex dynamics
of living things. Mathematical idealisations of biochemically reacting systems
must be able to capture stochastic phenomena. While robust theory exists to
describe such stochastic models, the computational challenges in exploring
these models can be a significant burden in practice since realistic models are
analytically intractable. Determining the expected behaviour and variability of
a stochastic biochemical reaction network requires many probabilistic
simulations of its evolution. Using a biochemical reaction network model to
assist in the interpretation of time course data from a biological experiment
is an even greater challenge due to the intractability of the likelihood
function for determining observation probabilities. These computational
challenges have been subjects of active research for over four decades. In this
review, we present an accessible discussion of the major historical
developments and state-of-the-art computational techniques relevant to
simulation and inference problems for stochastic biochemical reaction network
models. Detailed algorithms for particularly important methods are described
and complemented with MATLAB implementations. As a result, this review provides
a practical and accessible introduction to computational methods for stochastic
models within the life sciences community
An Online Method for the Data Driven Stochastic Optimal Control Problem with Unknown Model Parameters
In this work, an efficient sample-wise data driven control solver will be
developed to solve the stochastic optimal control problem with unknown model
parameters. A direct filter method will be applied as an online parameter
estimation method that dynamically estimates the target model parameters upon
receiving the data, and a sample-wise optimal control solver will be provided
to efficiently search for the optimal control. Then, an effective overarching
algorithm will be introduced to combine the parameter estimator and the optimal
control solver. Numerical experiments will be carried out to demonstrate the
effectiveness and the efficiency of the sample-wise data driven control method
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