16,034 research outputs found
Inference for Differential Equation Models using Relaxation via Dynamical Systems
Statistical regression models whose mean functions are represented by
ordinary differential equations (ODEs) can be used to describe phenomenons
dynamical in nature, which are abundant in areas such as biology, climatology
and genetics. The estimation of parameters of ODE based models is essential for
understanding its dynamics, but the lack of an analytical solution of the ODE
makes the parameter estimation challenging. The aim of this paper is to propose
a general and fast framework of statistical inference for ODE based models by
relaxation of the underlying ODE system. Relaxation is achieved by a properly
chosen numerical procedure, such as the Runge-Kutta, and by introducing
additive Gaussian noises with small variances. Consequently, filtering methods
can be applied to obtain the posterior distribution of the parameters in the
Bayesian framework. The main advantage of the proposed method is computation
speed. In a simulation study, the proposed method was at least 14 times faster
than the other methods. Theoretical results which guarantee the convergence of
the posterior of the approximated dynamical system to the posterior of true
model are presented. Explicit expressions are given that relate the order and
the mesh size of the Runge-Kutta procedure to the rate of convergence of the
approximated posterior as a function of sample size
Estimating the Expected Value of Partial Perfect Information in Health Economic Evaluations using Integrated Nested Laplace Approximation
The Expected Value of Perfect Partial Information (EVPPI) is a
decision-theoretic measure of the "cost" of parametric uncertainty in decision
making used principally in health economic decision making. Despite this
decision-theoretic grounding, the uptake of EVPPI calculations in practice has
been slow. This is in part due to the prohibitive computational time required
to estimate the EVPPI via Monte Carlo simulations. However, recent developments
have demonstrated that the EVPPI can be estimated by non-parametric regression
methods, which have significantly decreased the computation time required to
approximate the EVPPI. Under certain circumstances, high-dimensional Gaussian
Process regression is suggested, but this can still be prohibitively expensive.
Applying fast computation methods developed in spatial statistics using
Integrated Nested Laplace Approximations (INLA) and projecting from a
high-dimensional into a low-dimensional input space allows us to decrease the
computation time for fitting these high-dimensional Gaussian Processes, often
substantially. We demonstrate that the EVPPI calculated using our method for
Gaussian Process regression is in line with the standard Gaussian Process
regression method and that despite the apparent methodological complexity of
this new method, R functions are available in the package BCEA to implement it
simply and efficiently
Bayesian Analysis of ODE's: solver optimal accuracy and Bayes factors
In most relevant cases in the Bayesian analysis of ODE inverse problems, a
numerical solver needs to be used. Therefore, we cannot work with the exact
theoretical posterior distribution but only with an approximate posterior
deriving from the error in the numerical solver. To compare a numerical and the
theoretical posterior distributions we propose to use Bayes Factors (BF),
considering both of them as models for the data at hand. We prove that the
theoretical vs a numerical posterior BF tends to 1, in the same order (of the
step size used) as the numerical forward map solver does. For higher order
solvers (eg. Runge-Kutta) the Bayes Factor is already nearly 1 for step sizes
that would take far less computational effort. Considerable CPU time may be
saved by using coarser solvers that nevertheless produce practically error free
posteriors. Two examples are presented where nearly 90% CPU time is saved while
all inference results are identical to using a solver with a much finer time
step.Comment: 28 pages, 6 figure
Spectral Density-Based and Measure-Preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs
Approximate Bayesian Computation (ABC) has become one of the major tools of
likelihood-free statistical inference in complex mathematical models.
Simultaneously, stochastic differential equations (SDEs) have developed to an
established tool for modelling time dependent, real world phenomena with
underlying random effects. When applying ABC to stochastic models, two major
difficulties arise. First, the derivation of effective summary statistics and
proper distances is particularly challenging, since simulations from the
stochastic process under the same parameter configuration result in different
trajectories. Second, exact simulation schemes to generate trajectories from
the stochastic model are rarely available, requiring the derivation of suitable
numerical methods for the synthetic data generation. To obtain summaries that
are less sensitive to the intrinsic stochasticity of the model, we propose to
build up the statistical method (e.g., the choice of the summary statistics) on
the underlying structural properties of the model. Here, we focus on the
existence of an invariant measure and we map the data to their estimated
invariant density and invariant spectral density. Then, to ensure that these
model properties are kept in the synthetic data generation, we adopt
measure-preserving numerical splitting schemes. The derived property-based and
measure-preserving ABC method is illustrated on the broad class of partially
observed Hamiltonian type SDEs, both with simulated data and with real
electroencephalography (EEG) data. The proposed ingredients can be incorporated
into any type of ABC algorithm and directly applied to all SDEs that are
characterised by an invariant distribution and for which a measure-preserving
numerical method can be derived.Comment: 35 pages, 21 figure
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