96 research outputs found
An empirical Bayes procedure for the selection of Gaussian graphical models
A new methodology for model determination in decomposable graphical Gaussian
models is developed. The Bayesian paradigm is used and, for each given graph, a
hyper inverse Wishart prior distribution on the covariance matrix is
considered. This prior distribution depends on hyper-parameters. It is
well-known that the models's posterior distribution is sensitive to the
specification of these hyper-parameters and no completely satisfactory method
is registered. In order to avoid this problem, we suggest adopting an empirical
Bayes strategy, that is a strategy for which the values of the hyper-parameters
are determined using the data. Typically, the hyper-parameters are fixed to
their maximum likelihood estimations. In order to calculate these maximum
likelihood estimations, we suggest a Markov chain Monte Carlo version of the
Stochastic Approximation EM algorithm. Moreover, we introduce a new sampling
scheme in the space of graphs that improves the add and delete proposal of
Armstrong et al. (2009). We illustrate the efficiency of this new scheme on
simulated and real datasets
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
Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures
In this paper we provide general conditions to check on the model and the
prior to derive posterior concentration rates for data-dependent priors (or
empirical Bayes approaches). We aim at providing conditions that are close to
the conditions provided in the seminal paper by Ghosal and van der Vaart
(2007a). We then apply the general theorem to two different settings: the
estimation of a density using Dirichlet process mixtures of Gaussian random
variables with base measure depending on some empirical quantities and the
estimation of the intensity of a counting process under the Aalen model. A
simulation study for inhomogeneous Poisson processes also illustrates our
results. In the former case we also derive some results on the estimation of
the mixing density and on the deconvolution problem. In the latter, we provide
a general theorem on posterior concentration rates for counting processes with
Aalen multiplicative intensity with priors not depending on the data.Comment: With supplementary materia
Parametric inference for mixed models defined by stochastic differential equations
International audienceNon-linear mixed models defined by stochastic differential equations (SDEs) are consid- ered: the parameters of the diffusion process are random variables and vary among the individuals. A maximum likelihood estimation method based on the Stochastic Approximation EM algorithm, is proposed. This estimation method uses the Euler-Maruyama approximation of the diffusion, achieved using latent auxiliary data introduced to complete the diffusion process between each pair of measure- ment instants. A tuned hybrid Gibbs algorithm based on conditional Brownian bridges simulations of the unobserved process paths is included in this algorithm. The convergence is proved and the error induced on the likelihood by the Euler-Maruyama approximation is bounded as a function of the step size of the approximation. Results of a pharmacokinetic simulation study illustrate the accuracy of this estimation method. The analysis of the Theophyllin real dataset illustrates the relevance of the SDE approach relative to the deterministic approach
Estimation of parameters in incomplete data models defined by dynamical systems.
International audienceParametric incomplete data models defined by ordinary differential equa- tions (ODEs) are widely used in biostatistics to describe biological processes accurately. Their parameters are estimated on approximate models, whose regression functions are evaluated by a numerical integration method. Ac- curate and efficient estimations of these parameters are critical issues. This paper proposes parameter estimation methods involving either a stochas- tic approximation EM algorithm (SAEM) in the maximum likelihood es- timation, or a Gibbs sampler in the Bayesian approach. Both algorithms involve the simulation of non-observed data with conditional distributions using Hastings-Metropolis (H-M) algorithms. A modified H-M algorithm, including an original Local Linearization scheme to solve the ODEs, is pro- posed to reduce the computational time significantly. The convergence on the approximate model of all these algorithms is proved. The errors induced by the numerical solving method on the conditional distribution, the likelihood and the posterior distribution are bounded. The Bayesian and maximum likelihood estimation methods are illustrated on a simulated pharmacoki- netic nonlinear mixed-effects model defined by an ODE. Simulation results illustrate the ability of these algorithms to provide accurate estimates
A review on estimation of stochastic differential equations for pharmacokinetic/pharmacodynamic models
International audienceThis paper is a survey of existing estimation methods for pharmacokinetic/pharmacodynamic (PK/PD) models based on stochastic differential equations (SDEs). Most parametric estimation methods proposed for SDEs require high frequency data and are often poorly suited for PK/PD data which are usually sparse. Moreover, PK/PD experiments generally include not a single individual but a group of subjects, leading to a population estimation approach. This review concentrates on estimation methods which have been applied to PK/PD data, for SDEs observed with and without measurement noise, with a standard or a population approach. Besides, the adopted methodologies highly differ depending on the existence or not of an explicit transition density of the SDE
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