Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userEfficient inference for stochastic differential equation mixed-effects models using correlated particle pseudo-marginal algorithms
413
0
0.0
(
0
)
Added by
Samuel Wiqvist
Publication date
2019
fields
Mathematical Statistics
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that are able to account for random variability inherent in the underlying time-dynamics, as well as the variability between experimental units and, optionally, account for measurement error. Fully Bayesian inference for state-space SDEMEMs is performed, using data at discrete times that may be incomplete and subject to measurement error. However, the inference problem is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. A Gibbs sampler is proposed to target the marginal posterior of all parameter values of interest. The algorithm is made computationally efficient through careful use of blocking strategies and correlated pseudo-marginal Metropolis-Hastings steps within the Gibbs scheme. The resulting methodology is flexible and is able to deal with a large class of SDEMEMs. The methodology is demonstrated on three case studies, including tumor growth dynamics and neuronal data. The gains in terms of increased computational efficiency are model and data dependent, but unless bespoke sampling strategies requiring analytical derivations are possible for a given model, we generally observe an efficiency increase of one order of magnitude when using correlated particle methods together with our blocked-Gibbs strategy.
rate research
Read More
Motivated by penalized likelihood maximization in complex models, we study optimization problems where neither the function to optimize nor its gradient have an explicit expression, but its gradient can be approximated by a Monte Carlo technique. We propose a new algorithm based on a stochastic approximation of the Proximal-Gradient (PG) algorithm. This new algorithm, named Stochastic Approximation PG (SAPG) is the combination of a stochastic gradient descent step which - roughly speaking - computes a smoothed approximation of the past gradient along the iterations, and a proximal step. The choice of the step size and the Monte Carlo batch size for the stochastic gradient descent step in SAPG are discussed. Our convergence results cover the cases of biased and unbiased Monte Carlo approximations. While the convergence analysis of the Monte Carlo-PG is already addressed in the literature (see Atchade et al. [2016]), the convergence analysis of SAPG is new. The two algorithms are compared on a linear mixed effect model as a toy example. A more challenging application is proposed on non-linear mixed effect models in high dimension with a pharmacokinetic data set including genomic covariates. To our best knowledge, our work provides the first convergence result of a numerical method designed to solve penalized Maximum Likelihood in a non-linear mixed effect model.
Stochastic differential equations (SDEs) are established tools to model physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE intrinsic randomness of a system around its drift can be identified and separated from the drift itself. When it is of interest to model dynamics within a given population, i.e. to model simultaneously the performance of several experiments or subjects, mixed-effects modelling allows for the distinction of between and within experiment variability. A framework to model dynamics within a population using SDEs is proposed, representing simultaneously several sources of variation: variability between experiments using a mixed-effects approach and stochasticity in the individual dynamics using SDEs. These stochastic differential mixed-effects models have applications in e.g. pharmacokinetics/pharmacodynamics and biomedical modelling. A parameter estimation method is proposed and computational guidelines for an efficient implementation are given. Finally the method is evaluated using simulations from standard models like the two-dimensional Ornstein-Uhlenbeck (OU) and the square root models.
We consider Bayesian inference for stochastic differential equation mixed effects models (SDEMEMs) exemplifying tumor response to treatment and regrowth in mice. We produce an extensive study on how a SDEMEM can be fitted using both exact inference based on pseudo-marginal MCMC and approximate inference via Bayesian synthetic likelihoods (BSL). We investigate a two-compartments SDEMEM, these corresponding to the fractions of tumor cells killed by and survived to a treatment, respectively. Case study data considers a tumor xenography study with two treatment groups and one control, each containing 5-8 mice. Results from the case study and from simulations indicate that the SDEMEM is able to reproduce the observed growth patterns and that BSL is a robust tool for inference in SDEMEMs. Finally, we compare the fit of the SDEMEM to a similar ordinary differential equation model. Due to small sample sizes, strong prior information is needed to identify all model parameters in the SDEMEM and it cannot be determined which of the two models is the better in terms of predicting tumor growth curves. In a simulation study we find that with a sample of 17 mice per group BSL is able to identify all model parameters and distinguish treatment groups.
An ordinary differential equation (ODE) model, whose regression curves are a set of solution curves for some ODEs, poses a challenge in parameter estimation. The challenge, due to the frequent absence of analytic solutions and the complicated likelihood surface, tends to be more severe especially for larger models with many parameters and variables. Yang and Lee (2021) proposed a state-space model with variational Bayes (SSVB) for ODE, capable of fast and stable estimation in somewhat large ODE models. The method has shown excellent performance in parameter estimation but has a weakness of underestimation of the posterior covariance, which originates from the mean-field variational method. This paper proposes a way to overcome the weakness by using the Laplace approximation. In numerical experiments, the covariance modified by the Laplace approximation showed a high degree of improvement when checked against the covariances obtained by a standard Markov chain Monte Carlo method. With the improved covariance estimation, the SSVB renders fairly accurate posterior approximations.
This article addresses the problem of efficient Bayesian inference in dynamic systems using particle methods and makes a number of contributions. First, we develop a correlated pseudo-marginal (CPM) approach for Bayesian inference in state space (SS) models that is based on filtering the disturbances, rather than the states. This approach is useful when the state transition density is intractable or inefficient to compute, and also when the dimension of the disturbance is lower than the dimension of the state. Second, we propose a block pseudo-marginal (BPM) method that uses as the estimate of the likelihood the average of G independent unbiased estimates of the likelihood. We associate a set of underlying uniform of standard normal random numbers used to construct each of the individual unbiased likelihood estimates and then use component-wise Markov Chain Monte Carlo to update the parameter vector jointly with one set of these random numbers at a time. This induces a correlation of approximately 1-1/G between the logs of the estimated likelihood at the proposed and current values of the model parameters. Third, we show for some non-stationary state space models that the BPM approach is much more efficient than the CPM approach, because it is difficult to translate the high correlation in the underlying random numbers to high correlation between the logs of the likelihood estimates. Although our focus has been on applying the BPM method to state space models, our results and approach can be used in a wide range of applications of the PM method, such as panel data models, subsampling problems and approximate Bayesian computation.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
