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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.
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We consider the problem of model choice for stochastic epidemic models given partial observation of a disease outbreak through time. Our main focus is on the use of Bayes factors. Although Bayes factors have appeared in the epidemic modelling literat
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We study the class of state-space models and perform maximum likelihood estimation for the model parameters. We consider a stochastic approximation expectation-maximization (SAEM) algorithm to maximize the likelihood function with the novelty of usin