Modèles à espace d'états non linéaires/non gaussiens et inférence bayésienne par méthode {MCMC} -- Une application en évaluation des stocks halieutiques

National audienceDifference equations with delay are widely used to model the evolution of the biomass of a fish stock (delay difference models). Represented as a state- space model they allow, starting from the data of the annual catches, a relevant Bayesian analysis. For this purpose we can use an hybrid MCMC method combi- ning a Metropolis-Hastings algorithm within a Gibbs sampler, namely the single- component Metropolis-Hastings algorithm

Analyse bayésienne de modèles markoviens d'évolution de ressources naturelles

International audienceOne applies Monte Carlo methods to state sapce models with unknown parameters. The first one is a Monte Carlo Markov Chain algorithm. The second one is the particle filtering. We compare these methods applied to a biomass evolution model for fisheries

Bayesian numerical inference for hidden Markov models

International audienceIn many situations it is important to be able to propose N independent real- izations of a given distribution law. We propose a strategy for making N parallel Monte Carlo Markov Chains (MCMC) interact in order to get an approximation of an indepen- dent N-sample of a given target law. In this method each individual chain proposes can- didates for all other chains. We prove that the set of interacting chains is itself a MCMC method for the product of N target measures. Compared to independent parallel chains this method is more time consuming, but we show through examples that it possesses many advantages. This approach is applied to a biomass evolution model

{MCMC} for non linear/non {Gaussian} state-space models: Application to fishery stock assessment

International audienceWe consider a Monte Carlo Markov chain (MCMC) algorithm for fisheries stock assess- ment. The biomass of this stock at a given year could be modeled as a nonlinear function of the biomass and catch for the two previous years, of different parameters (recruitment, growth rate, nat- ural mortality rate). Given a time series of annual catch and effort data, we would like to achieve the best fitting between the data and a class of non linear/non Gaussian state-space models

A Markov model of land use dynamics

The application of the Markov chain to modeling agricultural succession is well known. In most cases, the main problem is the inference of the model, i.e. the estimation of the transition matrix. In this work we present methods to estimate the transition matrix from historical observations. In addition to the estimator of maximum likelihood (MLE), we also consider the Bayes estimator associated with the Jeffreys prior. This Bayes estimator will be approximated by a Markov chain Monte Carlo (MCMC) method. We also propose a method based on the sojourn time to test the adequation of Markov chain model to the dataset

Modèle markovien d'octroi de crédit en microfinance

ABSTRACT. Starting from the generalized model of Osman Khodr and Francine Diener [1], we present a new model that meets the expectations of the microfinance institution (MFI) and that of the borrowers and that incorporates all the characteristics of the poor, namely tolerance in case of partial default and the possibility of having a progressive loan automatically. This model will provide microfinance institutions with a decision support tool that is better adapted to the reality of microfinance. Our Markov chain consists of several statements associated with the economic status of the borrower including three types of recipients B 1 (state of being beneficiary at a time t = 0), B 2 (state to be beneficiary at a time t = 1) and I (state of financial inclusion: permanent beneficiary), an applicant state A 1 and A T −1 ((T − 1) excluded states). We modeled a borrower's behavior by a λ parameter that depends on the borrower's α probability of success. At the initial time, λ = 1+α 1−α , this quantity changes as soon as the borrower moves from one state to another with a probability of success different from α. The agency's decision to grant a credit depends entirely on the λ parameter which is compared to the set subjective threshold-values. The chance γ to have a loan (γ: probability of credit request granted) for a borrower depends on the parameter λ, with γ = 1 − 1 λ. keywords: Microfinance, Credit Grant Decision, Markov Chain, Individual Loan, Dynamic Incentive, Updated Expected ProfitEn partant du modèle généralisé de Osman Khodr et Francine Diener [1], nous présentons un nouveau modèle qui répond aux attentes de l'institution de microfinance (IMF) et celle des emprunteurs et qui incorpore toutes les caracté-ristiques des populations pauvres, à savoir la tolérance en cas de défaut partiel et la possibilité d'avoir un prêt progressif de façon automatique. Ce modèle offrira aux institutions de microfinance un outil d'aide à la décision plus adapté à la réalité de la microfinance. Notre chaîne de Markov comprend plusieurs états associés à la situation économique de l'emprunteur dont trois types de bénéficiaires B 1 (état d'être bénéficiaire au temps t = 0), B 2 (état d'être bénéficiaire au temps t = 1) et I (état d'inclusion financière: bénéficiaire permanent), un état de demandeur A 1 et A T −1 ((T − 1) états d'exclus). Nous avons modélisé le comportement d'un emprunteur par un paramètre λ qui dépend de la probabilité α de réussite de l'emprunteur. A l'instant initial, λ = 1+α 1−α , cette quantité change dès que l'emprunteur passe d'un état à un autre avec une probabilité de réussite différente de α. La décision de l'agence d'accorder un crédit dépend entièrement du paramètre λ qui est comparé aux valeurs-seuils subjectives fixées. La chance γ d'avoir un prêt (γ: probabilité de demande de crédit accordée) pour un emprunteur est fonction du paramètre λ, avec γ = 1 − 1 λ. MOTS-CLÉS : Microfinance, Décision d'octroi de crédit, chaîne de Markov, Prêt individuel, Incitation dynamique, Profit espéré actualis

Identification d'un systeme non-lineaire partiellement observe par la methode de la distance minimale

Nous considerons le probleme de l'estimation de parametres par la methode de la distance minimale, pour les processus de type diffusion partiellement observes. Nous construisons l'estimateur de la distance minimale (EDM) a partir de n observations independantes et nous montrons la consistance et la normalite asymptotique de l'EDM quand n tend vers l'infini. Nous etudions ensuite le cas particulier des systemes lineaires et nous presentons quelques resultats numeriques

Analyse bayésienne de modèles markoviens d'évolution de ressources naturelles

International audienceOne applies Monte Carlo methods to state sapce models with unknown parameters. The first one is a Monte Carlo Markov Chain algorithm. The second one is the particle filtering. We compare these methods applied to a biomass evolution model for fisheries

Un modèle markovien de transition agraire

International audienceWe present a Markov model for agricultural successions with 4 states. The application of Markov models to agricultural succession problems is not new, but relatively new tools of numerical Bayesian inference allow us to test general prior laws. I addition to the maximum likelihood estimate, we consider the Jeffreys prior (non-informative), and calculate the associated Bayesian estimator with a Metropolis-Hastings approximation procedure. We study the capabilities and limitations of this approach.Nous présentons un modèle de Markov de transition agraire à 4 états. L'application de ce genre de modèles aux problèmes de successions agraires n'est pas nouveau, mais les outils relativement récents de l'inférence bayésienne numérique permettent de tester des lois a priori plus générales. En plus de l'estimateur du maximum de vraisemblance, nous considérons la loi a priori de Jeffreys (non informative) et calculons l'estimateur bayésien associé à l'aide d'une approximation de Metropolis–Hastings. Nous étudions les capacités et les limites de cette approche

Particle and cell approximations for nonlinear filtering

We consider the nonlinear filtering problem for systems with noise--free state equation. First, we study a particle approximation of the a posteriori probability distribution, and we give an estimate of the approximation error. Then we show, and we illustrate with numerical examples, that this approximation can produce a non consistent estimation of the state of the system when the measurement noise tends to zero. Hence, we propose a histogram--like modification of the particle approximation, which is always consistent. Finally, we present an application to target motion analysis