The Simulation of Bayesian Dynamic Models

Abstract: This paper mainly deals with the stochastic simulation algorithm of dynamic linear models without normal assumptionsObservation equation:y_t=F_t~Tθ_t+v_t,v_t~[0,V_t];State equation:θ_t=G_tθ_(t-1)+ω_t,ω~[0,W_t]. ("without normal assumptions" means parameter vectorθand error vectorν、ωdo not obey normal distributions)and nonlinear dynamicmodelsObservation equation:(y_t|θ_t)~p(y_t|θ_t);State equation:(y_t|θ_(t-1))~p(y_t|θ_(t-1)). ("Nonlinearity" means observation vector y is not linear funct ion of parameter vectorθ),and then a kind of new Bayesian Dynamic Model is given.In the final the choice and monitoring of dynamic models are discussed.The main content contains:In the first chapter,the development of Bayes theory is introduced in which we mainly describe basic concepts of Bayesian Dynamic Model and these difficulties in current study.In the second section we summarize the whole framework of this article.In the second chapter,we introduce a wide range of stochastic simulation methods,mainly containing the importance sampling,Metropolis-Hastings sampling, Gibbs sampling.Since Metropolis-Hastings sampling and Gibbs sampling occupy an important position in simulations for dynamic models,a more detailed study including taking samples and convergence diagnosis of samples is done,and a specific diagnosis method:Riemann convergence diagnosis is given.In the third chapter,using stochastic simulation methods the above two kinds of models are resolved.The main idea is:The updating of sample density is used to replace the recursive density functions,furthermore to make inference;in this process, different models are given different recursive algorithms.Moreover,in this chapter,a kind of new dynamic model with log-concave density is given and proved satisfying with the "conjugate conditions" and we give some conclusions about this model based on Gibbs sampling.In the fourth chapter we discuss Monte Carlo methods in the selection of the model.The main idea is to use these Monte Carlo methods to obtain a Bayesian factor, which is useful for Bayesian model selection.As Bayesian factor which is mentioned in the literature[2]is effective in the complex circumstances no longer,we give a kind of new Bayesian factor that is very useful for these models discussed.In the fifth chapter,two kind of model-monitoring methods are given.The first method is to construct aχ~2-statistic using observation errors,then usingχ~2-statistic the cut-off point for given statistical accuracy could be obtained to monitor the model from local time to whole time;the second is a kind of new Bayesian factor which is the ratio of the marginal density with adjacent observation time,and it can realize the local monitoring of model at current time…
Key words: Bayesian Dynamic Models; random simulation; Bayesian factor; model monitoring

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