Bayesian Comparison between MCMC, PMCMC, and VI in Nonlinear Time Series

Abstract

Nonlinear time series models have gained significant attention in applied studies due to their ability to represent complex dynamical properties, such as fluctuation clustering, system transitions, and common non-Gaussian characteristics in financial data. This study aims to develop a general Bayesian framework for estimating and comparing nonlinear time series models within a state-space framework. The proposed framework integrates three key components: flexible model construction, coherent formulation of prior distributions, and modern computational inference techniques, including data augmentation particle-based Markov Monte Carlo (MCMC) algorithms and inverse methods (VI).To evaluate the performance of these methods under different sample sizes, a comprehensive simulation study was conducted using a stochastic fluctuation model for data generation. The results showed that the Bayesian methods provide high-precision parameter estimates, reliable latent state retrieval, and good uncertainty calibration. The study also demonstrated the superiority of the MCMC approach in small samples and in prediction accuracy, while the variance method exhibited high computational efficiency, despite its tendency to systematically underestimate subsequent uncertainty. To highlight the practical application of the proposed framework, it was applied to two sets of real data: one financial and the other numerical time series. The suitability of the models was evaluated using a range of Bayesian predictive diagnostic tools, such as one-step cross-validation, information criteria, and post-prediction tests. Overall, the proposed Bayesian framework offers a comprehensive and consistent methodology for addressing estimation and prediction issues in nonlinear time series analysis.

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