Understanding dynamic systems is a central goal in many scientific disciplines. State-space models provide a general framework for studying latent dynamic systems based on indirect observations. However, classical state-space methods require researchers to specify the parametric form of the system dynamics in advance, which can be challenging when the underlying processes are nonlinear and only partially explained by theory. Gaussian process state-space models address this by learning the system dynamics directly from data. However, estimating these models exactly can become computationally infeasible for moderately long time-series. In this paper, we therefore aim to improve the Bayesian estimation of approximate Gaussian process state-space models to make these models more accessible and facilitate the statistical learning of nonlinear dynamic systems in empirical research. To this end, we first propose two modifications to an existing Gibbs sampler for these models that considerably improve its sampling efficiency and convergence. Second, we use a confirmatory factor analysis measurement model, which reduces identifiability issues and allows researchers to impose a specific measurement structure on the model. Third, we provide a systematically validated software implementation of the model and sampler for applied use in empirical research. To validate the sampler, we conducted a simulation-based calibration which showed that the sampler converged reliably across many simulated data sets and produces well-calibrated posterior inferences. We further illustrate how the model can be applied and interpreted using two empirical examples. Together, these contributions provide a practical and validated workflow for learning nonlinear latent dynamics with Gaussian process state-space models.

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