Inferring the parameters of time series models from observed data is essential across many areas of science. Bayesian statistics provides a powerful framework for this purpose, but significant challenges arise when time series models are misspecified due to complexities in the underlying process (e.g., heterogeneity in the modelled population, or when parameter values fluctuate over time), inaccurate numerical approximation of the forward model (e.g., in models involving differential equations), or the presence of non-stationary, non-independent error terms. We introduce a series of models and computational strategies for dealing with misspecification in time series inference problems, with a particular focus on time series problems arising in epidemiology and problems involving ordinary differential equations.
The models and inference strategies discussed include: 1. A generalisation of the Poisson renewal model to allow heterogeneous behaviour between local and imported cases, which we use to show that accounting for such heterogeneous behaviour is essential for accurate inference of the time-varying reproduction number (Rt); 2. A Bayesian nonparametric approach to flexibly learn time variation in Rt, which we show is capable of learning accurate and precise estimates of the parameter; 3. Estimates of the gradient and the error in the log-likelihood arising from numerical approximation of differential equations derived from a posteriori error analysis; and 4. A flexible noise process accommodating correlated and heteroscedastic error terms and whose form can be inferred from time series data using kernel functions. We motivate our methodological innovation by a comprehensive examination of the biases in inference that result from insufficiently accurate numerical approximation of differential equations, as well as time series inverse problems and models drawn from epidemiology, hydrology, and cardiac electrophysiology
