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The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of theKalman filter for high dimensional linear Gaussian state space models. EnKFmethods have also been developed for parameter inference of static Bayesianmodels with a Gaussian likelihood, in a way that is analogous to likelihoodtempering sequential Monte Carlo (SMC). These methods are commonly referred toas ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is onlyasymptotically unbiased if the likelihood is linear Gaussian and the priors areGaussian. However, EKI is significantly faster to run. Currently, a largelimitation of EKI methods is that the covariance of the measurement error isassumed to be fully known. We develop a new method, which we callcomponent-wise iterative ensemble Kalman inversion (CW-IEKI), that allowselements of the covariance matrix to be inferred alongside the model parametersat negligible extra cost. This novel method is compared to SMC on threedifferent application examples: a model of nitrogen mineralisation in soil thatis based on the Agricultural Production Systems Simulator (APSIM), a modelpredicting seagrass decline due to stress from water temperature and light, anda model predicting coral calcification rates. On all of these examples, we findthat CW-IEKI has relatively similar predictive performance to SMC, albeit withgreater uncertainty, and it has a significantly faster run time.

Component-wise iterative ensemble Kalman inversion for static Bayesian models with unknown measurement error covariance