  The ensemble Kalman filter (EnKF) is a Monte Carlo approximation of the
Kalman filter for high dimensional linear Gaussian state space models. EnKF
methods have also been developed for parameter inference of static Bayesian
models with a Gaussian likelihood, in a way that is analogous to likelihood
tempering sequential Monte Carlo (SMC). These methods are commonly referred to
as ensemble Kalman inversion (EKI). Unlike SMC, the inference from EKI is only
asymptotically unbiased if the likelihood is linear Gaussian and the priors are
Gaussian. However, EKI is significantly faster to run. Currently, a large
limitation of EKI methods is that the covariance of the measurement error is
assumed to be fully known. We develop a new method, which we call
component-wise iterative ensemble Kalman inversion (CW-IEKI), that allows
elements of the covariance matrix to be inferred alongside the model parameters
at negligible extra cost. This novel method is compared to SMC on three
different application examples: a model of nitrogen mineralisation in soil that
is based on the Agricultural Production Systems Simulator (APSIM), a model
predicting seagrass decline due to stress from water temperature and light, and
a model predicting coral calcification rates. On all of these examples, we find
that CW-IEKI has relatively similar predictive performance to SMC, albeit with
greater uncertainty, and it has a significantly faster run time.
