Ensemble Kalman Inversion for General Likelihoods

Ensemble Kalman inversion represents a powerful technique for inference in statistical models with likelihoods of the form $y \mid x \sim \mathcal{N}(y \mid \mathcal{H}(x),\mathrm{R})$ where the forward operator $\mathcal{H}$ and covariance $\mathrm{R}$ are known. In this article, we generalise ensemble Kalman inversion to models with general likelihoods, $y \mid x \sim p(y \mid x)$ where the likelihood can be sampled from, but its density not necessarily evaluated. We examine the ensemble Kalman performance for both optimisation and uncertainty quantification against fully adaptive approximate Bayesian computation techniques.