Abstract: Bayes’ formula provides the centerpiece for ensemble data assimilation. However, beyond its conceptional simplicity and beauty Bayes’ formula is hardly ever directly applicable and this is true in particular when Bayes’ formula needs to be interfaced with complex scientific models. In this context it is better to talk of simulating Bayes’ formula within a McKean perspective. Bayes’ formula can be simulated in the setting of sequential Monte Carlo methods and general Markov chain Monte Carlo methods. However, while being unbiased these methods suffer from the bias-variance dichotomy in high dimensions. In my talk, I will start approaching Bayes’ formula from an entirely different perspective; namely that of coupling probability measures and optimal transportation. While this shift of focus in itself does not resolve the intractibility issue of high dimensional systems, it naturally puts the popular ensemble Kalman filters into context and suggests natural extensions to non-Gaussian data assimilation problems using linear programming which might lead to a better balance between bias and variance of the resulting estimators.