  We investigate the application of ensemble transform approaches to Bayesian
inference of logistic regression problems. Our approach relies on appropriate
extensions of the popular ensemble Kalman filter and the feedback particle
filter to the cross entropy loss function and is based on a well-established
homotopy approach to Bayesian inference. The arising finite particle evolution
equations as well as their mean-field limits are affine-invariant. Furthermore,
the proposed methods can be implemented in a gradient-free manner in case of
nonlinear logistic regression and the data can be randomly subsampled similar
to mini-batching of stochastic gradient descent. We also propose a closely
related SDE-based sampling method which again is affine-invariant and can
easily be made gradient-free. Numerical examples demonstrate the
appropriateness of the proposed methodologies.
