  Ensemble Kalman Inversion (EnKI) and Ensemble Square Root Filter (EnSRF) are
popular sampling methods for obtaining a target posterior distribution. They
can be seem as one step (the analysis step) in the data assimilation method
Ensemble Kalman Filter. Despite their popularity, they are, however, not
unbiased when the forward map is nonlinear. Important Sampling (IS), on the
other hand, obtains the unbiased sampling at the expense of large variance of
weights, leading to slow convergence of high moments.
  We propose WEnKI and WEnSRF, the weighted versions of EnKI and EnSRF in this
paper. It follows the same gradient flow as that of EnKI/EnSRF with weight
corrections. Compared to the classical methods, the new methods are unbiased,
and compared with IS, the method has bounded weight variance. Both properties
will be proved rigorously in this paper. We further discuss the stability of
the underlying Fokker-Planck equation. This partially explains why EnKI,
despite being inconsistent, performs well occasionally in nonlinear settings.
Numerical evidence will be demonstrated at the end.
