Exact particle flow for nonlinear filters: Seventeen dubious solutions to a first order linear underdetermined PDE

We solve the important and well known problem with particle filters, called “particle degeneracy” or “particle collapse”. This new filter theory is a radical departure from all other particle filters in five ways: (a) we do not use any proposal density; (b) we never resample particles; (c) we compute Bayes´ rule by particle flow rather than as a pointwise multiplication; (d) we compute the log of the unnormalized density rather than the normalized density; and (e) we do not use any MCMC methods.