We derive and test a new nonlinear filter that implements Bayes' rule using an ODE
rather than with a pointwise multiplication of two functions. This avoids one of the
fundamental and well known problems in particle filters, namely "particle collapse" as
a result of Bayes' rule. We use a log-homotopy to construct this ODE. Our new
algorithm is vastly superior to the classic particle filter, and we do not use any
proposal density supplied by an EKF or UKF or other outside source. This paper was
written for normal engineers, who do not have homotopy for breakfast.