We derive a new exact stochastic particle flow for Bayes' rule using a theorem of Gromov. We also show numerical experiments for high dimensional problems up to d=100. The accuracy of our new filter is many orders of magnitude better than standard particle filters, and our filter beats the EKF by orders of magnitude for difficult nonlinear problems. The new theoretical result is valid for arbitrary smooth nowhere vanishing densities, whereas our previous theory was derived for the special case of Gaussian densities with linear measurements. It is crucial to mitigate stiffness of the flow in order to achieve good numerical results.

One regularity condition for the classical Cramér-Rao lower bound (CRLB) of an unbiased estimator to hold is that the support of the likelihood function (LF) should be independent of the parameter to be estimated. This has been shown to be too stringent and the CRLB has been shown to be valid for the case of parameter-dependent support as long as the LF is continuous at the boundary of its support. For the case where the LF is not continuous at the boundary of its support, a new modified CRLB - designated as the Cramér-Rao-Leibniz lower bound (CRLLB) as it relies on the Leibniz integral rule - has been presented for the scalar parameter and measurement case in [3]. The CRLLB for multidimensional parameter and measurements has been developed in [8]. The present work applies the multidimensional CRLLB to n-dimensional measurement noise with the raised fractional cosine and the truncated Laplace distributions inside an (n - 1)-sphere.