A new factorisation scheme for the auxiliary particle filter to mitigate the effect of improper second stage weighting

The auxiliary particle filter improves over the conventional particle filter by using lookahead particles to identify and propagate only significant particles, allowing fewer particles while maintaining accuracy. However, scaling the final weights by ancestors of lookahead particles in the auxiliary particle filter can reduce the accurate placement final important particles and cause the filter to degenerate over time. This paper overcomes the problem by introducing a new auxiliary particle filter factorisation scheme for the joint density of the target state and the lookahead auxiliary variable, effectively implementing two consecutive particle filters connected in a serial network framework. This modification preserves the convergence of the auxiliary filter while reducing weight variance. The paper gives the theoretical analysis for the posterior convergence and the variance of the weights, and shows that the proposed factorisation scheme converges to the true posterior, in probability, as the sample size increases, with weight variance that is lesser than the conventional auxiliary particle filter. Simulations on active radar/sonar tracking demonstrate that the proposed framework matches the auxiliary filter's estimation accuracy but with lower Monte Carlo error variance, offering improved consistency.