Elucidating the Auxiliary Particle Filter via Multiple Importance Sampling Lecture Notes

Sequential Monte Carlo methods, also known as particle filtering, have seen an explosion of development both in theory and applications. The publication of 1 sparked huge interest in the area of sequential signal processing, particularly in sequential filtering. Ever since, the number of publications in which particle filtering plays a prominent role has continued to grow. An early reference of development is 2 and later tutorials include 3-9. With particle filtering, we estimate probability density functions (pdfs) of interest by probability mass functions, whose masses are placed at randomly chosen locations (particles) and whose weights are assigned to the particles. The particle filter (PF) proposed in 1 is often called the bootstrap PF (BPF), and although it is not optimal, it is the most often used filter by practitioners. A filter that became also popular is known as the auxiliary PF (APF) and was proposed in 10. With the APF, the objective is to generate better particles at each time step compared to those generated with the BPF, thereby improving filtering accuracy. In this article, we derive the APF from a new perspective, one based on interpreting the APF from the multiple importance sampling (MIS) paradigm. The derivation also shows its relationship with the BPF.