Elucidating the auxiliary particle filter via multiple importance sampling

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, and in particular in sequential filtering. Ever since, the number of publications where particle filtering plays a prominent role has continued to grow. An early reference of development is [2] and later tutorials include [3], [4], [5], [6], [7], [8], [9]. With particle filtering, we estimate probability density functions (pdfs) of interest by probability mass functions (pmfs) whose masses are placed at
randomly chosen locations (particles) and weights assigned to the particles. The particle filter (PF) proposed in [1] is often called the bootstrap particle filter (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 particle filter (APF) and was proposed in [10]. With the APF, the objective is to generate better particles at each time step than with the BPF and thereby improve the accuracy of the filtering. In these notes, 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.