Chapter 3 Recent Developments in Auxiliary Particle Filtering

State space models (SSMs; sometimes termed hidden Markov models, particularly in the discrete case) are very popular statistical models for time series. Such models describe the trajectory of some system of interest as an unobserved E-valued Markov chain, known as the signal process. Let X1 ∼ ν and Xn|(Xn−1 = xn−1) ∼ f(·|xn−1) denote this process. Indirect observations are available via an observation process, {Yn}n∈N. Conditional upon Xn, Yn is independent of the remainder of the observation and signal processes, with Yn|(Xn = xn) ∼ g(·|xn). For any sequence {zn}n∈N, we write zi:j = (zi, zi+1, ..., zj). In numerous applications, we are interested in estimating, recursively in time, an analytically intractable sequence of posterior distributions {p (x1:n| y1:n)}n∈N, of the form: