A Bayesian Uncertainty Analysis for Nonignorable Nonresponse

Abstract

The discrete time general state-space model is a flexible framework to deal with the nonlinear and/or non-Gaussian time series problems. However, the associated (Bayesian) inference problems are often intractable. Additionally, for many applications of interest, the inference solutions are required to be recursive over time. The particle filter (PF) is a popular class of Monte Carlo based numerical methods to deal with such problems in real time. However, PF is known to be computationally expensive and does not scale well with the problem dimensions. If a part of the state space is analytically tractable conditioned on the remaining part, the Monte Carlo based estimation is then confined to a space of lower dimension, resulting in an estimation method known as the Rao-Blackwellized particle filter (RBPF). In this chapter, we present a brief review of Rao-Blackwellized particle filtering. Especially, we outline a set of popular conditional tractable structures admitting such Rao-Blackwellization in practice. For some special and/or relatively new cases, we also provide reasonably detailed descriptions.We confine our presentation mostly to the practitioners’ point of view.<p>This is a review article on the models admitting Rao-Blackwellization in particle filtering.</p>COOP-LOCCADIC

Rao–Blackwellized Gaussian Smoothing

“…Note that Model 2 is not a special case of Model 1: The difference lies in the linear dynamics (4b) that depend on x n t rather than x n t−1 , while yt depends on the current state xt in both models. Also note that for Model 2, the state dynamics for x n t are sometimes given in terms of the transition density x n t ∼ p(x n t | x n t−1 ) (see [32]). Here, the functional form is chosen since it will simplify the derivations later on.…”

“…• Regular prediction: (31) • Posterior linearization prediction: (26) and (32) Furthermore, the conditional variablesx l t andP l t are again calculated as in (11). From SLR, Φt, Γt, and Σν t are given by…”