csmc

The authors introduce a class of controlled SMC algorithms where the proposal distributions are determined by approximating the solution of an associated optimal control problem using an iterative scheme.

Introduction

The authors extend existing methodology to cover static models. Secondly, the iterative procedure developed here approximates the optimal policy of a different control problem at each iteration.

Optimal controlled SMC

Notation

Sequences and sets

Given integers (n leq m) and a sequence ((xt)), we define the ordered sets ([n:m] = {n,...,m }) and write the subsequence (x{n:m} = (x_n,...,xm)). When (n < m), we will use the convention (Pi{t=n}^m x_t = 1)

Function spaces

Let ((E,mathcal{E})) be an arbitrary measureable space. The set of all real-valued, (mathcal{E})-neasureable functions on (E) are denoted by (mathcal{L}(E)) and (mathcal{B}(E)) respectively. We denote the set of real-valued continuous functions on (E) as (mathcal{C}(E)). Let (p in [1, infty)), (mu) be a probability measure on ((E,mathcal{E})) and (mathcal{L}^p(mu)) be the set of (mathcal{E})-measureable functions (varphi).

Measures and Markov Kernels

For any measureable space ((E,mathcal{E}))

Feynman-Kac models

Consider a nonhomogenous MC of length (T) on a measureable space ((X,mathcal{X})), associated with an initial probability measure (mu) and a sequence of Markov transition kernels (M_t). We write the law of the MC on path space (X^{T+1}), equipped with the product (sigma)-algerbra (mathcal{X}^{T+1}), as

[mathbb{Q}(dx_{0:T}) = mu(dx0)Pi{t=1}^T Mt(x{t-1}, dx_t)]

and denote expectations with respect to the law (mathbb{Q}).

SMC

Log-Gaussian Cox point process (d=900)