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)
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