Recursive Smoothers for General Functionals

From Proposition 4.1.1, one can easily infer a smoothing scheme that applies to the specific situation where the only quantity of interest is E[s(Xk) | Y0:n]

for a particular function s, and not the full conditional distribution φk,n|n. To

this aim, define the finite signed measure τn on (X, X ) by

τn(f ) =

Z

f (xn) s(xk) φk,n|n(dxk, dxn) , f ∈ Fb(X) ,

so that τn(X) = E[s(Xk) | Y0:n]. Proposition 4.1.1 then implies that

τk+1(f ) = c−1k+1 Z f (xk+1) Z s(xk) φk(dxk) Q(xk, dxk+1) gk+1(xk+1) , and τn+1(f ) = c−1n+1 Z f (xn+1) Z τn(dxn) Q(xn, dxn+1) gn+1(xn+1) (4.2)

for n ≥ k + 1 and f ∈ Fb(X). Equation (4.2) is certainly less informative

than Proposition 4.1.1, as one needs to fix the function s whose smoothed conditional expectation is to be updated recursively. On the other hand, this principle may be adapted to compute smoothed conditional expectations for a general class of functions that depend on the whole trajectory of the hidden states X0:nrather than on just a single particular hidden state Xk.

Before exposing the general framework, we first need to clarify a matter of terminology. In the literature on continuous time processes, and particularly in works that originate from the automatic control community, it is fairly common to refer to quantities similar to τn as filters—see for instance Elliott

80 4 Advanced Topics in Smoothing

defined as an object that may be evaluated recursively in n and is helpful in computing a quantity of interest that involves the observations up to index n. A more formal definition, which will also illustrate what is the precise meaning of the word recursive, is that a filter {τn}n≥0 is such that τ0 =

Rν(Y0) and τn+1 = Rn(τn, Yn+1) where Rν and {Rn}n≥0 are some non-

random operators. In the case discussed at the beginning of this section, Rn

is defined by (4.2) where Q is fixed (this is the transition kernel of the hidden chain) and Yn+1enters through gn+1(x) = g(x, Yn+1). Note that because the

normalizing constant c−1_n+1in (4.2) depends on φn, Q and gn+1, to be coherent

with our definition we should say that {φn, τn}n≥0jointly forms a filter. In this

book, we however prefer to reserve the use of the word filter to designate the state filter φn. We shall refer to quantities similar to {τn}n≥0as the recursive

smoother associated with the functional {tn}n≥0, where the previous example

corresponds to tn(x0, . . . , xn) = s(xk). It is not generally possible to derive a

recursive smoother without being more explicit about the family of functions {tn}n≥0. The device that we will use in the following consists in specifying

{tn}n≥0 using a recursive formula that involves a set of fixed-dimensional

functions.

Definition 4.1.2 (Smoothing Functional). A smoothing functional is a sequence {tn}n≥0of functions such that tn is a function Xn+1→ R, and which

may be defined recursively by

tn+1(x0:n+1) = mn(xn, xn+1)tn(x0:n) + sn(xn, xn+1) (4.3)

for all x0:n+1 ∈ Xn+2 and n ≥ 0, where {mn}n≥0 and {sn}n≥0 are two se-

quences of measurable functions X × X → R and t0 is a function X → R.

This definition can be extended to cases in which the functions tn are d-

dimensional vector-valued functions. In that case, {sn}n≥0 also are vector-

valued functions X × X → Rd while {mn}n≥0 are matrix-valued functions

X × X → Rd

× Rd_.

In simpler terms, a smoothing functional is such that the value of tn+1

in x0:n+1 differs from that of tn, applied to the sub-vector x0:n, only by a

multiplicative and an additive factor that both only depend on the last two components xn and xn+1. The whole family is thus entirely specified by t0

and the two sequences {mn}n≥0 and {sn}n≥0. This form has of course been

chosen because it reflects the structure observed in (4.1) for the joint smooth- ing distributions. It does however encompass some important functionals of interest. The first and most obvious example is when tn is a homogeneous

additive functional, that is, when tn(x0:n) =

n

X

k=0

s(xk)

for a given measurable function s. In that case, sn(x, x0) reduces to s(x0) and

4.1 Recursive Computation of Smoothed Functionals 81

The same strategy also applies for more complicated functions such as the squared sum (Pn

k=0s(xk))2. This time, we need to define two functions

tn,1(x0:n) = n X k=0 s(xk) , tn,2(x0:n) = " n X k=0 s(xk) #2 , (4.4)

for which we have the joint update formula tn+1,1(x0:n+1) = tn,1(x0:n) + s(xn+1) ,

tn+1,2(x0:n+1) = tn,2(x0:n) + s2(xn+1) + 2s(xn+1)tn,1(x0:n) .

Note that these equations can also be considered as an extension of Defini- tion 4.1.2 for the vector valued function tn= (tn,1, tn,2)t.

We now wish to compute E[tn(X0:n) | Y0:n] recursively in n, assuming that

the functions tnare such that these expectations are indeed finite. We proceed

as previously and define the family of finite signed measures {τn} on (X, X )

such that τn(f ) def = Z · · · Z f (xn) tn(x0:n) φ0:n|n(dx0, . . . , dxn) (4.5)

for all functions f ∈ Fb(X). Thus, τn(X) = E[tn(X0:n) | Y0:n]. We then have

the following direct consequence of (4.1).

Proposition 4.1.3. Let (tn)n≥0be a sequence of functions on Xn+1→ R pos-

sessing the structure of Definition 4.1.2. The finite signed measures {τn}n≥0

on (X, X ) defined by (4.5) may then be updated recursively according to τ0(f ) = {ν(g0)}−1 Z f (x0) ν(dx0) t0(x0) g0(x0) and τn+1(f ) = c−1n+1 Z Z f (xn+1)  τn(dxn) Q(xn, dxn+1) gn+1(xn+1)mn(xn, xn+1) +φn(dxn) Q(xn, dxn+1) gn+1(xn+1)sn(xn, xn+1)  (4.6) for n ≥ 0, where f denotes a generic function in Fb(X). At any index n,

E[tn(X0:n) | Y0:n] may be evaluated by computing τn(X).

In order to use (4.6), it is required that the standard filtering recursions (Proposition 3.2.5) be computed in parallel to (4.6). In particular, the nor- malizing constant cn+1is given by (3.22) as

82 4 Advanced Topics in Smoothing

As was the case for Definition 4.1.2, Proposition 4.1.3 can obviously be extended to cases where the functional (tn)n≥0is vector-valued, without any

additional difficulty. Because the general form of the recursion defined by Proposition 4.1.3 is quite complex, we first examine the simple case of homo- geneous additive functionals mentioned above.

Example 4.1.4 (First and Second Moment Functionals). Let s be a fixed function on X and assume that the functionals of interest are the sum and squared sum in (4.4). A typical example is when the base function s equals 1A for a some measurable set A. Then, E[tn,1(X0:n) | Y0:n] is the conditional

expected occupancy of the set A by the hidden chain {Xk}k≥0between indices

0 and n. Likewise, E[tn,2(X0:n) | Y0:n]−(E[tn,1(X0:n) | Y0:n])2is the conditional

variance of the occupancy of the set A.

We define the signed measures τn,1 and τn,2 associated to tn,1 and tn,2

by (4.5). We now apply the general formula given by Proposition 4.1.3 to obtain a recursive update for τn,1 and τn,2:

τ0,1(f ) = [ν(g0)]−1 Z f (x0) ν(dx0) s(x0)g0(x0) , τ0,2(f ) = [ν(g0)]−1 Z f (x0) ν(dx0) s2(x0)g0(x0) and, for n ≥ 0, τn+1,1(f ) = Z f (xn+1)  φn+1(dxn+1) s(xn+1) + c−1n+1 Z τn,1(dxn) Q(xn, dxn+1) gn+1(xn+1)  , τn+1,2(f ) = Z f (xn+1) h φn+1(dxn+1) s2(xn+1) + c−1n+1 Z τn,2(dxn) Q(xn, dxn+1) gn+1(xn+1) + 2c−1_n+1 Z τn,1(dxn) Q(xn, dxn+1) gn+1(xn+1)s(xn+1) i .