For any L≥1,m≥1 and any distributionχ on (X,X), define κχL,m(θ,Y)def= Φχ,Lθ,L,L+m−m (S,Y)−E
hΦχ,υ,0,m−m(S,Y)i
υ=θ . (6.41) We introduce theσ-algebra FeTn defined by
FeTn
def= σ{FTYn,HTn}, (6.42) where FTn is given by (6.9) and where HTn is independent from Y (the σ-algebraHTn is generated by the random variables independent from the ob-servationsYused to produce the Monte Carlo approximation of{Sk−1}nk=1).
Hence, for any positive integermand anyB∈ GTYn+m, sinceHTn is indepen-dent from B and from FTYn, P(B|FeTn) = P(B|FTYn). Therefore, the mixing coefficients defined in (6.10) are such that
β(GTYn+m,FeTn) =β(GTYn+m,FTnY).
Note thatθnis FeTn- measurable and that Sen is FeTn+1-measurable.
Lemma 6.4. Assume A2, A3-(¯p) and A4 for some p >¯ 2. Let p ∈(2,p).¯ There exists a constantC s.t. for any distribution χ on(X,X), anym≥1, k, ℓ≥0 and any Θ-valuedFe0Y-measurable r.v. θ,
Xk u=1
κχ2um+ℓ,m(θ,Y) p
≤C
#rk
m +kβm∆p
% ,
where ∆pdef= p¯p−p¯p andβ is given by A4.
Proof. For ease of notation χis dropped from the notation κχ2um,m. By the Berbee Lemma (see [Rio, 1990, Chapter 5]), for any m ≥ 1, there exists a Θ-valued r.v. υ⋆ on (Ω,F,P) independent fromGmY (see (6.9)) s.t.
P{θ 6=υ⋆}= sup
B∈GmY
|P(B|σ(θ))−P(B)|. (6.43)
SetLu def= 2um+ℓ. We write Xk
u=1
κLu,m(θ,Y) = Xk u=1
nΦχ,Lθ,Luu,L−mu+m(S,Y)−Φχ,Lυ⋆,Luu−,Lmu+m(S,Y)o
+ Xk u=1
κLu,m(υ⋆,Y) +kn Eh
Φχ,υ,0,m−m(S,Y)i
υ=υ⋆−Eh
Φχ,υ,0,m−m(S,Y)i
υ=θ
o .
(6.44)
By the Holder’s inequality withadef= ¯p/pand b−1 def= 1−a−1,
Φχ,Lθ,L,L+m−m (S,Y)−Φχ,Lυ⋆,L,L+m−m (S,Y)
p
≤Φχ,Lθ,L,L+m−m (S, ϑTY)−Φχ,Lυ⋆,L,L+m−m (S,Y)
¯ p
P{θ 6=υ⋆}∆p . By A3-(¯p), A4, (6.10) and (6.43), there exists a constant C1 s.t. for any m, L≥1, any distribution χand any Θ-valuedFe0Y-measurable r.v. θ,
Φχ,Lθ,L,L+m−m (S,Y)−Φχ,Lυ⋆,L,L+m−m (S,Y)
¯
p≤C1βm∆p .
Similarly, there exists a constantC2 s.t. for anym ≥1, any distribution χ and any Θ-valuedFe0Y-measurable r.v. θ,
Eh
Φχ,υ,0,m−m(S,Y)i
υ=υ⋆−Eh
Φχ,υ,0,m−m(S,Y)i
υ=θ
p≤C2βm∆p .
Let us consider the second term in (6.44). For any u ≥ 1 and any υ ∈Θ, the r.v. κLu,m(υ,Y) is a measurable function of Yi for all Lu −m+ 1 ≤ i ≤ Lu +m. Since Lu ≥ 2um, for any υ ∈ Θ, Pk
u=1κLu,m(υ,Y) is GmY -measurable. υ⋆ is independent from GmY so that:
Xk u=1
κLu,m(υ⋆,Y) p
=E
# E
# Xk u=1
κLu,m(υ,Y)
p%
υ=υ⋆
%1/p .
Define the strong mixing coefficient (see [Davidson, 1994]) αY(r)def= sup
u∈Z
sup
(A,B)∈FuY×Gu+rY
|P(A∩B)−P(A)P(B)|, r≥0.
Then, [Davidson, 1994, Theorem 14.1, p.210] implies that for anym≥1, the strong mixing coefficients of the sequenceκ(m) def= {κLu,m(υ,Y)}u≥1 satisfies ακ(m)(i)≤αY(2(i−1)m+ 1). Furthermore, by [Rio, 1990, Theorem 2.5],
Xk u=1
κLu,m(υ,Y) p
≤(2kp)1/2 Z 1
0
N(m)(t)∧kp/2
Qpυ,m(t)dt 1/p
,
whereN(m)(t)def= P
i≥11ακ(m)(i)>t and Qυ,m denotes the inverse of the tail function t7→ P(|κLu,m(υ,Y)| ≥ t). The sequence Y being stationary, this inverse function does not depend onu. By A4 and the inequality αY(r) ≤ βY(r) (see e.g. [Davidson, 1994, Chapter 13]), there exist β ∈ [0,1) and C∈(0,1) s.t. for anyu, m≥1,
N(m)(u)≤X
i≥1
1αY(2(i−1)m+1)>u≤X
i≥1
1Cβ2(i−1)m>u≤
logu−logC 2mlogβ
∨0.
Let U be a uniform r.v. on [0,1]. Observe that Cβ2mk <1. Then, by the Holder inequality applied with adef= ¯p/pand b−1 def= 1−a−1,
N(m)(U)∧k1/2
Qυ,m(U)
p def=
Z 1 0
N(m)(u)∧kp/2
Qpυ,m(u)du 1/p
≤
−1 2mlogβ
1/2Qυ,m(U)
−logU C
1/2
1(CβCmk,C)(U) p
+k1/2Qυ,m(U)1U≤Cβ2mk
p ,
≤
(Cβ2mk)∆pk1/2+
−1 2mlogβ
1/2
−logU C
1/2
1(CβCmk,C)(U) pb
× kQυ,m(U)kp¯ .
Since U is uniform on [0,1], Qυ,m(U) and |κLu,m(υ,Y)|have the same dis-tribution, see [Rio, 1990]. Then, by Lemma 6.3 and A3-(¯p), there exists a constantC s.t. for anyυ∈Θ, anym≥1,
sup
υ∈Θ kQυ,m(U)kp¯≤C sup
x,x′∈X2 |S(x, x′,Y0) p¯
, which concludes the proof.
Lemma 6.5. Assume A2, A3-(¯p) and A4 for some p >¯ 2. Let p ∈(2,p).¯ There exists a constant C s.t. for any n≥1, any 1≤mn≤τn+1 and any distributionχ on(X,X),
1 τn+1
2vXnmn
t=2mn
κχt,mn(θn, ϑTnY) p
≤C 1
√τn+1 +βmn∆p
,
where κχL,m and β are defined by (6.41) and A4, vn def
= jτ
n+1
2mn
k and ∆p def=
¯ p−p
p¯p .
Proof. We write,
2vXnmn
t=2mn
κχt,mn(θn, ϑTnY) p
≤
2mXn−1
ℓ=0
vXn−1
u=1
κχ2umn+ℓ,mn(θn, ϑTnY) p
.
Observe that by definitionθnisFeTYn-measurable. Then, by Lemma 6.4, there exists a constant C s.t. for anymn≥1 and anyℓ≥0,
vXn−1
u=1
κχ2umn+ℓ,mn(θn, ϑTnY) p
≤C rvn
mn+vnβmn∆p
. The proof is concluded upon noting that τn+1≥2mnvn.
Lemma 6.6. Assume A2, A3-(¯p) and A4 for somep >¯ 2. For anyp∈(2,p],¯ there exists a constant C s.t. for any n≥1, any 1≤mn ≤qn ≤τn+1 and any distribution χ on(X,X),
S¯χ,Tτn+1n(θn,Y)−¯S(θn)−ρen
p ≤C
ρmn+ mn τn+1
+τn+1−qn τn+1
, where ρendef
= τn+1−1 Pqn
t=2mnκχt,mn(θn, ϑTnY) andκχL,m is defined by (6.41).
Proof. By (6.3) and (6.22), ¯Sτχ,Tn+1n(θn,Y)−¯S(θn)−ρen=P4
i=1gi,nwhere g1,n def= 1
τn+1
τXn+1
t=1
Φχ,0θn,t,τ
n+1(S, ϑTnY)−Φχ,tθn,t,t+m−mn n(S, ϑTnY) ,
g2,n def= 1 τn+1
2mXn−1
t=1
Φχ,tθn,t,t+m−mn n(S, ϑTnY)−Eh
Φχ,θ,0,m−mnn(S,Y)i
θ=θn
,
g3,n def
= 1
τn+1
τXn+1
t=qn+1
Φχ,tθn,t,t+m−mn n(S, ϑTnY)−Eh
Φχ,θ,0,m−mnn(S,Y)i
θ=θn
, g4,n def= Eh
Φχ,θ,0,m−mn
n(S,Y)i
θ=θn−¯S(θn). In the caseτn+1>2mn, it holds
τn+1|g1,n| ≤
τXn+1
t=τn+1−mn+1
"
ρmn−1+ρτn+1−t
osc(St+Tn)
+
mn
X
t=1
"
ρmn+ρt−1
osc(St+Tn) + 2ρmn−1
τn+1X−mn
t=mn+1
osc(St+Tn), where we used Proposition 6.5(i) and Remark 6.7 in the last inequality. By A3-(¯p) and A4, there exists C s.t. kg1,nkp ≤ C"
ρmn+τn+1−1
. The same bound hold in the caseτn+1 ≤2mn. Forg2,n and g3,n, we use the bounds
Φχ,tθn,t,t+m−mn n(S, ϑTnY)−Eh
Φχ,θ,0,m−mnn(S,Y)i
θ=θn
≤ sup
(x,x′)∈X2
S(x, x′, YTn+t)+E
# sup
(x,x′)∈X2
S(x, x′, Y0)
% . Then, by A4,
Φχ,tθn,t,t+m−mn n(S, ϑTnY)−E h
Φχ,θ,0,m−mnn(S,Y)i
θ=θn
p
≤2 sup
(x,x′)∈X2
S(x, x′, Y0) p
,
and the RHS is finite under A3-(¯p). Finally,
|g4,n| ≤2ρmn−1E[osc(S0)] , where we used Theorem 6.1. This concludes the proof.
In´ egalit´ es de d´ eviation non asymptotiques pour
l’estimation de
fonctionnelles additives liss´ ees dans les mod` eles de Markov cach´ es (article)
The approximation of fixed-interval smoothing distributions is a key is-sue in inference for general state-space hidden Markov models (HMM). This contribution establishes non-asymptotic bounds for the Forward Filtering Backward Smoothing (FFBS) and the Forward Filtering Backward Simula-tion (FFBSi) estimators of fixed-interval smoothing funcSimula-tionals. We show that the rate of convergence of the Lq-mean errors of both methods de-pends on the number of observationsT and the number of particlesN only through the ratio T /N for additive functionals. In the case of the FFBS, this improves recent results providing bounds depending onT /√
N.
7.1 Introduction
State-space models play a key role in statistics, engineering and econo-metrics; see for example [Capp´eet al., 2005], [Durbin et Koopman, 2000], and [West et Harrison, 1989]. Consider a process {Xt}t≥0 taking values in a general state-space X. This hidden process can be observed only through the observation process {Yt}t≥0 taking values in Y. Statistical inference in general state-space models involves the computation of expectations of
117
additive functionals of the form ST =
XT t=1
ht(Xt−1, Xt),
conditionally to{Yt}Tt=0, whereT is a positive integer and{ht}Tt=1 are func-tions defined onX2. These smoothed additive functionals appear naturally for maximum likelihood parameter inference in hidden Markov models. The computation of the gradient of the log-likelihood function (Fisher score) or of the intermediate quantity of the Expectation Maximization algorithm in-volves the estimation of such smoothed functionals, see [Capp´eet al., 2005, Chapter 10 and 11] and [Doucetet al., 2011].
Except for linear Gaussian state-spaces or for finite state-spaces, these smoothed additive functionals cannot be computed explicitly. In this paper, we consider Sequential Monte Carlo algorithms, henceforth referred to as particle methods, to approximate these quantities. These methods combine sequential importance sampling and sampling importance resampling steps to produce a set of random particles with associated importance weights to approximate the fixed-interval smoothing distributions.
The most straightforward implementation is based on the so-called path-space method. The complexity of this algorithm per time-step grows only linearly with the number N of particles, see [Del Moral, 2004]. However, a well-known shortcoming of this algorithm is known in the literature as the path degeneracy; see [Doucetet al., 2011] for a discussion.
Several solutions have been proposed to solve this degeneracy prob-lem. In this paper, we consider the Forward Filtering Backward Smoothing algorithm (FFBS) and the Forward Filtering Backward Simulation algo-rithm (FFBSi) introduced in [Doucetet al., 2000] and further developed in [Godsillet al., 2004]. Both algorithms proceed in two passes. In the forward pass, a set of particles and weights is stored. In the Backward pass of the FFBS the weights are modified but the particles are kept fixed. The FFBSi draws independently different particle trajectories among all possible paths.
Since they use a backward step, these algorithms are mainly adapted for batch estimation problems. However, as shown in [Del Moralet al., 2010a], when applied to additive functionals, the FFBS algorithm can be imple-mented forward in time, but its complexity grows quadratically with the number of particles. As shown in [Doucet al., 2011a], it is possible to im-plement the FFBSi with a complexity growing only linearly with the number of particles.
The control of the Lq-norm of the deviation between the smoothed ad-ditive functional and its particle approximation has been studied recently in [Del Moralet al., 2010a, Del Moral et al., 2010b]. In the unpublished paper by [Del Moralet al., 2010b], it is shown that the FFBS estimator variance of any smoothed additive functional is upper bounded by terms
depending on T and N only through the ratio T /N. Furthermore, in [Del Moralet al., 2010a], for anyq >2, a Lq-mean error bound for smoothed functionals computed with the FFBS is established. When applied to strongly mixing kernels, this bound amounts to be of orderT /√
N either for (i) uniformly bounded in time general path-dependent functionals, (ii) unnormalized additive functionals (see [Del Moralet al., 2010a,
Equa-tion (3.8), pp. 957]).
In this paper, we establish Lq-mean error and exponential deviation in-equalities of both the FFBS and FFBSi smoothed functionals estimators.
We show that, for anyq ≥2, the Lq-mean error for both algorithms is up-per bounded by terms depending onT and N only through the ratioT /N under the strong mixing conditions for (i) and (ii). We also establish an exponential deviation inequality with the same functional dependence inT and N.
This paper is organized as follows. Section 7.2 introduces further defi-nitions and notations and the FFBS and FFBSi algorithms. In Section 7.3, upper bounds for the Lq-mean error and exponential deviation inequalities of these two algorithms are presented. In Section 7.4, some Monte Carlo experiments are presented to support our theoretical claims. The proofs are presented in Sections 7.5 and 7.6.