Análisis estadístico

The SIS method is an efficient way of implementing importance sampling sequentially. However; unless the proposal distribution is very close to the true distribution, the im- portance weight step will lead over a number of iterations to a small number of particles with very large weights compared to the rest of the particles. This will eventually result in one of the normalised weights to being 1 and the others being 0, effectively leading to a particle approximation with a single particle, see Kong et al. [1994] and Doucet et al. [2000b]. This problem is called the weight degeneracy problem.

In order to address the weight degeneracy problem, a resampling step is introduced at iterations of the SIS method, leading to the sequential importance sampling resampling (SISR) algorithm. Generally, we can describe resampling as a method by which a weighted empirical distribution is replaced with an equally weighted distribution, where the samples of the equally weighted distribution are drawn from the weighted empirical distribution. Here, resampling is applied to πN

n−1before proceeding to approximate πn. Assume, again,

that πn−1 is approximated with N particles X1:n−1(1) , . . . , X (N )

1:n−1 with normalised weights

W_n−1(i) as in equation (2.10). We draw N independent samples from πN

n−1, namely eX (i) 1:n−1,

i = 1, . . . , N such that

Obviously, this corresponds to drawing N independent samples from a multinomial dis- tribution, therefore this particular resampling scheme is called multinomial resampling. After resampling, for each i = 1, . . . , N we sample Xn(i) from Qn( eX1:n−1(i) ,·), weight the

particles X_1:n(i) = ( eX_1:n−1(i) , Xn(i)) using

W_n(i) ∝ dπn d(πn−1⊗ Qn) (X1:n(i)), N X i=1 W_n(i) = 1.

The SISR method, also known as the particle filter, is summarised in Algorithm 2.7. Algorithm 2.7. Sequential importance sampling resampling (SISR)

For n = 1; for i = 1, . . . , N sample X₁(i) _{∼ q}1, set W1(i) ∝ dπdq11(X

(i) 1 ).

For n = 2, 3, . . .

• Resample {X1:n−1(i) }1≤i≤N according to the weights {Wn−1(i) }1≤i≤N to get resampled

particles{ eX_1:n−1(i) }1≤i≤N with weight 1/N.

• For i = 1, . . . , N; sample Xn(i) ∼ Qn( eX1:n−1(i) ,·), set X (i) 1:n = ( eX (i) 1:n−1, X (i) n ), and set Wn(i) ∝ dπn d(πn−1⊗ Qn) (X1:n(i)).

The importance of resampling in the context of SMC was first demonstrated by Gor- don et al. [1993] based on the ideas of Rubin [1987]. Although the resampling step alleviates the weight degeneracy problem, it has two drawbacks. Firstly, since after suc- cessive resampling steps some of the distinct particles for X1:n are dropped in favour of

more copies of highly-weighted particles. This leads to the impoverishment of particles such that for k << n, very few particles represent the marginal distribution of Xk un-

der πn [Andrieu et al., 2005; Del Moral and Doucet, 2003; Olsson et al., 2008]. Hence,

whatever being the number of particles, πn(dx1:k) will eventually be approximated by a

single unique particle for all (sufficiently large) n. As a result, any attempt to perform integrations over the path space will suffer from this form of degeneracy, which is called

path degeneracy. The second drawback is the extra variance introduced by the resampling

step. There are a few ways of reducing the effects of resampling.

• One way is adaptive resampling i.e. resampling only at iterations where the effective sample size drops below a certain proportion of N. For a practical implementation, the effective sample size at time n itself should be estimated from particles as well. One particle estimate of Neff,n is given in Liu [2001, pp. 35-36]

e Neff,n = 1 PN i=1W (i)2 n .

• Another way to reduce the effects of resampling is to use alternative resampling methods to multinomial resampling. Let In(i) is the number of times the i’th

particle is drawn from πN

n in a resampling scheme. A number of resampling methods

have been proposed in the literature that satisfy E [In(i)] = NWn(i)but have different

var [In(i)]. The idea behind E [In(i)] = NWn(i) is that the mean of the particle

approximation to πn(ϕn) remains the same after resampling. Standard resampling

schemes include multinomial resampling [Gordon et al., 1993], residual resampling [Liu and Chen, 1998; Whitley, 1994], stratified resampling [Kitagawa, 1996], and systematic resampling [Carpenter et al., 1999; Whitley, 1994]. There are also some non-standard resampling algorithms such that the particle size varies (randomly) after resampling (e.g. Crisan et al. [1999]; Fearnhead and Liu [2007]), or the weights are not constrained to be equal after resampling (e.g. Fearnhead and Clifford [2003]; Fearnhead and Liu [2007]).

• A third way of avoiding path degeneracy is provided by the resample-move al- gorithm [Gilks and Berzuini, 2001], where each resampled particle eX_1:n(i) is moved according to a MCMC kernel Kn: Xn → P(En) whose invariant distribution is πn.

In fact we could have included this MCMC move step in Algorithm 2.7 to make the algorithm more generic. However, the resample-move algorithm is a useful de- generacy reduction technique usually in a much more general setting. Although possible in principle, it is computationally infeasible to apply a kernel to the path space on which current particles exist as the state space grows at evert iteration of SISR. The resample-move algorithm will be revisited in Section 2.5.4, where it is considered as a special case of a wide class of sequential sampling methods that operate on sequences of arbitrary spaces.

• The final method we will mention here that is used to reduce path degeneracy is

block sampling [Doucet et al., 2006], where at time n one samples components

Xn−L+1:n for L > 1, and previously sampled values for Xn−L+1:n−1 are simply

discarded. In return of the computational cost introduced by L, this procedure reduces the variance of weights and hence reduces the number of resampling steps (if an adaptive resampling strategy is used) dramatically. Therefore, path degeneracy is reduced.