Discrete Particle Filter

If the number of elements in Xt is finite (and sufficiently small),

sir

algorithms are wasteful because there is usually a positive probability that two available particle paths at Stept are identical, i.e. for anym; n2 Kt,

P˚XB1nWtjt

1Wt DX Bm

1Wtjt

1Wt >0: (2.15)

Thediscrete particle filter(

dpf

) introduced by (Fearnhead, 1998; Fearnhead

& Clifford, 2003) tackles this problem by propagating particles in a way that reduces the probability on the left hand side in Equation 2.15 to zero. This is done by extending each available particle trajectory once in every possible direction at Stept. To keep the computational cost from growing

exponentially int, the resulting trajectories are then stochastically pruned

in a way that is optimal in the sense that it minimises the variance of the sum of the self-normalised importance weights.

In this subsection, we show that by using particular choices of the kernelsSt ₁,Rt ₁,Qt, andt, the

dpf

can be viewed as a special case

of the generic

smc

algorithm. To our knowledge, this is a new result. It immediately implies the validity of

csmc

algorithms, backward sampling, or ancestor sampling (see Section 3.4 in the next chapter), as described in Whiteley, Andrieu and Doucet (2010), for the

dpf

. However, the

dpf

cannot be viewed as a special case of

sir

due to the dependence in the proposal kernels and the use of a biased resampling scheme. Here, we recall that in the terminology of Definition 2.3, a resampling scheme is termed ‘biased’ if it does not lead to an evenly weighted (i.e. unweighted) set of particles after resampling. We reiterate that any estimates of in- tegrals of the formt.ft/will still be unbiased as long as the resampled

particles are suitably weighted.

Without loss of generality, assume a finite state spaceXt D NK, for

anyt 2Tfor some (usually not too large)K 2N. At thetth step of the

algorithm, we have Nt WD MK ^Kt particles, where M 2 N can be

chosen to control the computational cost of the algorithm. As described below, at Step t, we selectMt WD Nt=K particle trajectories from the

previous step and extend each of them in allK possible directions.

Resampling Scheme. In this case, we do not make use of the auxiliary

kernelsSs ₁. The kernelRs ₁D zRs ₁is then given by z Rs ₁.z₁Ws ₁;das ₁/ D zR?_{s 1}.z₁Ws ₁;da1sWM₁s/ K Y nD2 •_a1WMs s 1 .da .n ₁/MsC1WnMs s ₁ /;

whereRz?_{s 1}.z_1Ws 1;da1sWM₁s/denotes the resampling scheme (for Ms off-

spring) developed in Fearnhead (1998) which summarised in the following. A more formal description of the entire kernel _Rz_{s 1 can be found in}

Section A.5 of the appendix, for completeness.

At Steps, we use Fearnhead (1998, Algorithm 5.2) to solve Ns ₁ X nD1 1^Cs ₁Wsn₁.z1Ws ₁/ DMs;

forCs 1 > 0. The idea is that particles whose self-normalised weights

exceed the threshold 1=Cs ₁get exactly one offspring. The remaining

particles have at most one offspring.

Collect the indices of the former particles in the set

Ls WD#˚n2Ks ₁

ˇ ˇW

n

s 1.z1Ws 1/ >1=Cs 1

and letlsW f1; : : : ;#Lsg !Ls be the function which mapsnto thenth

largest element inLs. We then set the first #Lsparent indices determinist-

ically viaA1W#Ls

s ₁ WD.ls.1/; : : : ; ls.#Ls//. The remainingMs #Lsparent

indices take values in Ks ₁nLs. They are generated using systematic

resampling based on the weights.Wsn₁.z1Ws ₁//n2Ks ₁nLs, after these have

been re-normalised to sum to 1.

Note thatMs DNs ₁impliesCs ₁1=Œmaxn2Ks 1W

n

s 1.z1Ws 1/and

thusLs DNMs, i.e. in this case, we propagate all existing particle paths

without any pruning.

Proposal Kernel. The proposal kernels are completely deterministic, i.e.

q₁.dx₁/D•.₁;:::;K/.dx₁/, and

Qs..z₁Ws ₁;as ₁/;dxs/D•.₁Ms;2Ms;:::;KMs/.dxs/;

wheremdenotes anm-component vector of 1s. In other words, each of the Ms particle trajectories chosen as parents by the resampling distribution

2.3 Some Important

smc

Algorithms

Importance Weights. To ensure that the Radon–Nikodým derivative

x

wt exists, let₁.u₁; /WD•u₁and, fors >1, lets.u₁Ws; /be the uniform

distribution onZ.us 1/MsC1;usMs DWD

us

s . Ifsis the time-reversal kernel

from Assumption 2.9, then by the properties of the resampling scheme employed here (see Section A.5 of the appendix),

s..u₁Ws;z₁Ws ₁; bs ₁/;fbsg/•bs ₁.fa bs s 1g/ D R m s ₁..z1Ws ₁; bs/;fabss1g/ 1^Cs ₁Wsbs₁1.z1Ws ₁/ 1_fabs s 1g.bs 1/1Dsus.bs/:

Hence, thenth Step-t particle weight,wnt.z1Wt/, can be written as

wnt.z1Wt/D t.fx bn 1Wtjt 1Wt g/ Qt sD2Œ1^Cs ₁W bn s 1jt s ₁ .z1Ws ₁/ Dwbtn 1jt t ₁ .z1Wt ₁/ t.x bn 1Wt 1jt 1Wt ₁ ;fxtng/ 1^Ct ₁W bn t ₁jt t ₁ .z1Wt ₁/ ;

for anyz_1Wt in the support of Qt andwnt.z1Wt/D0, otherwise.

In document On extended state space constructions for monte carlo methods (Page 76-78)