Examples

A main thread throughout this work is that most seemingly-complicated Monte Carlo algorithms can be viewed as special cases of

is

on a suitably extended space (and thus as instances oftheMonte Carlo method). For

example, it is well known thatsequential importance sampling (the idea

of which dates at least as far back as (Hammersley & Morton, 1954; M. N. Rosenbluth & Rosenbluth, 1955) and special cases of it such asannealed importance sampling (Jarzynski, 1997b, 1997a; Neal, 2001) may be viewed

as standard

is

.

In this subsection, we show, by example, that this also applies to many other algorithms. First, as shown in Example 1.8,rejection sampling (also

referred to asaccept–reject method) first described in Kahn (1949), may

be viewed as a special case of (self-normalised)

is

on an extended space. This was pointed out in Y. Chen (2005), for instance.

1.8 Example (rejection sampling). Rejection sampling is usually viewed as generating a random number of

iid

samples from some distribution

2M₁.X/, as follows.

(1) Propose

iid

samplesX1; : : : ; XN from some distribution 2M₁.X/ satisfying .

(2) Assume there existsz>0such thatwWDzd=d 1and such that

wcan be evaluated.

(3) Forn 2 N_N WD fn 2 N j n Ng, independently ‘accept’Xnwith probabilityw.Xn/and setK WD fn2 N_N jXnis ‘accepted’g. Then marginally,.Xn/_n2K is an

iid

sample from.

Rejection sampling thus entails generating

iid

samplesXx1; : : : ;XxN from an extended proposal distribution

N _{WD ˝UnifŒ0;1 2}M₁.Xx/;

whereXxWDXŒ0;1. These proposals are then used to form an

is

approx- imationNis;N of the extended measure

N

_WD ˝L2M.xX/;

where L.x;dz/ WD 1_Œ0;w.x/.z/dz. The importance weights are therefore defined byw.x x/N WDŒd =N d .x; z/N D1_Œ0;w.x/.z/. The resulting marginal

1.3 State-Space Extension and Reduction

self-normalised

is

approximation of can then easily be seen to be

N _WD.#K/ 1X n2K

•Xn;

where#K denotes the cardinality of the setK. In particular,Nis;N.1/is an unbiased estimate of the marginal acceptance probability,z.

Finally, a standard

is

approximation of Dz on the marginal spaceX (with proposal distribution ) can be viewed as a Rao–Blackwellisation of the rejection-sampling approximation. That is,

is;N.f /D 1 N N X nD1 wf .Xn/DENis;N.f ˝1Œ₀;₁/ ˇ ˇX :

Note that we are fixing the number of proposed samples, N, in the rejection-sampling scheme. A Rao–Blackwellisation in the case where rejec- tion sampling is performed until a certain number of acceptedsamples has been obtained was developed in Casella and Robert (1996).

Many other algorithms that seem to be generalisations of

is

, at a first glance, can actually also be viewed as standard

is

on an extended space, as shown in Examples 1.9 and 1.10.

1.9 Example (generalised importance sampling). Let x 2 K₁.X;X/ be some-invariant stochastic kernel, i.e. such that x D. As shown in MacEachern, Clyde and Liu (1999, Theorem 6.1) it is possible to apply such a kernel to the weighted sample used to construct an

is

approximation of without having to adjust the weights.

Even though this procedure is sometimes referred to as ‘generalised’ im- portance sampling (e.g. Robert & Casella, 2004, Section 14.2), as pointed out in Doucet and Johansen (2011) (see also Del Moral, Doucet & Jasra, 2006b), it may be viewed as standard importance sampling on the extended space

x

X _WDX2(i.e.ZDXin the notation of this section), with extended proposal distribution N WD ˝ and extended target measureN WD˝˘, where

˘.x0;dx/ _WD d .x; /

d .x

0_{/.dx/ D} d .x; /

d .x

0_/.dx/

represents thetime-reversal kernelof associated with. Indeed writing

x

X D .X; X0_/, the weightsw.x x/N D Œd =N d .N x/N D Œd=d .x/ do not depend on the second component.

1.10 Example (dynamic weighting). Thedynamically weighted Monte

Carlo-framework (Wong & Liang, 1997; Liu, Liang & Wong, 2001; Liang, 2002) designs an extended measure

Q

.dxdv/ WDvg.dx;dv/;

on Xz WD X Œ0;1/. Here, g 2 P.zX/, Xz WD XŒ0;1/, is said to be

correctly weighted with respect to ifQ admits as a marginal, i.e. if

.A/D Q .AŒ0;1//for anyA2B.X/. In this case, an

iid

sample

z

X1; : : : ;XzN Q WDg;

whereXznD.Xn; Vn/, can be used to approximate by standard

is

. The method is called ‘dynamic’ importance sampling because the nth importance weight, w.z Xzn/ D Vn, is not necessarily deterministic given

Xn_{. It is also referred to as ‘generalised’ importance sampling in Liu (2001,}

pp. 36–37), Liang (2002) because taking

g.dxdv/_WD .dx/•w.x/.dv/;

wherew WDd=d , leads back to a direct

is

approximation of the marginal

using as a proposal distribution.

However, this approach is clearly no more than standard

is

on an extended space. Indeed, let N 2 M.xX/be some other extended measure on a space

x

X DXZsuch that (1)N admits as a marginal, (2)N has a densitywx with respect to some probability measure N 2M₁.Xx/. Using the proposal distribution N, we can then construct an

is

approximation

N is;N _WD 1 N N X nD1 x w.Xxn/•Xxn

ofN and hence obtain an approximationN of the marginal.

Note, however, thatZ and thusXxD XZis often a high-dimensional space which can render the preceding

is

scheme inefficient. Since we are only interested in the marginal measure, the key insight here is that we can turn this

is

scheme into an

is

scheme on the potentially lower-dimensional spacezXD XV, withVWDŒ0;1/, with extended targetQ and proposal

In document On extended state space constructions for monte carlo methods (Page 37-40)