Non-Centred Parametrisation - Non-Centred Metropolis-Within-Gibbs Algorithm

5.2 Non-Centred Metropolis-Within-Gibbs Algorithm

5.2.2 Non-Centred Parametrisation

The reparametrisation adopted in this paper is based on a representation derived in Ferguson and Klass (1972). The use of this representation to derive a (partially) non-centred parametrisation for compound Poisson

5.2 Non-Centred Metropolis-Within-Gibbs Algorithm processes was suggested by Roberts et al. (2004) and such reparametrisa- tions were also extensively used by Griffin and Steel (2006) in the context of Lévy-driven stochastic volatility models. Alternative reparametrisa- tions were suggested in Roberts et al. (2004) but were found to be inferior in the presence of efficient updates of the latent point process.

The basic idea is to take points distributed according to a unit-intensity

ppp

on T Œ0;/N , denoted z, whereN is some value in Œ;1/. First,

we discard those points whose second components exceedsand divide

the second component of the remaining points by. This is sometimes

calledthinningand leaves a set of points distributed according to a

ppp

onTŒ0;1with intensity measureLebjTŒ₀;₁. These points are then

transformed into realisations of points ˘ _{by applying the inverse}

cumulative distribution function(

cdf

)-method to the second component.

For completeness and to set up some notation, we outline the formal justification of this reparametrisation below.

Define the space

z ΨWD 1 [ kD0 .fkg TkŒ0;//N

and let˘ .z d /z be the distribution of points z D.K;z Sz₁_{W z}_K;Ez₁_{W z}_K/which

are generated by a unit-intensity

ppp

onT.0;N . Again these points are

taken to be ordered according to their first components. In the following, we describe how z _{is first thinned and then transformed to obtain the}

desired points ˘.

WriteNn WD fk2N jkng. Let H WD˚

j 2N_K_z ˇˇEzj

be the indices of points in z _{whose second component does not exceed} Dl. /and setK WD#H. Similarly, let

y H WD˚

j 2N_K_z ˇˇEzj > DNKznH

be the set of the remaining indices and setKy WD#Hy. Collect the elements

ofH andHy in vectorshDh₁_WK andhO D Oh₁_{W O}_K, in increasing order. Note

Points ˘ are then the result of the one-to-one reparametrisation

.; /z ! .; ; /;y (5.2)

where the left hand side represents an

ncp

and the right hand side represents acentred parametrisation(

cp

), i.e.

(1) z _WD.K;z Sz₁_{W z}_K;Ez₁_{W z}_K/represents all the points under the

ncp

(2) D.K; S₁WK; E₁WK/is a sample from the desired

ppp

under the

cp

(3) y _WD.K;y Sy₁_{W y}_K;Ey₁_{W y}_K/are ‘artificial’ points added to under the

cp

In the following, we specify the one-to-one transformation involved in Equation 5.2.

Forj 2 N_K_y, the points in y are related to those under the

ncp

via .Syj;Eyj/WD zShOj;EzhOj

Forj 2 NK, the points in are related to those under the

ncp

via .Sj; Ej/D Szhj;Ezhj

Above, letting Fx denote the (generalised) inverse of the

cdf

associated with, the functionW T.0; !TEis defined by

.s;Q e/Q 7!.s;Q e/Q _WDŒs;Q Fx.1 e=/:Q

This function transforms the thinned points from the unit-intensity

ppp

(under the

ncp

) – specifically, those points whose second component does not exceedN _{– into the desired points} _{which make up the compound}

Poisson process (under the

cp

). SinceFx _{is the (generalised) inverse of}

the

cdf

associated with, the functioncan be interpreted as applying

the inverse

cdf

method.

Letting and z _{be related as in Equation 5.2, we obtain the following}

proposition, stated here for completeness.

5.2 Non-Centred Metropolis-Within-Gibbs Algorithm

Proof. The fact that is the set of ordered points distributed according

to a

ppp

onTEfollows by the independence property and the Mapping

Theorem for

ppp

s (Kingman, 1992, p. 18). In particular, forA2B.TE/,

letting D denote the Jacobian of, ŒLeb˝2jT.₀;/ı. / 1.A/ D Z A jdet.D/../ 1.s; e//j 1dsde D ŒLebjT˝ _.A/ D.A/:

Thus, the is the set of ordered points distributed according to a

ppp

TEwith intensity measure.

5.2.3 Extended Target Distribution

In this subsection, we specify an extended distribution.Q dd /. First,

this distribution admits the distribution of interest, .d d /, as a

marginal. Second, a Gibbs sampler or Metropolis-within-Gibbs algorithm targeting this distribution makes use of an

ncp

based on the representation outlined above.

Centred Parametrisation. The original formulation of the model in

Equation 5.1 uses a

cp

, i.e. it implies prior dependence between and

. To permit the reparametrisation from Equation 5.2, we augment the target distribution.dd /with

y _y˘ _{WD z}˘jT.;N :

Here,˘zjT.;N is the distribution of points – again ordered according to

the first component – that have been generated by a unit-intensity

ppp

onT.;N . We thus obtain a distributionO / O over the parameters

on the right hand side in Equation 5.2, defined by

Non-Centred Parametrisation. Recall thaty_Trepresents the collection

of all available observations (in the intervalTDŒ0; T ). With the abuse

of notation induced by writing

g.yTj Q /Dg.yTj /

whenever and z _{are related according to Equation 5.2), we can equi-}

valently define an extended target distributionQ / Q parametrised via

the left hand side in Equation 5.2, i.e. by

.dd /Q D$ .d /˘ .z d /gQ .yTj Q /:

This parametrisation represents an

ncp

becauseand z are independent

a-priori. If the observations are not too informative, then the dependence betweenand zunderQ should be smaller than the dependence between and underO. Hence, such a parametrisation is often beneficial in

the context of Gibbs sampling.

5.2.4 The Algorithm

A single iteration of the non-centred

mcmc

algorithm proposed by Roberts et al. (2004) is given in Algorithm 5.3.

5.3 Algorithm (non-centred Metropolis-within-Gibbs).

(1) Update z by sampling // using the

cp

(i) (components of ) using a.d j /-invariant

mcmc

kernel, (ii) y O.dOj; /D z˘j

T.;N .d /O .

(2) Updateusing a.Q dj Q /-invariant

mcmc

kernel. // using the

ncp

5.4 Remark. Note that Step 1 is independent of the previously sampled points in yas these are sampled again from their full conditional distribution. Furthermore, assuming that anMetropolis–Hastings(

mh

) kernel is used to update the parameters, Step 2 is independent of those points in y that lie in the set

T._?;:N

Here,?Dl.?/, where?is the value forproposed as part of the

mh

In document On extended state space constructions for monte carlo methods (Page 167-172)