Influencia de la diversidad de ingresos en la cooperación comunitaria en el CDR #

points inT._?;N (under the

ncp

) actually have to be sampled. Thus, the algorithm can also deal with models for whichDl./is not bounded but only almost-surely finite.

The key reason for why this sampling scheme is often preferable to Algorithm 5.1 is that in Step 1, we are not fixing the points under our actual target distribution . Instead, is a function of and z. This

will often allow greater movement in the-direction in Step 2.

Even more so as pointed out by Roberts et al. (2004), Griffin and Steel (2006), the particular

ncp

outlined above has another advantage: a de- crease in D l. /coincides with the removal of those points from

that are associated with the smallest jump sizes. Similarly, an increase incoincides with adding points to that have relatively small jumps.

The above construction is therefore termeddependent thinning in Griffin

and Steel (2006). Usually, adding or removing a single small jump has little impact on the posterior density. This property can further facilitate movement in the-direction compared the other

ncp

s from Roberts et al.

(2004) which add or remove jumps with arbitrary jump size.

5.3

Non-Centred Particle Gibbs Sampler

5.3.1

Motivation

The

ncp

discussed in the previous section can help reduce the impact of correlationbetweenand on the efficiency of Algorithm 5.3 but it

does not alleviate inefficiencies resulting from the correlation between individual components of if these are still updated one-at-a-time.

A strategy for systematically updating in one block is offered by the

conditional sequential Monte Carlo(

csmc

) kernels introduced by Andrieu

et al. (2010) and described in Section 3.4 of this work. Simplesequential Monte Carlo(

smc

) algorithms have been applied to latent point processes

in Godsill and Vermaak (2004), Chopin et al. (2013). More sophisticated

smc

algorithms based around the

smc

-sampler framework (Del Moral et al., 2006b) have been developed in Del Moral et al. (2007), Whiteley et al. (2011), Martin et al. (2013) and in Chapter 4 of this work. As pointed out in Whiteley et al. (2011) and further analysed in Chapter 4, the latter

class of

smc

algorithms can introduce a substantial bias in the case of exponentially-distributed interjump times (as is the case here).

We therefore employ simple

smc

algorithms even though these are potentially very inefficient (in the sense that sample impoverishment is severe). Our

smc

algorithm slightly differs from that described in Chopin et al. (2013) in two ways. Firstly, following Chopin (2002), we allow for more than one observation to be included per

smc

step in order to speed up the algorithm. Secondly, we employ a slightly different parametrisation which permits the use of the variance-reduction techniques:backward sampling (

bs

) andancestor sampling (

as

) (Whiteley, 2010; Lindsten et al.,

2012) within

pg

samplers. These were described in Section 3.4.

Alternatives. There are, of course, alternatives to

pg

samplers for con-

ducting inference in the models described here. For instance, we could use

smc

-based pseudo-marginal

mh

algorithms known asparticle mar- ginal Metropolis–Hastings (

pmmh

) algorithms (Andrieu et al., 2010) or

pseudo-marginal

smc

algorithms based around

pmmh

updates known as

smc

-squared(Chopin et al., 2013).

By construction, these methods are robust to strong correlation of

and under. However, these methods tend to require large numbers

of particles. For instance, Chopin et al. (2013) report the need for around 500 to 3;000 particles for a moderately-long time series in the simplest

version of the Lévy-driven stochastic volatility model discussed in Sec- tion 5.4. We have found such numbers of particles to be prohibitively high for implementations in high-level programming languages such as R (R Development Core Team, 2014) or Matlab (The MathWorks, Inc., 2015). With smaller numbers of particles, pseudo-marginal

mh

kernels are well known to suffer from the so called ‘stickiness’ problem, i.e. from long periods of high rejection rates.

5.3.2

Conditional

smc

Kernel

In this subsection, we describe some of the details of the (conditional)

smc

algorithms which are needed to deal with the specific class of models analysed here (and in the previous chapter).

Step Size. ForI 2N we let 0Dt₀ < t₁ < : : : < t_I DT. Here,.t_{i 1}; t_i

5.3 Non-Centred Particle Gibbs Sampler

ith

smc

step. That is, at theith

smc

step, we both assimilate observations

and propose jumps in the interval.ti ₁; ti.

Without loss of generality and to simplify the presentation, we assume that.ti/i2NI is a subsequence of.tQp/p2NP. Note that the commonly-used

strategy of assimilating one observation per

smc

step corresponds to the special case.ti/i2NI D.tQp/p2NP.

If the weights do not deteriorate too quickly over

smc

steps, i.e. if the effective sample size does not decrease too steeply after a single

smc

step, it can be preferable to increase this step size to reduce the computational cost of the algorithm (Chopin, 2002).

Reparametrisation. For the

csmc

kernel, we need to apply a further

reparametrisation to ensure that the computational cost of performing a single step of

bs

or

as

does not grow withT (on average). This can be

achieved by parametrising the compound Poisson process not in terms of jump sizes but in terms of the values of the process at the jump times. The latter coincides with the representation used in the previous chapter. Recall that the compound Poisson process is denotedLD.Lt/t2T. We

can apply another one-to-one reparametrisation of the form

.; / !.; P_1WI/; (5.3)

where

P_{i WD}.Ki; Si;1WKi;Lzi;1WKi/

denotes the points (as well as their number) of the latent compound Pois- son process whose first component (the jump time) falls in the interval

.ti ₁; ti. These points are again ordered according to their first compon-

ent, i.e. ti ₁< Si;₁< : : : < Si;Ki ti. The second components, Lzi;1WKi,

no longer represent the actual jump sizes but are now taken to be the values of the compound Poisson processLat the jump times. That is,

z

Li;j WDLSi;j, for anyj 2NKi. This is corresponds to the terminology

‘jump size’ used in the previous chapter.

With this reparametrisation, we may write the distribution targeted by the (conditional)

smc

algorithm as.dP₁WI//.dP₁WI/, where

.dP_1WI/WD I Y iD1 P ˘_i.dPij P₁Wi ₁/g.y.ti ₁;tij P1Wi; y.t0;ti ₁/:

Here, ˘P_i.dPij P₁Wi ₁/denotes the conditional prior distribution of the

points in the interval.ti ₁; ti, Pi, and we again slightly abuse notation by

writingg.yTj P1WI/Dg.yTj /if and P are related as in Equation 5.3.

Note that the observations taken in disjoint intervals are not necessarily assumed to be independent given the

ppp

and given. Indeed, in the

example considered in Section 5.4, we analytically integrate out a subset of the static parameters which means that the observations in disjoint intervals are no longer conditionally independent given the

ppp

and given the remaining parameters.

The

smc

algorithm then targets.d /P using a sequence of interme-

diate distributions _i.dP_1Wi//i.dP1Wi/; where i.dP1Wi/WD i Y jD1 P ˘j.dPjj P₁Wj ₁/g.y.tj 1;tjj P1Wj; y.t0;tj 1/:

5.3.3

Full Algorithm

The full

pg

sampler is outlined in Algorithm 5.5. Note that the comments made in Remark 5.4 fully apply to this algorithm, too. That is,Dl./

only needs to be almost-surely bounded.

5.5 Algorithm (non-centred particle Gibbs).

(1) Update z by // using the

cp

(i) reparametrising.; /!.; P_1WI/,

(ii) updating P using a

csmc

kernel (with

bs

/

as

, if possible), (iii) reparametrising.; P_1WI/!.; /,

(iv) sampling y O.dOj; /D z˘j

T.;N .d /O .

In document Influencia de la diversidad de ingresos en la cooperación comunitaria en el CDR # 2 Orestes Acosta del municipio de Remedios de la provincia de Villa Clara (página 62-92)