Shot-Noise Cox-Process Model

For the shot-noise Cox-process example, we used the simulated data set shown in Figure 4.2. We chose a Gaussian prior for the vector of static parameters, with covariance matrix diag.10;10;102/ and truncated to .0;1/3. For the static-parameter updates we switched to a partially non-

centred parametrisation of the jump sizes to improve mixing of the decay parameter.

As shown in Figure 4.6, the estimated marginal posterior densities from all three algorithms have similar modes. However, those obtained from the

rsmc

-based

pg

sampler are more concentrated. This difference is possibly due to the approximation described in Subsection 4.3.2 which restricts the number of jumps in any particular interval. In this model, it produces visibly different results because the exponential prior on the

2 y 2 De ns it y De ns it y De ns it y 0 50 100 0 5 10 0:4 0:6 0 2 4 0 1 2 0 0:05 0 0:5 1 0 10 20 0 0:5 1 1:5 0 5 0 0:05 0 0:5 1 0 10 20 0 0:5 1 1:5 0 5 0 0:05 0 0:5 1 0 10 20 0 0:5 1 1:5 0 5

Figure 4.4Kernel density estimates for the marginal posterior densities of the static parameters in the elementary change-point model.Top row: rsmc-based

pgsampler algorithm with 100 particles (solid line), 50 particles (dashed line), 25 particles (dash-dotted line). Middle row: vrpf-based pgsampler with 100 particles (solid line), 50 particles (dashed line), 25 particles (dash-dotted line).

Bottom row: tworjmcmc chains. Vertical lines indicate the true parameters;

4.6 Summary Iteration 20;000 40;000 60;000 0 50 100 0 50 100 0 50 100

Figure 4.5 Trace plots for the scale-parameter estimates in the elementary change-point model. Top: rsmc-basedpgsampler with 100 particles.Middle:

vrpf-basedpgsampler with 100 particles.Bottom: rjmcmcsampler.

interjump times allows large numbers of jumps to be placed close to each other with non-negligible probability. Thus, the posterior distribution of this model has tail regions with large numbers of jumps which the

rsmc

- based

pg

sampler algorithm rarely enters. This could also contribute to the differences in the autocorrelations in Figure 4.6. Note that the effect of this approximation can be reduced by decreasing the step sizetn tn ₁.

4.6

Summary

In this chapter, we have demonstrated that

pg

samplers can be applied to piecewise deterministic processes and have presented a number of methodological developments in doing so. Numerical studies provide a comparative illustration of the performance of the proposed methods.

One of the methodological developments presented in this work in- volves a novel representation of the

smc

sampler from Whiteley et al. (2011). This kind of representation, which embeds a ‘variable-dimension’ problem within a ‘fixed-dimension’ problem, may be useful more gen- erally, e.g. for applying quasi-

smc

methods (Gerber & Chopin, 2015) to variable-dimension problems. An extension of this representation to allow

Lag (of 2 y) Lag (of 2 ) A u to co rr el at io n Lag (of) 2 y 2 De ns it y 0 25 50 0 25 50 0 25 50 0 1 2 3 0 0:05 0:1 0 0:01 0:02 0 1 0 1 0 20 40 0 150 300

Figure 4.6Static-parameter estimates for the shot-noise Cox-process example. Based on thersmc-basedpgsampler algorithm with 100 particles (solid line), thevrpf-basedpgsampler with 100 particles (dashed line), and an rjmcmc

sampler (dash-dotted line). Top row: kernel density estimates of the marginal posterior densities. Vertical lines indicate true parameters; dotted lines represent prior densities.Bottom row:autocorrelations.

5

Non-Centred Particle Gibbs

Samplers for Compound

Poisson-Process Models

5.1

Introduction

5.1.1

Motivation

In this chapter, we devise a particle Gibbs sampler for static-parameter es- timation in partially or noisily observed compound Poisson processes. The algorithm is based on a non-centred parametrisation, described in Section 5.2, in order to reduce the impact of the correlation between the latent point process and the parameters on the mixing of the Gibbs sampler. Some modi- fications for enhancing the efficiency of the (conditional) sequential Monte Carlo algorithm at the heart of the particle Gibbs sampler are mentioned in Section 5.3. Finally, in Section 5.4, we provide illustrative results demon- strating the performance of the algorithm on a challenging Lévy-driven stochastic volatility model.

This paper considers a class of statistical models which are based around a compound Poisson processLD.Lt/t2Œ₀;T , for someT 2 .0;1/. Such

a process can be represented as

Lt WD K

X

jD1

Ej 1Œ₀;t .Sj/;

for t 2 Œ0; T  DW T. Here, the number of jumps, K, and the ordered jump times, 0 < S₁ < S₂ < : : : < S_K, are generated by a Poisson

process on T with finite intensity D l. / which we assume to be

constant, for simplicity. Here,lW Θ! .0;1/is some known function

which determines the intensity, and thejump sizes E₁; E₂; : : : ; E_K are

iid

random variables distributed according to some distribution on

Throughout, we assume that the process is latent, i.e. can only be partially or noisily observed. The aim is to conduct inference about the parametersthat parametrise both the compound Poisson process

and the likelihood function of the observations,g. j /. In a Bayesian

framework, this entails computing or at least approximating themarginal

posterior distribution of.

Usually, Markov chain Monte Carlo (

mcmc

) algorithms are used to

approximate such posterior distributions. In addition, to circumvent intractable integrals, we usually have to work on an extended space and approximate the joint posterior distribution of and , denoted

.d d /. Due to the difficulty of constructing efficient global updates

for.; /, the

mcmc

transitions are almost always based around a con-

volution of local, component-wise updates, resulting in so called Gibbs samplers, Metropolis-within-Gibbs algorithms, or combinations of the two. Within these algorithms, the components of are usually updated individually using a particular type of

mcmc

update known asreversible- jump Markov chain Monte Carlo (

rjmcmc

) kernel (Green, 1995). A single

sweep of such a sampler is outlined in Algorithm 5.1, where.d j /and .dj /denote, respectively, the full conditional distributions of and

ofunder.

5.1 Algorithm (centred Metropolis-within-Gibbs).

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

mcmc

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

mcmc

kernel.

Unfortunately, as is well known, such component-wise updates impede mixing of the

mcmc

chain whenever components are highly correlated. To reduce the impact of correlation betweenand on the mixing of the

mcmc

chain, Roberts, Papaspiliopoulos and Dellaportas (2004) propose Metropolis-Within-Gibbs samplers with single-site

rjmcmc

updates for based on variousnon-centred parametrisations(

ncp

s). Roughly speak-

ing,

ncp

s are parametrisations under which certain subsets of paramet- ers/latent variables are independent a-priori.

One of these

ncp

s is based on a representation by Ferguson and Klass (1972) which was also used by Griffin and Steel (2006). It was found to be highly effective in Roberts et al. (2004) as soon as the latent point process

could be updated as a single block.

5.2 Non-Centred Metropolis-Within-Gibbs Algorithm

In document On extended state space constructions for monte carlo methods (Page 160-166)