Gschlößl, Czado: Does a Gibbs sampler approach to spatial Poisson regression models outperform a single site MH sampler?

Transcription

1 Gschlößl, Czado: Does a Gbbs sampler approach to spatal Posson regresson models outperform a sngle ste MH sampler? Sonderforschungsberech 386, Paper 46 (25) Onlne unter: Projektpartner

2 Does a Gbbs sampler approach to spatal Posson regresson models outperform a sngle ste MH sampler? Susanne Gschlößl Clauda Czado December 6, 25 Abstract In ths paper we present and evaluate a Gbbs sampler for a Posson regresson model ncludng spatal effects. The approach s based on Frühwrth-Schnatter and Wagner (24b) who show that by data augmentaton usng the ntroducton of two sequences of latent varables a Posson regresson model can be transformed nto an approxmate normal lnear model. We show how ths methodology can be extended to spatal Posson regresson models and gve detals of the resultng Gbbs sampler. In partcular, the nfluence of model parametersaton and dfferent update strateges on the mxng of the MCMC chans s dscussed. The developed Gbbs samplers are analysed n two smulaton studes and appled to model the expected number of clams for polcyholders of a German car nsurance company. The mxng of the Gbbs samplers depends crucally on the model parametersaton and the update schemes. The best mxng s acheved when collapsed algorthms are used, reasonable low autocorrelatons for the spatal effects are obtaned n ths case. For the regresson effects however, autocorrelatons are rather hgh, especally for data wth very low heterogenety. For comparson a sngle component Metropols Hastngs algorthms s appled whch dsplays very good mxng for all components. Although the Metropols Hastngs sampler requres a hgher computatonal effort, t outperforms the Gbbs samplers whch would have to be run consderably longer n order to obtan the same precson of the parameters. Key words: block updates, collapsng, data augmentaton, Gbbs sampler, model parametersaton, spatal Posson count data Both at Center of Mathematcal Scences, Munch Unversty of Technology, Boltzmannstr.3, D Garchng, Germany,emal: susanne@ma.tum.de, cczado@ma.tum.de,

3 Introducton In ths paper we present a straghtforward Gbbs sampler for spatal Posson regresson models usng data augmentaton technques. In partcular, we am to nvestgate whether ths Gbbs sampler s found to be superor to a conventonal sngle ste Metropols Hastngs (MH) sampler. The ssue of model parametersaton and several update schemes for the parameters n the Gbbs sampler s thoroughly addressed. The performance of the developed Gbbs sampler schemes and the MH sampler s nvestgated n two smulaton studes as well as on real data from a German car nsurance company. Performance of the samplers s measured n the computatonal costs requred to obtan the same precson of the posteror means of the parameters. Snce the full condtonal dstrbutons of a spatal Posson regresson model do not follow any standard dstrbuton, often sngle ste MH steps are performed n a MCMC settng, see for example Dggle et al. (998), Dmakos and Frgess (22) or Gschlößl and Czado (25). However, ths requres the choce of approprate proposal dstrbutons n order to acheve reasonable acceptance rates and a good mxng of the MCMC chans. Advanced ndependence proposals, lke for example a normal proposal wth the same mode and nverse curvature at the mode as the target dstrbuton, can lead to hgh acceptance rates and low autocorrelatons but nvolve consderable computatonal efforts. Frühwrth-Schnatter and Wagner (24a) developed a Gbbs sampler for Posson regresson models for small counts. They show that by data augmentaton va the ntroducton of two sequences of latent varables a lnear normal model s obtaned. In Frühwrth-Schnatter and Wagner (24b) an applcaton of ths Gbbs sampler to state space models s gven, n Frühwrth-Schnatter and Wagner (24a) the same methodology s appled for standard Posson regresson models and Posson regresson models wth overdsperson. Usng smlar technques, a Gbbs sampler for logstc models s developed n Frühwrth-Schnatter and Waldl (24). The am of ths paper s to show that ths methodology can be extended to spatal Posson regresson models n a straghtforward manner allowng for a Gbbs update of both regresson parameters and spatal effects. Although we only consder spatal Posson data dstrbuted on regons n ths paper, the presented methodology could also be appled on geostatstcal Posson models, see Dggle et al. (998). It s well known, that mxng and convergence of the Gbbs sampler depends crucally on several mplementaton ssues, see for example Roberts and Sahu (997) for a detaled dscusson. Hgh autocorrelatons can be reduced by updatng several parameters n one block or usng collapsed algorthms, another mportant ssue s model parametersaton. Gelfand et al. (995) dscuss the effcency of centered and non-centered parametersatons for herarchcal normal lnear models, Papasplopoulos et al. (23) address parametersaton ssues for several classes of herarchcal models and ntroduce partally non-centered parametersatons. Chrstensen et al. (25) propose the standardzaton and orthogonalzaton of all model components leadng to effcent and robust MCMC algorthms. In ths paper both centered and non-centered model parametersatons are consdered, varous 2

4 algorthmc schemes, n partcular a jont block update of the ntercept and the spatal effects as well as collapsed algorthms, see Lu et al. (994), are dscussed. The performance of the samplers s examned and compared to a sngle ste MH sampler wth ndependence proposals n two smulaton studes. In the frst study, the samplers are appled on data wth both large and small spatal effects, whle the second study consders the nfluence of the data heterogenety on the performance of the samplers. The performance of the samplers s measured n the computatonal costs requred n order to obtan a certan precson of the posteror means of the regresson parameters and spatal effects. Ths s done by takng both the Monte Carlo error of the posteror means of the parameters and the computatonal tme requred for one teraton nto account. A very smlar approach for comparng the performance of MCMC samplers s conducted by Chrstensen and Waagepetersen (22). Among the Gbbs samplers collapsed algorthms perform best. In partcular for data wth small spatal effects, the Monte Carlo errors of the spatal effects are consderably reduced when collapsed samplers and model parametersatons wth non-centered scale or varance are used. The Monte Carlo errors of the regresson parameters however are rather hgh, especally for data wth low heterogenety. The MH ndependence sampler n contrast, exhbts very low Monte Carlo errors and good mxng for both regresson and spatal effects n all settngs. Although the MH sampler requres a hgher computatonal effort, ths drawback s compensated by the hgh precson of the posteror means of the parameters. In order to obtan the same precson the Gbbs samplers would have to be run consderably longer, dmnshng the computatonal advantage n comparson to the MH sampler. Therefore we have to conclude that the proposed Gbbs sampler for spatal Posson regresson models can not outperform a sngle ste MH sampler usng ndependence proposals. Ths paper s organzed as follows. In Secton 2 the spatal Posson regresson model s specfed and the two steps of the data augmentaton scheme are descrbed for ths specfc model. Detals on several algorthmc schemes for updatng the regresson and spatal effects are gven n Secton 3. In Secton 4 the developed Gbbs sampler schemes are examned and compared to a sngle component MH sampler wth ndependence proposals n two smulaton studys. We also apply the Gbbs samplers to model the expected number of clams n a real data set from a German car nsurance company. Secton 5 gves a summary and draws conclusons. 2 Data augmentaton and Gbbs sampler for spatal Posson regresson models We assume that observatons Y, =,..,n observed at J regons follow a Posson model y Posson(µ ). (2.) The mean µ s specfed by µ = t exp(z α) := t exp(x β + v γ) = t exp(x β + γ R() ) (2.2) 3

5 where z = (x,v ) denotes the covarate vector x = (,x,..,x p ) and the ncdence vector v = (v,..,v J ), f R() = j for the regons,.e. v j =, wth R(),..,J} denotng otherwse the regon of the -th observaton. Note, that we do not only observe one sngle but several observatons n each regon. Further α = (β,γ) denotes the vector of regresson parameters β = (β,β,..,β p ) and spatal random effects γ = (γ,..γ J ). By the ncluson of spatal effects we allow for geographcal dfferences n the J regons. The quantty t gves the exposure tme for the -th observaton. We assume a normal pror dstrbuton centered around zero wth a large standard devaton for the regresson parameters β, n partcular β N p+ (,V ) where V = τ 2 I p+ wth τ 2 =. Here N p (µ,σ) denotes the p-varate Normal dstrbuton wth mean µ and covarance matrx Σ. For the spatal effects a condtonal autoregressve (CAR) pror based on Petttt et al. (22) s used. In partcular, we assume γ ψ,σ 2 N J (,σ 2 Q ) where the elements of the precson matrx Q = (Q j ),,j =,..,J are gven by + ψ N = j Q j = ψ j, j otherwse. (2.3) We wrte j for regons and j whch are contguous and assume regons to be neghbours f they share a common border. N denotes the number of neghbours of regon. The spatal hyperparameter ψ determnes the degree of spatal dependence, for ψ = ndependence of the spatal effects s obtaned whereas for ψ the degree of spatal dependency ncreases. Note, that ths pror s a proper dstrbuton n contrast to the well known ntrnsc CAR model ntroduced by Besag and Kooperberg (995). Other proper spatal pror dstrbutons have been consdered, see for example Czado and Prokopenko (24) who use a modfcaton of Model (2.3) and Sun et al. (2). Therefore, we have a multvarate normal pror dstrbuton for the regresson and spatal parameters α whch s gven by wth Σ =. For the spatal hyperparameters θ = (ψ,σ 2 ) the proper pror dstrbutons ( V σ 2 Q ) ψ α θ N p++j (,Σ) (2.4) ( + ψ) 2 and σ 2 IGamma(,.5) 4

6 are assumed. The parametersaton of ths model descrbed by Observaton Equaton (2.2) and Pror Specfcaton (2.4) s called non-centered n the mean, snce the ntercept β appears n the observaton equaton, but not n the spatal pror formulaton. Other possble model parametersatons nclude parametersatons addtonally non-centered n the scale or varance of the spatal pror as well as a centered parametersaton, where the ntercept β only appears as the mean of the spatal pror. These parametersatons are summarzed n Table. For a summary on exstng parametersaton technques see for example Frühwrth-Schnatter (24). Intally, our nvestgatons are based on the non-centered mean parametersaton gven by (2.2) parametersaton spatal pror observaton equaton centered γ c N(β,σ 2 Q ) µ = t exp(x β + v γc ) non-centered mean γ N(,σ 2 Q ) µ = t exp(β + x β + v γ) non-centered mean and scale γ N(,Q ) µ = t exp(β + x β + σv γ ) non-centered mean and varance γ N(,I) µ = t exp(β + x β + σv Lγ ) where LL = Q Table : Spatal pror and observaton equaton for dfferent model parametersatons, where x := (x,..,x p ) and β := (β,..,β p ) and (2.4). Necessary changes when other parametersatons are used wll be ndcated specfcally. 2. Step : Introducton of hdden nter-arrval tmes The basc dea of the data augmentaton scheme developed by Frühwrth-Schnatter and Wagner (24b) s to regard the Posson observatons y, =,..,n, as the number of jumps of an unobserved Posson process wth ntensty µ wthn the unt nterval. The frst step of the data augmentaton conssts n the ntroducton of y + hdden nter-arrval tmes τ j,j =,..,y + for each observaton y. Usng that the nter-arrval tmes are ndependent and follow an exponental dstrbuton wth parameter µ, see for example Mkosch (24),.e. τ j α Exponental(µ ) = Exponental() µ, we obtan log τ j α = log t z α + ǫ j, ǫ j log(exponental()). (2.5) Denote by τ = τ j, =,..,n,j =,..,y + } the collecton of all nter-arrval tmes. Snce the posteror dstrbuton of α condtonal on τ s ndependent of y, condtonal on τ we are now dealng wth model (2.5) whch s lnear n the parameters α, but stll has a non-normal error term. 2.2 Step 2: Mxture approxmaton for error term The second step of the data augmentaton scheme elmnates the non-normalty of model (2.5). As shown by Frühwrth-Schnatter and Wagner (24b), the error term n (2.5) can be approx- 5

7 mated suffcently close to a normal dstrbuton by a mxture of fve normal dstrbutons,.e. p(ǫ j ) = exp(ǫ j exp(ǫ j )) 5 w r f N (ǫ j ;m r,s 2 r), where f N ( ;m r,s 2 r ) denotes the densty of the normal dstrbuton wth mean m r and varance s 2 r. Frühwrth-Schnatter and Wagner (24b) also gve the correspondng values for m r,s 2 r and the weghts w r. In the second step of the data augmentaton the component ndcators r j,..,5} are ntroduced as latent varables. Denotng the set of all component ndcators by R = r j, =,..,n,j =,..,y + }, we have condtonal on R r= log τ j α,r j = log t z α + m rj + ǫ j, ǫ j N(,s 2 r j ),.e. we are dealng wth a normal model whch s lnear n α now. Snce the pror dstrbuton π(α θ) s normal as well, the resultng posteror dstrbuton s multvarate normal and a Gbbs sampler can be appled. Note, that by performng ths data augmentaton we are no longer dealng wth n but wth n = (y + ) observatons. Therefore ths Gbbs Sampler s manly useful for count data wth small counts only, otherwse the data set mght get very large. 2.3 Algorthmc scheme The algorthmc scheme for the above Gbbs Sampler s the followng: Choose approprate startng values for the component ndcators R and the nter-arrval tmes τ. () sample regresson and spatal parameters α = (β,γ) gven τ,r,θ (2) sample spatal hyperparameters θ = (ψ,σ 2 ) gven α (3) sample the nter-arrval tmes τ j gven α,y (4) sample the component ndcators r j gven τ,α Step () conssts of samplng from a multvarate normal dstrbuton. Ths can be done n one block, however t mght be computatonally more effcent to perform an update n several smaller blocks. We wll consder several update strateges for step () n Secton 3 n more detal. The spatal hyperparameter ψ s updated usng a MH step, whereas σ 2 can be updated usng a Gbbs step. Steps (3) and (4), elaborated n Frühwrth-Schnatter and Wagner (24b), are descrbed n the Appendx A and B, for the choce of startng values see Appendx C. 6

8 3 Updatng schemes for the regresson and spatal parameters n the Gbbs Sampler For α several update schemes are possble and wll be dscussed n ths secton. For notatonal convenence we defne wth N := n = (y + ) τ = ( τ,.., τ N ) := (τ,..,τ,y +,τ 2,..,τ 2,y2 +,..,τ n,..,τ n,yn+), ǫ = ( ǫ,.., ǫ N ) := (ǫ,..,ǫ,y +,ǫ 2,..,ǫ 2,y2 +,..,ǫ n,..,ǫ n,yn+), m = ( m,.., m N ) := (m r,..,m,m r,y + r 2,..,m,..,m r2,y2 + r n,..,m rn,yn+ ) and s 2 = ( s 2,.., s 2 N) := (s 2 r,..,s 2 r,y +,s2 r 2,..,s 2 r 2,y2 +,..,s2 r n,..,s 2 r n,yn+ ). Let t = ( t,.., t N ) denote the vector where t s repeated y + tmes. Further defne ỹ = (ỹ,..,ỹ N ) := (log τ m + log t,..,log τ N m N + log t N ). Usng ths notaton we have accordng to (2.5) where z = z. z N ỹ α,r N( z α, s 2 ) s a N (p + + J)-matrx where z s repeated y + tmes. 3. Block update of α = (β, γ) For a jont update of the regresson parameters β and the spatal effects γ n one block we have to consder the full condtonal of α = (β,γ) whch s gven by p(α θ, τ, R) π(α θ) exp 2 exp 2 N = ( exp 2 s 2 (ỹ + z α)2) [ α Σ α + N = (ỹ + z α) 2]} s 2 [α Σ α α 2α µ α ]}, where Σ α := Σ + N = s z 2 z and µ α := N = s z 2 ỹ. Hence, α θ,τ,r N p++j (Σ α µ α,σ α ). 7

9 3.2 Separate update of β and γ The calculaton of the posteror covarance matrx Σ α n Secton 3. can be computatonally expensve f the number of regresson parameters and spatal effects s large as s the case n most spatal applcatons. Therefore t mght be more effcent to update β and γ n two separate blocks. The full condtonal dstrbutons of β and γ are gven by β γ,θ,τ,r N p+ (Σ β µ β,σ β ) and γ β,θ,τ,r N J(Σ γ µ γ,σ γ ). The explct formulas for Σ β, µ β, Σ γ and µ γ are gven n Table Block update of the ntercept β and γ (block) Due to dentfablty problems between the ntercept β and the spatal effects γ mxng and convergence s not very good when β and γ are updated n two separate blocks. Better results are acheved f a jont block update of β and γ s performed, whereas the remanng parameters β = (β,..,β p ) are stll updated n one separate block. Wth ths settng the posteror dstrbutons are gven by and β β,γ,θ,τ,r N p (Σ β µ β,σ β ) γ,β β,θ,τ,r N J+ (Σ γβ µ γβ,σ γβ ) wth Σ β, µ β, Σ γβ and µ γβ as gven n Table Collapsed algorthm for a model parametersaton wth a non-centered mean (coll) Another possblty s to use a collapsed algorthm. Ths means, that partcular components of the posteror are ntegrated out and an update based on the margnal dstrbuton s performed. In our context the jont posteror dstrbuton of β and γ can be wrtten as p(β,γ θ,τ,r) p(β τ,r)p(γ β,θ,τ,r) where p(β τ,r) = p(β,γ θ,τ,r)dγ s the margnalsed posteror densty of β wth γ ntegrated out. It s shown n the Appendx D that wth Σ col and µ col as gven n Table 2. β τ,r N p+ (Σ col µ col,σ col ) Step () n the algorthmc scheme presented n Secton 2.3 s then the followng for the collapsed algorthm: sample β from N p+ (Σ col µ col,σ col ) sample γ β, θ, τ, R as n Secton 3.2 8

10 Secton 3.2 Σ β := V + N = s x x 2 µ β := N = s x 2 (ỹ + γ R() ) Σ γ := σ 2 Q + N = s 2 ṽ ṽ µ γ := N = s 2 ṽ (ỹ + x β) 3.3 Σ β := V β + N = s x 2 β x β (block) µ β := N x β (ỹ + γ R() + β ) Σ γβ := ( = s 2 τ 2 Q σ 2 µ γβ := N = s 2 ) + N = s (,ṽ 2 )(,ṽ ) (,ṽ )(ỹ + x β β ) V β = τ 2 I p x β = ( x,.., x p ) 3.4 Σ col := τ 2 I + N = s x x 2 ( N = s ṽ x 2 ) A ( N = s ṽ x 2 ) (coll) µ col := ( N = s ṽ x 2 ) A ( N = s ṽ 2 ỹ ) N = s x 2 ỹ A := N = s ṽ 2 ṽ + σ 2 Q 3.5 Σ col := τ 2 I + N = s x x 2 (σ N = s ṽ x 2 ) (A ) (σ N = s ṽ x 2 ) (coll2) µ col := (σ N = s ṽ x 2 )(A ) (σ N = s ṽ 2 ỹ ) N = s x 2 ỹ A := σ 2 N = s ṽ 2 ṽ + Q Σ γ := σ2 N = s ṽ 2 ṽ + Q µ γ := σ N = s ṽ 2 (ỹ + x β) Σ σ := N = s (γ 2 R() )2 + τσ 2 µ σ := N (ỹ + x β) = γ R() s Σ col := τ 2 I + N = s x x 2 (σ N = s 2 (coll3) µ col := (σ N = s L ṽ 2 x )(A ) (σ N = s 2 L v x ) (A ) (σ N A := σ 2 N = s 2 L ṽ ṽ L + I Σ γ := σ 2 N = s 2 L ṽ ṽ L + I µ γ := σ N = s 2 L ṽ (ỹ + x β) = s 2 L v ỹ ) N = s x 2 ỹ Σ σ := N = s (γ ( N 2 = s L ṽ 2 ṽ L)γ + τσ 2 µ σ := N = v Lγ s (ỹ 2 + x β) 3.7 µ cent γ := β σ 2 Q N = s 2 ṽ (ỹ + x β β ) (centered) Σ β := σ 2 J,j= Q j + τ 2 µ β := σ 2 Qγ c L v x ) Table 2: Covarance and mean specfcatons for the update strateges n Sectons

11 3.5 Collapsed algorthm for a model parametersaton wth a non-centered mean and scale (coll2) Up to now, we only consdered models wth the non-centered mean parametersaton specfed by (2.2) and the spatal pror γ ψ,σ N J (,σ 2 Q ). In ths secton we consder a model where the pror of the spatal effects s not only non-centered n the mean, but n the scale as well,.e. the thrd model parametersaton gven n Table. By assumng γ ψ N J (,Q ), σ appears as an unknown parameter n the observaton equaton, n partcular we have µ = t exp(x β + σγ R() ). For ths parametersaton and π( ) denotng the pror dstrbutons, the jont posteror of β,γ,ψ and σ s gven by p(β,γ,ψ,σ ỹ,τ,r) exp 2 n = (ỹ + x β + σṽ γ ) 2} π(β)π(γ ψ)π(ψ)π(σ). s 2 Followng the lnes of Secton 3.4 we obtan for β the margnalzed posteror dstrbuton β σ,τ,r N p+ ((Σ col ) µ col,(σ col ) ). The full condtonal dstrbuton for γ s gven by γ β,τ,r,σ,ψ N J ((Σ γ) µ γ,(σ γ) ). The defntons of Σ col, µ col, Σ γ and µ γ can be found n Table 2. The spatal hyperparameter ψ s agan updated usng a MH step snce the full condtonal dstrbuton can not be sampled from drectly. For ths model parametersaton we choose a normal pror for σ, n partcular σ N(,τ 2 σ). Note, that σ s not restrcted to take postve values, leadng to nondentfablty, snce the same lkelhood results for (σ,γ ) and ( σ, γ ). However, as ponted out by Frühwrth- Schnatter (24), ths leads to an mproved mxng for models wth small scales σ 2 snce boundary problems for σ are avoded. The full condtonal dstrbuton of σ s then agan normal, n partcular see Table 2 for detals on Σ σ and µ σ. σ β,γ,τ,r N((Σ σ) µ σ,(σ σ) ), 3.6 Collapsed algorthm for a model parametersaton wth a non-centered mean and varance (coll3) In ths secton we consder the model parametersaton non-centered n both mean and varance, also gven n Table. In contrast to the non-centered parametersaton n scale only consdered n the prevous secton, we now assume the pror γ N J (,I).

12 The spatal structure ncorporated n the precson matrx Q s now moved to the observaton equaton gven by µ = t exp(x β + σv Lγ ), where L s a lower trangular matrx resultng from the Cholesky decomposton Q = LL. The resultng jont posteror dstrbuton of β,γ,ψ and σ s gven by p(β,γ,ψ,σ ỹ,τ,r) exp 2 n = The margnalzed posteror dstrbuton of β changes to (ỹ + x β + σṽ Lγ ) 2} π(β)π(γ )π(ψ)π(σ). s 2 β σ,τ,r N p+ ((Σ col ) µ col,(σ col ) ), the full condtonal dstrbuton of γ s gven by γ β,τ,r,σ,ψ N J ((Σ γ ) µ γ,(σ γ ) ), wth Σ col, µ col, Σ γ and µ γ as gven n Table 2. Whle ψ s agan updated usng a MH step, the full condtonal dstrbuton of σ s gven by σ β,γ,ψ,τ,r N((Σ σ ) µ σ,(σ σ ) ), see Table 2 for detals on Σ σ and µ σ. Here agan the normal pror σ N(,τ2 σ ) s assumed. 3.7 Centered CAR-Model (centered) Alternatvely, the centered spatal pror γ c β N(β,σ 2 Q ) wth β N(,τ 2 ) and β N(,τ 2 I p ) can be used. For ths model the posteror dstrbuton for β s the same as n Secton 3.3 but wth µ β replaced by N = s x 2 β (ỹ + γr() c ). The posteror dstrbuton for γ c s gven by γ c β,β,θ,τ,r,y N J (Σ γ µcent γ,σ γ ) where Σ γ s the same as n Secton 3.2 and µ cent γ s gven n Table 2. β s updated n an extra Gbbs step, n partcular wth Σ β and µ β defned as n Table 2. β β,γ,θ,τ,r,y N(Σ β µ β,σ β ) 4 Smulaton studes and applcaton We am to apply the developed Gbbs samplers to analyse the expected number of clams n a data set from a German car nsurance company. The data nclude 637 polcyholders n Bavara wth

13 full comprehensve car nsurance wthn one year and contan nformaton on several covarates lke age and gender of the polcyholders, klometers drven per year and the geographcal regon each polcyholder s lvng n. Bavara s dvded nto 96 regons. The varablty of these data s very small, 95% of the observatons are zero observatons, the hghest number of clams observed s only four. The data have been already analysed by Gschlößl and Czado (25) who consdered both a spatal Posson regresson model as well as spatal models takng overdsperson nto account. They show that the spatal effects are very small for these data and have no sgnfcant contrbuton to explanng the expected clam number. In ths secton, the performance of the Gbbs sampler schemes developed n Sectons 2 and 3 wll be examned on smulated data frst. For comparson, we addtonally use a sngle ste Metropols Hastngs algorthm for spatal Posson regresson models wth an ndependence proposal where both β and γ are updated component by component. In partcular, we use a t-dstrbuton wth 2 degrees of freedom as proposal whch has the same mode and nverse curvature at the mode as the target dstrbuton. The performance of the samplers s measured n terms of the computaton tme requred n order to obtan a certan precson of the estmated posteror means of the parameters. The posteror mean of a varable θ s gven by θ := R ˆθ j= j wth ˆθ j,j =,..,R denotng the MCMC terates of θ after burnn. The precson of θ s gven by the Monte Carlo standard error of θ whch s defned as σ MC ( θ) := σasy( θ) R where ( σasy( θ) 2 := V ar(θ) + 2 k= ) ρ k (θ) denotes the asymptotc varance of θ, V ar(θ) the sample varance and ρ k (θ) the autocorrelaton of the MCMC terates ˆθ,.., ˆθ R at lag k. The asymptotc varance wll be estmated usng the ntal monotone sequence estmator (see Geyer (992)), defned by 2m+ ˆσ asy( θ) 2 := V ˆ ar(θ)( + 2 j= ˆρ k (θ)), where m s chosen to be the largest nteger such that the sequence Γ m = ˆρ 2m (θ) + ˆρ 2m+ (θ) s postve and monotone. Here Vˆar(θ) := ˆγ, ˆρ k (θ) := ˆγ k, ˆγ k := R k ˆγ R j= (ˆθ j θ)(ˆθ j+k θ). We addtonally requre the estmated emprcal autocorrelatons ˆρ 2m+ (θ) to fall below.. In order to obtan a certan precson k, R = ˆσ2 asy samples are needed. Hence, the computaton k 2 tme requred to obtan a precson k for an algorthm wth computatonal costs m per teraton, s gven by R m. For a drect comparson of the Gbbs sampler schemes to the MH ndependence sampler we consder the computatonal costs relatve to the costs of the MH sampler requred to obtan the same precson of the posteror means of the parameters. Ths s gven by R rel m rel := ˆσ2 asy m ˆσ asy,nd 2 m nd, where ˆσ asy,nd 2 and m nd denote the estmated asymptotc varance and the computatonal costs for one teraton of the MH ndependence sampler. We consder two studes. In the frst study the nfluence of the sze of the spatal effects on mxng behavour s examned, whle n the second study the mpact of data heterogenety s 2

14 nvestgated. In both studes the Gbbs samplers descrbed n Sectons ,.e. the followng model parametersatons and update schemes are assumed: non-centered mean: block update of β β,γ and (β,γ) β gven n Secton 3.3 (block) collapsed algorthm gven n Secton 3.4 (coll) non-centered mean and scale: collapsed algorthm gven n Secton 3.5 (coll2) non-centered mean and varance: collapsed algorthm gven n Secton 3.6 (coll3) centered parametersaton: algorthm gven n Secton 3.7 (centered) In the followng we wll refer to these samplers as block, coll, coll2, coll3 and centered. 4. Computatonal costs Recall, that by usng the data augmentaton scheme descrbed above, we are no longer dealng wth n observatons, but wth N = n = (y + ) latent nter-arrval tmes τ j and mxture component ndcators r j. Both τ and R have to be updated, therefore the number of varables to sample from n each teraton s 2N +J +p+(+2 hyperparameters) n comparson to J +p+(+2 hyperparameters) varables n the MH ndependence sampler. The MH ndependence sampler n contrast requres the calculaton of the posteror mode and the nverse curvature at the posteror mode for each of the J + p + components n every teraton. The posteror mode may be obtaned usng the bsecton method for example. In our smulaton studes, except the sampler coll3, the Gbbs samplers are always faster than the MH ndependence sampler. However, the computatonal advantage of the Gbbs samplers depends on the complexty of the model. The computatonal costs m rel relatve to the costs of the MH sampler for one teraton are reported n Table 3. For the settng n Study wth 5 observatons, an ntercept and two covarates for example, the centered Gbbs sampler only takes.86 tmes as long as the MH ndependence sampler. For the settng n Study 2 wth a larger data set the centered Gbbs sampler even takes only.26 tmes as long. Among the Gbbs samplers the centered Gbbs sampler s the fastest, followed closely by the Gbbs sampler usng a block update. The collapsed Gbbs samplers noncentered n the mean (coll) and non-centered n mean and scale (coll2) requre slghtly more tme than the centered Gbbs sampler. The computatonal effort for the Gbbs sampler n the model parametersaton non-centered n the mean and the varance (coll3) however s more than twce as large. In ths algorthm a Cholesky decomposton of the precson matrx Q has to be performed n every teraton. 4.2 Study : Influence of the sze of the spatal effects We consder two smulated data sets of sze 5 wth y Posson(µ ), =,..,5. For both data sets the mean µ s specfed by µ = exp(β + x β + x 2 β 2 + γ R() ) 3

15 sampler Study Study 2 ndependence block centered coll.96.3 coll coll Table 3: Computaton tmes m rel for the dfferent samplers relatve to the MH ndependence sampler for the settngs n Study and Study 2. where x s an ndcator varable and x 2 a contnuous standardzed varable. The exposure s assumed to be t = for all observatons. We assume a smple spatal structure, namely regons on a grd. The spatal effects γ are generated accordng to the CAR pror γ N(,σ 2 Q ) wth spatal dependence parameter ψ = 3. For the frst smulated data set y we assume σ 2 = resultng n a range of [mn(γ)max(γ)] = [.86,.85] for the spatal effects, whereas for the second data set y 2 we take σ 2 =. resultng n a range of [mn(γ)max(γ)] = [.8,.8]. The Gbbs samplers block, coll, coll2, coll3 and centered as well as the ndependence MH sampler are run for 5 teratons, a burnn of teratons s taken. As descrbed above, the performance of the samplers s measured n terms of the Monte Carlo standard error of the posteror means of the parameters and the requred computaton tmes. Snce estmaton of the Monte Carlo error s based on the estmated emprcal autocorrelatons, ths quantty also depends on the mxng of the samplers. For a far comparson of the Monte Carlo error of the spatal effects the model parametersaton of each sampler has to be taken nto account. Therefore we compute the Monte Carlo error for β + γ for the MH ndependence sampler and the samplers block and collapsed, whle for the centered sampler the standard error of γ s consdered snce here the ntercept s the spatal pror mean and therefore already ncluded n γ. For the coll2 and coll3 samplers the Monte Carlo errors for β + σγ and β + σlγ, respectvely, are computed. In the left panel of Table 4, for each sampler the Monte Carlo standard errors and the performance relatve to the MH ndependence sampler R rel m rel are reported for the regresson parameters β,β 2 and the spatal effects n data set y. For the spatal effects the average error, taken over all J components, s gven. Addtonally plots of the emprcal estmated autocorrelatons are presented n Fgure. In the left panel the autocorrelatons for 25 of the spatal effects, n the rght panel autocorrelatons for the regresson effects are plotted. Mxng for all Gbbs samplers s reasonable well, n average autocorrelatons of the spatal effects are below. at a lag of about 6 to 8. The average Monte Carlo error for the spatal effects s around. for all Gbbs samplers. The Monte Carlo error of the regresson parameters however s lower for the collapsed Gbbs samplers, for the block and the centered Gbbs sampler especally the autocorrelatons of β decrease rather slowly. The ndependence MH sampler n contrast, dsplays the smallest Monte Carlo error for both 4

16 spatal effects and regresson parameters. In average the autocorrelatons of β + γ j are below. at a lag 3 already, the autocorrelatons for the regresson parameters decrease rapdly as well. Consderng the computatonal effort relatve to the MH ndependence sampler, gven by R rel m rel, the MH ndependence sampler outperforms the Gbbs samplers consderably. The computatonal effort requred to obtan the same precson of the posteror means of the spatal effects s more than 5 tmes as large for the Gbbs samplers compared to the ndependence sampler. Data set y Data set y 2 sampler spatal β β 2 spatal β β 2 effects effects ndependence block centered coll coll coll Table 4: Estmated ˆσ MC (upper row) for the regresson parameters β,β 2 and average estmated ˆσ MC for the spatal effects γ + β n the ndependence, block, coll sampler, γ n the centered, β + σγ n the coll2 and β + σlγ n the coll3 sampler, as well as R rel m rel (lower row) for all parameters for data set y and y 2 usng dfferent update strateges n Study. The correspondng results for data set y 2 wth small spatal effects are reported n the rght panel n Table 4, plots of the estmated emprcal autocorrelatons are gven n Fgure. Here, clearly the lowest precson and worst mxng s obtaned f the Gbbs sampler based on the centered model parametersaton s used. Ths confrms the results gven n Gelfand et al. (995). They show that for a herarchcal normal lnear model wth random effects the centered parametersaton s effcent f the varance of the random effects domnates the varance n the data. However, f the varance of the random effects s very small n contrast to the varablty of the data (as t s the case n data set y 2 ), hgh posteror correlatons result. For the block and partcularly the collapsed Gbbs samplers a consderably lower Monte Carlo error s obtaned. The average Monte Carlo error of the spatal effects n the collapsed sampler coll s almost as small as n the MH ndependence sampler. For the regresson effects however, the MH ndependence sampler 5

17 exhbts lower Monte Carlo standard errors. The computatonal costs R rel m rel relatve to the MH sampler, whch are requred to obtan the same precson of the posteror means of the parameters are greater for all Gbbs samplers for both spatal effects and regresson parameters. Hence, the ndependence sampler gves the best performance for data set y 2 as well. The varance of the two smulated data sets y and y 2 takes the values var(y ) =.5 and var(y 2 ) =.49. However, the varablty of our real data from a car nsurance company s very small, the varance of these data s only.5. Therefore we conduct a second smulaton study where we examne whether the heterogenety of the data nfluences the performance of the samplers. 4.3 Study 2: Influence of data heterogenety We smulate two data sets based on the desgn of the real data where, accordng to Gschlößl and Czado (25), eght covarates sgnfcant for explanng the expected clam number y were observed,.e. y Posson(µ ), =,..,637 wth µ = t exp(x β + γ R()). Here x = (,x,..,x 8 ) and x k,k =,..,8 are standardzed categorcal and metrcal covarates, the observaton specfc exposure t takes values up to one year. In ths settng we have 96 rregular regons n Bavara. The spatal effects γ agan are generated accordng to the CAR pror γ N(,σ 2 Q ) wth ψ = 8 and σ 2 =.. Ths results n small spatal effects wth a range of [.6.8],.e. spatal effects smlar to the ones observed n our real data set. For the frst data set y 3 the ntercept β s taken to be, whereas for the second data set y 4 we take β = 2.5. For the remanng regresson parameters the same values are assumed for both data sets. The resultng varances of y 3 and y 4 are V ar(y 3 ) =.46 and V ar(y 4 ) =.5,.e. data set y 4 has very low heterogenety and s close to our real data. The varance of data set y 3 s not partcularly hgh ether, but n comparson to data set y 4 we wll refer to ths data set as data wth hgh heterogenety. The block, centered, coll, coll2 and coll3 Gbbs samplers are run for 5 teratons, the frst teratons are dscarded for burnn. For comparson agan the MH ndependence sampler s appled. The Monte Carlo errors for the posteror means of the regresson parameters β,..,β 8, the spatal effects γ n the centered, β +γ n the non-centered mean, β +σγ n the non-centered mean and scale and β + σlγ n the non-centered mean and varance model parametersaton and the quanttes R rel m rel are reported n Table 5. For the hgh heterogenety data set y 3 the collapsed Gbbs samplers coll2 and coll3 exhbt the lowest Monte Carlo errors for the spatal effects among the Gbbs samplers. The sampler coll2 even only requres 38 % of the computatonal effort of the MH sampler n order to obtan the same precson for the spatal effects. The precson and autocorrelatons of the regresson effects however are consderably smaller n the ndependence sampler compared to all Gbbs samplers. The precson and autocorrelatons (see Fgure 2) of the regresson effects however are consderably smaller n the ndependence sampler compared to all Gbbs samplers. In order to acheve a hgh precson lke n the MH sampler 6

18 for all parameters, for each Gbbs sampler the maxmum relatve effort R rel m rel, occurrng for spatal and regresson parameters, s requred. Snce the maxmum values R rel m rel are consderably greater than for each Gbbs sampler, the MH sampler s clearly superor to the Gbbs samplers. The average Monte Carlo error for the spatal effects n data set y 4 wth low heterogenety s rather hgh for the three Gbbs sampler schemes block, centered and coll for both spatal effects and regresson parameters, the estmated emprcal autocorrelatons plotted n Fgure 2 decrease very slowly. Whle for the hgh heterogenety data y 3 the computatonal costs n order to obtan the same precson for the spatal effects of the block Gbbs sampler are only.65 tmes as large as of the MH sampler, for the data y 4 the performance of the Gbbs sampler s clearly worse wth R rel m rel = Results are mproved for the collapsed algorthms based on the model parametersatons non-centered n the scale (coll2) and n the varance (coll3). The sampler coll2 performs even better than the MH sampler (R rel m rel =.45). As ndcated n Secton 3.5, the model parametersaton wth non-centered scale s supposed to mprove mxng partcularly for models wth small scale σ 2 whch s the case for data sets y 3 and y 4. However, the Monte Carlo errors for the regresson parameters are rather hgh for all Gbbs samplers and n partcular consderably hgher than for the hgh heterogenety data y 3. The MH ndependence sampler n contrast exhbts a hgh precson for all parameters agan. Compared to data set y 3, the standard errors for all parameters resultng from the MH sampler are about twce as large for data set y 4, ths loss of precson however s much smaller than for the Gbbs samplers. Accordng to the performance measure R rel m rel for the regresson parameters, the MH sampler outperforms the Gbbs samplers consderably. For example, although the Gbbs sampler coll2 sampler only requres 3 % of the computaton tme of the MH sampler for one teraton (see Table 3), 3.33 (R rel m rel for β 2 ) tmes the effort of the MH sampler for data set y 4 would be needed n order to obtan for all parameters a precson comparable to the MH sampler. Note that, compared to the collapsed algorthm coll2, the collapsed algorthm coll3 does not dsplay sgnfcantly lower standard errors, nether n Study nor n Study 2. The addtonal computatonal effort requred for coll3 whch s more than twce as large as for coll2, see Table 3, does not pay off. 4.4 Applcaton to car nsurance data Fnally we apply the dscussed Gbbs samplers as well as the ndependence MH sampler on the car nsurance data set descrbed at the begnnng of ths secton. The Monte Carlo errors for the posteror means of the regresson and the spatal effects as well as the correspondng values of R rel m rel are reported n Table 6. Smlar results as for data set y 4 whch s very close to our real data, are observed. In partcular for the regresson parameters, the performance of all Gbbs samplers s consderably worse than the performance of the MH ndependence sampler. When usng the non-centered scale and varance parametersatons at least for the spatal effects reasonable low errors are obtaned, however, accordng to the relatve effort R rel m rel the MH sampler s stll superor. 7

19 data sampler spatal β β 2 β 3 β 4 β 5 β 6 β 7 β 8 effects nd y 3 block centered coll coll coll nd y 4 block centered coll coll coll Table 5: Estmated ˆσ MC 2 (upper row) for the regresson parameters β,..,β 8 and estmated average ˆσ MC 2 for the spatal effects γ + β n the ndependence, block, coll sampler, γ n the centered, β + σγ n the coll2 and β + σlγ n the coll3 sampler, as well as R rel m rel (lower row) for data set y 3 and y 4 usng dfferent update strateges n Study 2. 5 Summary and conclusons We have presented a new MCMC methodology for spatal Posson regresson models, extendng the approach by Frühwrth-Schnatter and Wagner (24b). Usng data augmentaton we have shown that a straghtforward Gbbs sampler for spatal Posson models s avalable. Several update schemes lke a jont block update of the ntercept and the spatal effects as well as collapsed algorthms have been dscussed. Further we have addressed the ssue of model parametersaton, centered as well as non-centered model parametersatons n the mean, the scale and the varance have been consdered. The performance of the Gbbs sampler based on dfferent model parametersatons and update schemes has been compared to a sngle ste MH ndependence sampler on smulated and real data. Performance s measured n terms of the computatonal 8

20 sampler spatal β β 2 β 3 β 4 β 5 β 6 β 7 β 8 effects ndependence block centered coll coll coll Table 6: Estmated ˆσ MC (upper row) for the regresson parameters β,..,β 8 and average estmated ˆσ MC for the spatal effects γ + β n the ndependence, block, coll sampler, γ n the centered, β + σγ n the coll2 and β + σlγ n the coll3 sampler, as well as R rel m rel (lower row) for the car nsurance data usng dfferent update strateges. costs requred n order to obtan the same precson of the posteror means of the parameters. For data whch are not too homogeneous, the Gbbs samplers dsplay good mxng and reasonable small Monte Carlo errors. In partcular for data wth small spatal random effects, the performance s mproved when collapsed Gbbs samplers are used, whle the centered parametersaton s not very effcent any more n ths case. The MH ndependence sampler however exhbts the smallest Monte Carlo errors for all parameters for data wth both small and large spatal effects. Takng addtonally the requred computaton tmes of the samplers nto account, the MH sampler gves the best performance. For data wth low heterogenety the Monte Carlo errors ncrease sgnfcantly for all Gbbs samplers, mxng of the samplers s much worse. The MH sampler n contrast also mxes well for low heterogenety data, the precson of the posteror means of the parameters s consderably hgher than for the Gbbs samplers. Consderng the computaton tmes of the samplers and the requred MCMC teratons n order to obtan the same precson for all parameters, the MH sampler clearly outperforms the Gbbs samplers for low heterogenety data. Smlar results are observed for the real data whch also dsplay low heterogenety. In the lterature varous approaches for MCMC estmaton n spatal Posson models are provded. Knorr-Held and Rue (22) dscuss effcent block samplng MH algorthms for Markov random feld models n dsease mappng, based on the methodology developed n Rue (2). Haran et al. (23) study MH algorthms wth proposal dstrbutons based on Structured MCMC, ntroduced by Sargent et al. (2), for spatal Posson models, whle Chrstensen et al. (25) dscuss Langevn-Hastngs updates n spatal GLMM s. Rue et al. (24) present non-gaussan 9

21 approxmatons to hdden Markov random felds and gve applcatons n dsease mappng and geostatstcal models. These methods have been found to be superor to a conventonal MH sampler only performng ndvdual updates of the parameters. Therefore, snce a sngle ste MH sampler clearly outperformed the Gbbs samplers developed n ths paper, a comparson of the Gbbs samplers to these methods seems to be unnecessary. However, the performance of the Gbbs samplers mght be mproved by applyng the reparametersaton technques presented n Chrstensen et al. (25), whch s subject of current research. A Samplng the nter-arrval tmes Gven y and α, the nter-arrval tmes for dfferent observatons =,.., n are ndependent. For fxed however, τ,..,τ,y + are stochastcally dependent, but ndependent of the component ndcators R. The nter-arrval tmes τ,..,τ y are ndependent of α and only depend on the number of jumps, whereas τ,y + depends on the model parameters. Usng ths we have p(τ y,α,r) = n p(τ,y + y,α,τ,..,τ y )p(τ,..,τ y y ) = Gven y = n, the n arrval tmes of a Posson process are dstrbuted as the order statstcs of n U([, ]) dstrbuted random varables, see for example Mkosch (24). The last nterarrval tme τ,y +, gven y,τ,..,τ y, s exponentally dstrbuted wth mean µ = t exp(z α) condtonally on beng greater than y j= τ j. Usng the lack of memory property of the exponental dstrbuton ths corresponds to samplng τ,y + from an exponental dstrbuton wth mean µ plus an offset y j= τ j. Therefore the nter-arrval tmes can be sampled as follows: If y > sample y random numbers u,..,u y U([,]) sort these random numbers: u,(),..,u,(y ) defne τ j as the ncrements τ j = u,(j) u,(j ), j =,..,y where u j,() := sample τ,y + = y j= τ j + ζ, where ζ Exponental(µ ) If y = sample τ = + ζ, where ζ Exponental(µ ) B Samplng the component ndcators The component ndcators R are mutually ndependent gven τ,α, therefore p(r τ,α) = n = y + j= p(r j τ j,α). Further p(r j = k τ j,α) = p(r j = k,τ j,α) p(τ j,α) = p(τ j r j = k,α)p(r j = k) p(τ j α) p(τ j r j = k,α)w k (2.) 2

22 snce w k = p(r j = k). Snce log τ j α,r j N( log µ +m rj,s 2 r j ), τ j s log normal dstrbuted,.e. p(τ j r j = k,α) [ exp ( log(τj ) + log µ m ) k 2 ]. s k τ j 2 s k r j can therefore be sampled from the dscrete dstrbuton (2.) wth fve categores. C Startng values Startng values for the component ndcators r j are obtaned by drawng random numbers from to 5. For τ j startng values are generated accordng to the samplng procedure descrbed n Appendx A. For observatons equal to zero we sample ζ Exponental(.), for observatons greater than zero ζ Exponental(y ), as suggested n Frühwrth-Schnatter and Wagner (24b). D Detals on algorthm n Secton 3.4 For the collapsed algorthm n Secton 3.4 we consder p(β τ,r) = p(β,γ θ,τ,r)dγ. We have p(β,γ θ,τ,r) exp [ N 2 = = exp 2 s 2 [ β τ 2 Iβ + exp [γ ( N 2 = := c(β) exp 2 ]} (ỹ + x β + ṽ γ)2 + γ σ 2 Qγ + β τ 2 Iβ s 2 N = (ỹ + x β) 2]} s 2 ṽ ṽ + σ 2 Q )γ N + 2γ ]} ṽ (ỹ + x β) [ γ Aγ + 2γ a ]} s 2 = (4.2) where A := N = s 2 ṽ ṽ + σ 2 Q. Further exp [ ]} γ Aγ + 2γ a 2 exp [ ]} γ Aγ + 2γ A(A a) + (A a) A(A a) (A a) A(A a) 2 exp [ ]} (γ + A a) A(γ + A a) (A a) A(A a) 2 and therefore exp [ ]} γ Aγ + 2γ a dγ (2π) J 2 A } 2 exp 2 2 (A a) A(A a) } exp 2 (A a) A(A a) (4.3) 2

23 From (4.2) and (4.3) t then follows that Fnally, wth p(β,γ θ,τ,r)dγ } c(β)exp 2 (A a) A(A a) exp [ β (τ 2 I + 2 N = x x )β + 2β s 2 N = s 2 ]} x ỹ a A a a A a = ( N ṽ ỹ + = s 2 = s 2 N ) A ( ṽ x β N ṽ ỹ + = s 2 = s 2 = s 2 = N ) ṽ x β β ( N ) A ( N ṽ x ṽ x )β + 2β ( N ) A ( N ṽ x ) ṽ ỹ = s 2 s 2 = s 2 t follows that p(β τ, R) exp [ β ( τ 2 I + 2 = s 2 N N x x ( N ṽ x ) A ( ṽ x )β ) s 2 = 2β ( N N ( ṽ x ) A ( ṽ ỹ ) = s 2 = N = s 2 x ỹ )]}, s 2 = s 2.e. β τ,r N(Σ col µ col,σ col ) wth N Σ col := τ 2 N I + x x N ( ṽ x ) A ( ṽ x ) = s 2 = s 2 = s 2 and Acknowledgement N N µ col := ( ṽ x ) A ( ṽ ỹ ) s 2 = = s 2 N x ỹ. We would lke to thank Sylva Frühwrth-Schnatter for frutful dscussons and helpful comments and suggestons. The frst author s supported by a doctoral fellowshp wthn the Graduertenkolleg Angewandte Algorthmsche Mathematk, whle the second author s supported by Sonderforschungsberech 386 Statstsche Analyse Dskreter Strukturen, both sponsored by the Deutsche Forschungsgemenschaft. = s 2 22

24 References Besag, J. and C. Kooperberg (995). On condtonal and ntrnsc autoregressons. Bometrka 82, Chrstensen, O., G. Roberts, and M. Sköld (25). Robust MCMC methods for spatal GLMM s. to appear n Journal of Computatonal and Graphcal Statstcs. Chrstensen, O. and R. Waagepetersen (22). Bayesan predcton of spatal count data usng generalzed lnear mxed models. Bometrcs 58, Czado, C. and S. Prokopenko (24). Modelng transport mode decsons usng herarchcal bnary spatal regresson models wth cluster effects. Dscusson paper 46, SFB 386 Statstsche Analyse dskreter Strukturen. Dggle, P. J., J. A. Tawn, and R. A. Moyeed (998). Model-based geostatstcs. J. Roy. Statst. Soc. Ser. C 47(3), Wth dscusson and a reply by the authors. Dmakos, X. and A. Frgess (22). Bayesan premum ratng wth latent structure. Scandnavan Actuaral Journal (3), Frühwrth-Schnatter, S. (24). Effcent Bayesan parameter estmaton. In S. K. A.C. Harvey and N. Shephard (Eds.), State space and unobserved component models, pp Cambrdge: Cambrdge Unv. Press. Frühwrth-Schnatter, S. and H. Wagner (24a). Data augmentaton and Gbbs samplng for regresson models of small counts. IFAS Research Paper Seres Frühwrth-Schnatter, S. and H. Wagner (24b). Gbbs samplng for parameter-drven models of tme seres of small counts wth applcatons to state space modellng. IFAS Research Paper Seres 24-. Frühwrth-Schnatter, S. and H. Waldl (24). Data augmentaton and Gbbs samplng for logstc models. IFAS Research Paper Seres Gelfand, A., S. Sahu, and B.P.Carln (995). Effcent parametrsatons for normal lnear mxed models. Bometrka 82 (3), Geyer, C. (992). Practcal Markov Chan Monte Carlo. Statstcal Scence 7(4), Gschlößl, S. and C. Czado (25). Modellng count data wth overdsperson and spatal effects. Dscusson paper 42, SFB 386 Statstsche Analyse dskreter Strukturen. Haran, M., J. Hodges, and B. Carln (23). Acceleratng computaton n Markov random feld models for spatal data va structured MCMC. Journal of Computatonal and Graphcal Statstcs 2(2), Knorr-Held, L. and H. Rue (22). On block updatng n Markov random feld models for dsease mappng. Scandnavan Journal of Statstcs. Theory and Applcatons 29,

25 Lu, J., W. Wong, and A. Kong (994). Covarance structure of the Gbbs sampler wth applcatons to the comparsons of estmators and augmentaton schemes. Bometrka 8 (), Mkosch, T. (24). Non-Lfe Insurance Mathematcs. An Introducton wth Stochastc Processes. New York: Sprnger. Papasplopoulos, O., G. O. Roberts, and M. Sköld (23). Non-centered parameterzatons for herarchcal models and data augmentaton. In Bayesan statstcs, 7 (Tenerfe, 22), pp New York: Oxford Unv. Press. Wth a dscusson by Alan E. Gelfand, Ole F. Chrstensen and Darren J. Wlknson, and a reply by the authors. Petttt, A., I. Wer, and A. Hart (22). A condtonal autoregressve Gaussan process for rregularly spaced multvarate data wth applcaton to modellng large sets of bnary data. Statstcs and Computng 2 (4), Roberts, G. and S. Sahu (997). Updatng schemes, correlaton structure, blockng and parameterzaton for the Gbbs sampler. Journal of the Royal Statstcal Socety, B 59 (2), Rue, H. (2). Fast samplng of Gaussan Markov random felds. Journal of the Royal Statstcal Socety. Seres B. Statstcal Methodology 63(4), Rue, H., I. Stensland, and S. Erland (24). Approxmatng hdden Gaussan Markov random felds. Journal of the Royal Statstcal Socety. Seres B. Statstcal Methodology 66(4), Sargent, D., J. Hodges, and B. Carln (2). Structured Markov Chan Monte Carlo. Journal of Computatonal and Graphcal Statstcs 9, Sun, D., R. K. Tsutakawa, H. Km, and Z. He (2). Bayesan analyss of mortalty rates wth dsease maps. Statstcs n Medcne 9,