Small area estimation for unemployment using latent Markov models
Section 6. Final remarks

In this paper we develop a new area level SAE method that uses a Latent Markov Model (LMM) as the linking model. In LMMs (Bartolucci et al., 2013), the characteristic of interest, and its evolution in time, is represented by a latent process that follows a Markov chain, usually of first order. Under the assumption of normality for the conditional distribution of the response variables given the latent variables, the model is estimated using an augmented data Gibbs sampler. The proposed model has been applied to quarterly data from the Italian LFS from 2004 to 2014. The model-based method has been found to be effective for developing LMAs level estimates of unemployment incidence and the reduction in the coefficient of variation compared to the direct estimator is quite evident. The proposed approach is also more accurate than the direct and the time-series model-based estimator proposed by You et al. (2003) in reproducing census data. An advantage of this methodology is that it also provides a clustering of the small areas in homogeneous groups.

LMMs can be seen as an extension of latent class models to longitudinal data. In this regard, our approach represents an extension of the latent class SAE model proposed by Fabrizi et al. (2016). Moreover, LMMs may be seen as an extension of Markov chain models to control for measurement errors and can easily handle multivariate data, providing a very flexible modeling framework. The approach could be extended using spatial correlation information, and it could consider different distributions for the manifest variables, such as Poisson, Binomial, and Multinomial responses. In this scenario, we could fit unmatched sampling and linking models and handle departures from the normality assumption, but a Gibbs sampler cannot be used any longer, and Metropolis-Hastings sampling is an option. The proposed univariate model can account for measurement errors, but the extension to multivariate framework could be also possible, taking into account the conditional independence assumption.

In this application we have not accounted explicitly for the serial correlation induced by the rotating panel design. A natural way to take the different features of this design into account, such as the rotating group bias and the autocorrelation of the survey errors, is to use state space-model specifications, as in Pfeffermann (1991), Pfeffermann and Rubin-Bleuer (1993) and, more recently, Van den Brakel and Krieg (2015) and Boonstra and Van den Brakel (2016). In this context, it would also be interesting to extend to SAE the LMM with serial correlation in the measurement model proposed by Bartolucci and Farcomeni (2009). State space-model specifications can also be a useful tool to capture and model the strong trend and seasonality of this type of data.

Acknowledgements

The work of Bertarelli, Ranalli, D’Aló and Solari has been developed under the support of the project PRIN-2012F42NS8 “Household wealth and youth unemployment: new survey methods to meet current challenges”. The Authors are grateful to the Associated Editor and two anonymous Referees who provided very useful comments on earlier versions of this paper.

Appendix A

Model estimation

In the following we first illustrate Bayesian estimation and model selection based on a MCMC algorithm which is implemented in a data augmentation framework (Tanner and Wong, 1987).

A.1  Data augmentation method

In order to estimate the small area parameters Θ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHyoGaaiilaaaa@3340@ the measurement parameters ϕ obs , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaWgaaWcbaGaaGjbVl aab+gacaqGIbGaae4CaaqabaGccaaMb8Uaaiilaaaa@3A06@ and the latent parameters ϕ lat , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaWgaaWcbaGaaGjbVl aabYgacaqGHbGaaeiDaaqabaGccaaMb8Uaaiilaaaa@3A03@ we follow a data augmentation approach. We recall that the observed data consist of the direct estimates θ ^ i t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaBaaaleaacaWGPb GaamiDaaqabaGccaGGSaaaaa@35FF@ the corresponding smoothed MSE ^ i t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaqiaaqaaiaab2eacaqGtbGaaeyraa GaayPadaWaaSbaaSqaaiaadMgacaWG0baabeaakiaacYcaaaa@3769@ and the covariate vectors x i t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWH4bWaaSbaaSqaaiaadMgacaWG0b aabeaakiaacYcaaaa@353A@ with i = 1, , m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8UaamyBaaaa@3A76@ and t = 1, , T . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8Uaamivaiaac6caaaa@3B1A@ Moreover, the data augmentation approach explicitly introduces the latent variables U i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@3459@ treated as missing data, the values of which are updated during the MCMC algorithm that is, therefore, based on a complete data likelihood. In this context, the use of conjugate priors to the complete data likelihood allows us to sample from the conditional posterior of the latent states in a straightforward way. Since the state space is finite, sampling the latent states conditionally given the model parameters is also simple.

To generate samples from the joint posterior distribution of the model parameters and latent states, the proposed MCMC algorithm proceeds as follows. Let Θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWHyoGbaKaaaaa@32A0@ be the matrix of realizations of the available direct estimates that is defined as in (4.1), with each θ i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaWgaaWcbaGaamyAaiaads haaeqaaaaa@3535@ replaced by θ ^ i t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaBaaaleaacaWGPb GaamiDaaqabaGccaGGSaaaaa@35FF@ and let U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHvbaaaa@324A@ be the matrix of the latent variable U i t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgacaWG0b aabeaakiaacYcaaaa@3513@ with elements organized as in Θ ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWHyoGbaKaacaGGUaaaaa@3352@ Then the posterior distribution of all model parameters and latent variables, given the observed data, has the following expression:

p ( U , ϕ lat , ϕ obs , Θ | Θ ^ ) p ( U | ϕ lat ) π ( ϕ lat ) π ( ϕ obs ) p ( Θ | U , ϕ obs ) p ( Θ ^ | Θ ) . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaacaWHvbGaaGilai aaysW7iiWacqWFvpGzdaWgaaWcbaGaaGjbVlaabYgacaqGHbGaaeiD aaqabaGccaaISaGaaGjbVlab=v9aMnaaBaaaleaacaaMe8Uaae4Bai aabkgacaqGZbaabeaakiaaiYcacaaMe8UaaCiMdiaaykW7daabbaqa aiaaykW7ceWHyoGbaKaaaiaawEa7aaGaayjkaiaawMcaaiabg2Hi1k aadchadaqadaqaaiaahwfacaaMc8+aaqqaaeaacaaMc8Uae8x1dy2a aSbaaSqaaiaaysW7caqGSbGaaeyyaiaabshaaeqaaaGccaGLhWoaai aawIcacaGLPaaacqaHapaCdaqadaqaaiab=v9aMnaaBaaaleaacaaM e8UaaeiBaiaabggacaqG0baabeaaaOGaayjkaiaawMcaaiabec8aWn aabmaabaGae8x1dy2aaSbaaSqaaiaaysW7caqGVbGaaeOyaiaaboha aeqaaaGccaGLOaGaayzkaaGaamiCamaabmaabaGaaCiMdiaaykW7ca aI8bGaaGPaVlaahwfacaaISaGaaGjbVlab=v9aMnaaBaaaleaacaaM e8Uaae4BaiaabkgacaqGZbaabeaaaOGaayjkaiaawMcaaiaadchada qadaqaaiqahI5agaqcaiaaykW7daabbaqaaiaaykW7caWHyoaacaGL hWoaaiaawIcacaGLPaaacaaIUaaaaa@8DB1@

The MCMC algorithm alternates between sampling the latent variables and the parameters from the corresponding full conditional distribution. This scheme is repeated for R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbaaaa@3243@ iterations. At the end of each iteration r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGYbGaaiilaaaa@3313@ r = 1, , R , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGYbGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8UaamOuaiaacYcaaaa@3B14@ the sampled model parameters and latent variables are obtained and are denoted by U ( r ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHvbWaaWbaaSqabeaadaqadaqaai aadkhaaiaawIcacaGLPaaaaaGccaaMb8Uaaiilaaaa@373B@ ϕ lat ( r ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaqhaaWcbaGaaGjbVl aabYgacaqGHbGaaeiDaaqaamaabmaabaGaamOCaaGaayjkaiaawMca aaaakiaaygW7caGGSaaaaa@3C84@ ϕ obs ( r ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaqhaaWcbaGaaGjbVl aab+gacaqGIbGaae4CaaqaamaabmaabaGaamOCaaGaayjkaiaawMca aaaakiaaygW7caGGSaaaaa@3C87@ and Θ ( r ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHyoWaaWbaaSqabeaadaqadaqaai aadkhaaiaawIcacaGLPaaaaaGccaaMb8UaaiOlaaaa@3783@ More precisely, each iteration consists in:

  1. drawing U ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHvbWaaWbaaSqabeaadaqadaqaai aadkhaaiaawIcacaGLPaaaaaaaaa@34F7@ from p ( U | ϕ lat ( r 1 ) , ϕ obs ( r 1 ) , Θ ( r 1 ) ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaacaWHvbGaaGPaVp aaeeaabaGaaGPaVJGadiab=v9aMnaaDaaaleaacaaMe8UaaeiBaiaa bggacaqG0baabaWaaeWaaeaacaWGYbGaeyOeI0IaaGymaaGaayjkai aawMcaaaaaaOGaay5bSdGaaGzaVlaaiYcacaaMe8Uae8x1dy2aa0ba aSqaaiaaysW7caqGVbGaaeOyaiaabohaaeaadaqadaqaaiaadkhacq GHsislcaaIXaaacaGLOaGaayzkaaaaaOGaaGzaVlaaiYcacaaMe8Ua aCiMdmaaCaaaleqabaWaaeWaaeaacaWGYbGaeyOeI0IaaGymaaGaay jkaiaawMcaaaaaaOGaayjkaiaawMcaaiaacUdaaaa@5C50@
  2. drawing ϕ lat ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaqhaaWcbaGaaGjbVl aabYgacaqGHbGaaeiDaaqaamaabmaabaGaamOCaaGaayjkaiaawMca aaaaaaa@3A40@ from p ( ϕ lat | U ( r ) ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaiiWacqWFvpGzda WgaaWcbaGaaGjbVlaabYgacaqGHbGaaeiDaaqabaGccaaMc8+aaqqa aeaacaaMc8UaaCyvamaaCaaaleqabaWaaeWaaeaacaWGYbaacaGLOa GaayzkaaaaaaGccaGLhWoaaiaawIcacaGLPaaacaGG7aaaaa@4345@
  3. drawing ϕ obs ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaqhaaWcbaGaaGjbVl aab+gacaqGIbGaae4CaaqaamaabmaabaGaamOCaaGaayjkaiaawMca aaaaaaa@3A43@ from p ( ϕ obs | U ( r ) , Θ ( r 1 ) ) ; MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaiiWacqWFvpGzda WgaaWcbaGaaGjbVlaab+gacaqGIbGaae4CaaqabaGccaaMc8+aaqqa aeaacaaMc8UaaCyvamaaCaaaleqabaWaaeWaaeaacaWGYbaacaGLOa GaayzkaaaaaOGaaGzaVlaacYcaaiaawEa7aiaaysW7caWHyoWaaWba aSqabeaadaqadaqaaiaadkhacqGHsislcaaIXaaacaGLOaGaayzkaa aaaaGccaGLOaGaayzkaaGaai4oaaaa@4C92@
  4. drawing Θ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHyoWaaWbaaSqabeaadaqadaqaai aadkhaaiaawIcacaGLPaaaaaaaaa@353D@ from p ( Θ | U ( r ) , ϕ obs ( r ) , Θ ^ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaacaWHyoGaaGPaVp aaeeaabaGaaGPaVlaahwfadaahaaWcbeqaamaabmaabaGaamOCaaGa ayjkaiaawMcaaaaakiaaygW7caGGSaaacaGLhWoacaaMe8occmGae8 x1dy2aa0baaSqaaiaaysW7caqGVbGaaeOyaiaabohaaeaadaqadaqa aiaadkhaaiaawIcacaGLPaaaaaGccaaISaGaaGjbVlqahI5agaqcaa GaayjkaiaawMcaaiaac6caaaa@4E1E@

In the following we illustrate in details each of the above steps. In this regard, note that our illustration is referred to the case where all elements of Θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWHyoGbaKaaaaa@32A0@ are available. However, in our application, some elements of this matrix are missing. This requires minor adjustments to the MCMC algorithm, consisting in imputing the missing values by a Gibbs sampler and sampling directly from its full conditional distribution.

A.1.1  Simulation of U ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWHvbWaaWbaaSqabeaadaqadaqaai aadkhaaiaawIcacaGLPaaaaaaaaa@3501@

Each latent variable U i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@3459@ is drawn separately from the corresponding full conditional distribution, which is of multinomial type with specific parameters. In particular, we have that

U i t | U i , t 1 ( r ) , U i , t + 1 ( r 1 ) , ϕ lat ( r 1 ) , ϕ obs ( r 1 ) , Θ ( r 1 ) Multi k ( q i t ) , t = 1, , T , i = 1, , m , ( A .1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaaSbaaSqaaiaadMgacaWG0b aabeaakiaaykW7daabbaqaaiaaykW7caWGvbWaa0baaSqaaiaadMga caaMb8UaaGilaiaaykW7caWG0bGaeyOeI0IaaGymaaqaamaabmaaba GaamOCaaGaayjkaiaawMcaaaaakiaaygW7caGGSaaacaGLhWoacaaM e8UaamyvamaaDaaaleaacaWGPbGaaGzaVlaaiYcacaaMc8UaamiDai abgUcaRiaaigdaaeaadaqadaqaaiaadkhacqGHsislcaaIXaaacaGL OaGaayzkaaaaaOGaaGzaVlaaiYcacaaMe8occmGae8x1dy2aa0baaS qaaiaaysW7caqGSbGaaeyyaiaabshaaeaadaqadaqaaiaadkhacqGH sislcaaIXaaacaGLOaGaayzkaaaaaOGaaGzaVlaaiYcacaaMe8Uae8 x1dy2aa0baaSqaaiaaysW7caqGVbGaaeOyaiaabohaaeaadaqadaqa aiaadkhacqGHsislcaaIXaaacaGLOaGaayzkaaaaaOGaaGzaVlaaiY cacaaMe8UaaCiMdmaaCaaaleqabaWaaeWaaeaacaWGYbGaeyOeI0Ia aGymaaGaayjkaiaawMcaaaaarqqr1ngBPrgifHhDYfgaiqaakiab+X Ji6iaab2eacaqG1bGaaeiBaiaabshacaqGPbWaaSbaaSqaaiaadUga aeqaaOWaaeWaaeaacaWHXbWaaSbaaSqaaiaadMgacaWG0baabeaaaO GaayjkaiaawMcaaiaaiYcacaaMf8UaamiDaiaai2dacaaIXaGaaGil aiaaysW7cqWIMaYscaaISaGaaGjbVlaadsfacaaISaGaaGzbVlaadM gacaaI9aGaaGymaiaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7caWG TbGaaGilaiaaywW7caaMf8UaaiikaiaabgeacaqGUaGaaeymaiaacM caaaa@A82B@

where U i , t 1 ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaa0baaSqaaiaadMgacaaMb8 UaaGilaiaaykW7caWG0bGaeyOeI0IaaGymaaqaamaabmaabaGaamOC aaGaayjkaiaawMcaaaaaaaa@3C4D@ disappears for t = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdaaaa@33E7@ and U i , t + 1 ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbWaa0baaSqaaiaadMgacaaMb8 UaaGilaiaaykW7caWG0bGaey4kaSIaaGymaaqaamaabmaabaGaamOC aaGaayjkaiaawMcaaaaaaaa@3C42@ disappears for t = T . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaadsfacaGGUaaaaa@34B7@ Moreover, the probability vector q i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHXbWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@3479@ is defined as follows:

π u ( r 1 ) π u | U i 2 ( r 1 ) ( r 1 ) , u = 1, , k ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaGaamyDaaqaam aabmaabaGaamOCaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGccqaH apaCdaqhaaWcbaWaaqGaaeaacaWG1bGaaGPaVdGaayjcSdGaaGPaVl aadwfadaqhaaadbaGaamyAaiaaikdaaeaadaqadaqaaiaadkhacqGH sislcaaIXaaacaGLOaGaayzkaaaaaaWcbaWaaeWaaeaacaWGYbGaey OeI0IaaGymaaGaayjkaiaawMcaaaaakiaaygW7caaISaGaaGzbVlaa dwhacaaI9aGaaGymaiaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7ca WGRbGaaG4oaaaa@58CF@

π u | U i , t 1 ( r ) ( r 1 ) π U i , t + 1 ( r 1 ) | u ( r 1 ) , u = 1, , k ; MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaWaaqGaaeaaca WG1bGaaGPaVdGaayjcSdGaaGPaVlaaykW7caWGvbWaa0baaWqaaiaa dMgacaaMb8UaaGilaiaaykW7caWG0bGaeyOeI0IaaGymaaqaamaabm aabaGaamOCaaGaayjkaiaawMcaaaaaaSqaamaabmaabaGaamOCaiab gkHiTiaaigdaaiaawIcacaGLPaaaaaGccqaHapaCdaqhaaWcbaGaam yvamaaDaaameaacaWGPbGaaGzaVlaaiYcacaaMc8UaamiDaiabgUca RiaaigdaaeaadaqadaqaaiaadkhacqGHsislcaaIXaaacaGLOaGaay zkaaaaaSGaaGPaVpaaeeaabaGaaGPaVlaadwhaaiaawEa7aaqaamaa bmaabaGaamOCaiabgkHiTiaaigdaaiaawIcacaGLPaaaaaGccaaMb8 UaaGilaiaaywW7caWG1bGaaGypaiaaigdacaaISaGaaGjbVlablAci ljaaiYcacaaMe8Uaam4AaiaaiUdaaaa@6F96@

π u | U i , T 1 ( r ) ( r 1 ) , u = 1, , k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHapaCdaqhaaWcbaWaaqGaaeaaca WG1bGaaGPaVdGaayjcSdGaaGPaVlaadwfadaqhaaadbaGaamyAaiaa ygW7caaISaGaaGPaVlaadsfacqGHsislcaaIXaaabaWaaeWaaeaaca WGYbaacaGLOaGaayzkaaaaaaWcbaWaaeWaaeaacaWGYbGaeyOeI0Ia aGymaaGaayjkaiaawMcaaaaakiaaygW7caaISaGaaGzbVlaadwhaca aI9aGaaGymaiaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7caWGRbGa aGOlaaaa@5594@

A.1.2  Simulation of ϕ lat ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaaiiWacqWFvpGzdaqhaaWcbaGaaGjbVl aabYgacaqGHbGaaeiDaaqaamaabmaabaGaamOCaaGaayjkaiaawMca aaaaaaa@3A4A@

Recalling that ϕ lat = { π , Π } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaWgaaWcbaGaaGjbVl aabYgacaqGHbGaaeiDaaqabaGccaaI9aWaaiWaaeaacaWHapGaaGil aiaaysW7caWHGoaacaGL7bGaayzFaaGaaGjcVlaacYcaaaa@41BD@ we first draw π ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHapWaaWbaaSqabeaadaqadaqaai aadkhaaiaawIcacaGLPaaaaaaaaa@3565@ from the full conditional distribution:

π | U ( r ) Dirichlet ( 1 k + n 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHapGaaGPaVpaaeeaabaGaaGPaVl aahwfadaahaaWcbeqaamaabmaabaGaamOCaaGaayjkaiaawMcaaaaa aOGaay5bSdqeeuuDJXwAKbsr4rNCHbaceaGae8hpIOJaaeiraiaabM gacaqGYbGaaeyAaiaabogacaqGObGaaeiBaiaabwgacaqG0bWaaeWa aeaacaWHXaWaaSbaaSqaaiaadUgaaeqaaOGaey4kaSIaaCOBamaaBa aaleaacaaIXaaabeaaaOGaayjkaiaawMcaaiaaiYcaaaa@4FCA@

where n 1 = ( n 11 , , n 1 k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHUbWaaSbaaSqaaiaaigdaaeqaaO GaaGypamaabmaabaGaamOBamaaBaaaleaacaaIXaGaaGymaaqabaGc caaISaGaaGjbVlablAciljaaiYcacaaMe8UaamOBamaaBaaaleaaca aIXaGaam4AaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaOGamai2 gkdiIcaaaaa@43D6@ and n 1 u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaaigdacaWG1b aabeaaaaa@3440@ is the number of areas in state u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1baaaa@3266@ at time 1, with u = 1, , k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8Uaam4Aaiaac6caaaa@3B32@ Moreover, we draw each row of matrix Π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHGoaaaa@3298@ from the distribution

π u ¯ | U ( r ) Dirichlet ( 1 k + n u ¯ , t ) , t = 2, , T , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHapWaaSbaaSqaaiqadwhagaqeaa qabaGccaaMc8+aaqqaaeaacaaMc8UaaCyvamaaCaaaleqabaWaaeWa aeaacaWGYbaacaGLOaGaayzkaaaaaaGccaGLhWoarqqr1ngBPrgifH hDYfgaiqaacqWF8iIocaqGebGaaeyAaiaabkhacaqGPbGaae4yaiaa bIgacaqGSbGaaeyzaiaabshadaqadaqaaiaahgdadaWgaaWcbaGaam 4AaaqabaGccqGHRaWkcaWHUbWaaSbaaSqaaiqadwhagaqeaiaaygW7 caaISaGaaGPaVlaadshaaeqaaaGccaGLOaGaayzkaaGaaGilaiaayw W7caWG0bGaaGypaiaaikdacaaISaGaaGjbVlablAciljaaiYcacaaM e8UaamivaiaaygW7caaISaaaaa@62F8@

where n u ¯ , t = ( n u ¯ , t 1 , , n u ¯ , t k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHUbWaaSbaaSqaaiqadwhagaqeai aaygW7caaISaGaaGPaVlaadshaaeqaaOGaaGypamaabmaabaGaamOB amaaBaaaleaaceWG1bGbaebacaaMb8UaaGilaiaaykW7caWG0bWaaS baaWqaaiaaigdaaeqaaaWcbeaakiaaygW7caaISaGaaGjbVlablAci ljaaiYcacaaMe8UaamOBamaaBaaaleaaceWG1bGbaebacaaMb8UaaG ilaiaaykW7caWG0bWaaSbaaWqaaiaadUgaaeqaaaWcbeaaaOGaayjk aiaawMcaamaaCaaaleqabaGccWaGyBOmGikaaaaa@5521@ and n u ¯ , t u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiqadwhagaqeai aaygW7caaISaGaaGPaVlaadshadaWgaaadbaGaamyDaaqabaaaleqa aaaa@3993@ is the number of areas moving from state u ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWG1bGbaebaaaa@327E@ to state u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1baaaa@3266@ at time t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiilaaaa@3315@ with t = 2, , T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaikdacaaISaGaaG jbVlablAciljaaiYcacaaMe8Uaamivaaaa@3A69@ and u , u ¯ = 1, , k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGilaiaaysW7ceWG1bGbae bacaaI9aGaaGymaiaaiYcacaaMe8UaeSOjGSKaaGilaiaaysW7caWG RbGaaiOlaaaa@3E87@

A.1.3  Simulation of ϕ obs ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaaiiWacqWFvpGzdaqhaaWcbaGaaGjbVl aab+gacaqGIbGaae4CaaqaamaabmaabaGaamOCaaGaayjkaiaawMca aaaaaaa@3A4D@

Considering that ϕ obs = { β 1 , , β k , σ 1 2 , , σ k 2 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaWgaaWcbaGaaGjbVl aab+gacaqGIbGaae4CaaqabaGccaaI9aWaaiWaaeaacaWHYoWaaSba aSqaaiaaigdaaeqaaOGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVl aahk7adaWgaaWcbaGaam4AaaqabaGccaaMb8UaaGilaiaaysW7cqaH dpWCdaqhaaWcbaGaaGymaaqaaiaaikdaaaGccaaMb8UaaGilaiaays W7cqWIMaYscaaISaGaaGjbVlabeo8aZnaaDaaaleaacaWGRbaabaGa aGOmaaaaaOGaay5Eaiaaw2haaaaa@5715@ , we first draw each β u , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHYoWaaSbaaSqaaiaadwhaaeqaaO Gaaiilaaaa@348A@ u = 1, , k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8Uaam4AaiaacYcaaaa@3B30@ from the full conditional distribution:

β u | U ( r ) , Θ ( r ) N p ( η 1, u , Σ 1, u ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHYoWaaSbaaSqaaiaadwhaaeqaaO GaaGPaVpaaeeaabaGaaGPaVlaahwfadaahaaWcbeqaamaabmaabaGa amOCaaGaayjkaiaawMcaaaaakiaaygW7caGGSaaacaGLhWoacaaMe8 UaaCiMdmaaCaaaleqabaWaaeWaaeaacaWGYbaacaGLOaGaayzkaaaa aebbfv3ySLgzGueE0jxyaGabaOGae8hpIOJaamOtamaaBaaaleaaca WGWbaabeaakmaabmaabaGaaC4TdmaaBaaaleaacaaIXaGaaGilaiaa ykW7caWG1baabeaakiaaiYcacaaMe8UaaC4OdmaaBaaaleaacaaIXa GaaGilaiaaykW7caWG1baabeaaaOGaayjkaiaawMcaaiaaiYcaaaa@5ABC@

where

η 1, u = Λ 1, u 1 i = 1 m t = 1 T θ i t I ( U i t = u ) x i t , Σ 1, u = σ u 2 Λ 1, u 1 , Λ 1, u = i = 1 m t = 1 T x i t x i t I ( U i t = u ) + Λ 0 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaafaqaaeWacaaabaGaaC4TdmaaBaaale aacaaIXaGaaGilaiaaykW7caWG1baabeaaaOqaaiaai2dacaWHBoWa a0baaSqaaiaaigdacaaISaGaaGPaVlaadwhaaeaacqGHsislcaaIXa aaaOWaaabCaeqaleaacaWGPbGaaGypaiaaigdaaeaacaWGTbaaniab ggHiLdGcdaaeWbqaaiabeI7aXnaaBaaaleaacaWGPbGaamiDaaqaba GccaWGjbWaaeWaaeaacaWGvbWaaSbaaSqaaiaadMgacaWG0baabeaa kiaai2dacaWG1baacaGLOaGaayzkaaGaaCiEamaaBaaaleaacaWGPb GaamiDaaqabaGccaaISaaaleaacaWG0bGaaGypaiaaigdaaeaacaWG ubaaniabggHiLdaakeaacaWHJoWaaSbaaSqaaiaaigdacaaISaGaaG PaVlaadwhaaeqaaaGcbaGaaGypaiabeo8aZnaaDaaaleaacaWG1baa baGaaGOmaaaakiaahU5adaqhaaWcbaGaaGymaiaaiYcacaaMc8Uaam yDaaqaaiabgkHiTiaaigdaaaGccaaISaaabaGaaC4MdmaaBaaaleaa caaIXaGaaGilaiaaykW7caWG1baabeaaaOqaaiaai2dadaaeWbqabS qaaiaadMgacaaI9aGaaGymaaqaaiaad2gaa0GaeyyeIuoakmaaqaha baGaaCiEamaaBaaaleaacaWGPbGaamiDaaqabaGccaWH4bWaa0baaS qaaiaadMgacaWG0baabaqcLbwacWaGyBOmGikaaOGaamysamaabmaa baGaamyvamaaBaaaleaacaWGPbGaamiDaaqabaGccaaI9aGaamyDaa GaayjkaiaawMcaaaWcbaGaamiDaiaai2dacaaIXaaabaGaamivaaqd cqGHris5aOGaey4kaSIaaC4MdmaaBaaaleaacaaIWaaabeaakiaaiY caaaaaaa@904B@

with I ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGjbWaaeWaaeaacqGHflY1aiaawI cacaGLPaaaaaa@360D@ denoting the indicator function equal to 1 if its argument is true and to 0 otherwise. Then, we draw each σ u 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyDaaqaai aaikdaaaaaaa@3512@ from

σ u 2 | U ( r ) , Θ ( r ) IG ( a 1, u , b 1, u ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHdpWCdaqhaaWcbaGaamyDaaqaai aaikdaaaGccaaMc8+aaqqaaeaacaaMc8UaaCyvamaaCaaaleqabaWa aeWaaeaacaWGYbaacaGLOaGaayzkaaaaaOGaaGzaVlaacYcaaiaawE a7aiaaysW7caWHyoWaaWbaaSqabeaadaqadaqaaiaadkhaaiaawIca caGLPaaaaaqeeuuDJXwAKbsr4rNCHbaceaGccqWF8iIocaqGjbGaae 4ramaabmaabaGaamyyamaaBaaaleaacaaIXaGaaGilaiaaykW7caWG 1baabeaakiaaygW7caaISaGaaGjbVlaadkgadaWgaaWcbaGaaGymai aaiYcacaaMc8UaamyDaaqabaaakiaawIcacaGLPaaacaaISaaaaa@5C7B@

with

a 1, u = a 0 + n . u 2 , b 1, u = b 0 + 1 2 ( i = 1 m t = 1 T θ i t 2 I ( U i t = u ) + η 0 Λ 0 η 0 η 1, u Λ 1, u η 1, u ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaafaqaaeGacaaabaGaamyyamaaBaaale aacaaIXaGaaGilaiaaykW7caWG1baabeaaaOqaaiaai2dacaWGHbWa aSbaaSqaaiaaicdaaeqaaOGaey4kaSYaaSaaaeaacaWGUbWaaSbaaS qaaiaai6cacaWG1baabeaaaOqaaiaaikdaaaGaaGilaaqaaiaadkga daWgaaWcbaGaaGymaiaaiYcacaaMc8UaamyDaaqabaaakeaacaaI9a GaamOyamaaBaaaleaacaaIWaaabeaakiabgUcaRmaalaaabaGaaGym aaqaaiaaikdaaaWaaeWaaeaadaaeWbqabSqaaiaadMgacaaI9aGaaG ymaaqaaiaad2gaa0GaeyyeIuoakmaaqahabaGaeqiUde3aa0baaSqa aiaadMgacaWG0baabaGaaGOmaaaakiaadMeadaqadaqaaiaadwfada WgaaWcbaGaamyAaiaadshaaeqaaOGaaGypaiaadwhaaiaawIcacaGL PaaaaSqaaiaadshacaaI9aGaaGymaaqaaiaadsfaa0GaeyyeIuoaki abgUcaRiaahE7adaqhaaWcbaGaaGimaaqaaKqzGfGamai2gkdiIcaa kiaahU5adaWgaaWcbaGaaGimaaqabaGccaWH3oWaaSbaaSqaaiaaic daaeqaaOGaeyOeI0IaaC4TdmaaDaaaleaacaaIXaGaaGilaiaaykW7 caWG1baabaqcLbwacWaGyBOmGikaaOGaaC4MdmaaBaaaleaacaaIXa GaaGilaiaaykW7caWG1baabeaakiaahE7adaWgaaWcbaGaaGymaiaa iYcacaaMc8UaamyDaaqabaaakiaawIcacaGLPaaacaaISaaaaaaa@818C@

where n . u = t = 1 T n t u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGUbWaaSbaaSqaaiaai6cacaWG1b aabeaakiaai2dadaaeWaqabSqaaiaadshacaaI9aGaaGymaaqaaiaa dsfaa0GaeyyeIuoakiaaykW7caWGUbWaaSbaaSqaaiaadshacaWG1b aabeaaaaa@3F0C@ is the number of areas in state u MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1baaaa@3266@ regardless of the specific time occasion.

A.1.4  Simulation of Θ ( r ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeqabeqadiWa ceGabeqabeqabeqadeaakeaacaWHyoWaaWbaaSqabeaadaqadaqaai aadkhaaiaawIcacaGLPaaaaaaaaa@3547@

The goal of SAE is to predict each θ i t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaWgaaWcbaGaamyAaiaads haaeqaaOGaaiilaaaa@35EF@ i = 1, , m , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8UaamyBaiaacYcaaaa@3B26@ t = 1, , T , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaGypaiaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8UaamivaiaacYcaaaa@3B18@ based on the model and the observed data. This amounts to draw these elements from

θ i t | U ( r ) , ϕ obs ( r ) , θ ^ i t N ( θ ^ i t ( r ) , γ i t ( r ) ψ i t ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaWgaaWcbaGaamyAaiaads haaeqaaOGaaGPaVpaaeeaabaGaaGPaVlaahwfadaahaaWcbeqaamaa bmaabaGaamOCaaGaayjkaiaawMcaaaaakiaaygW7caGGSaaacaGLhW oacaaMe8occmGae8x1dy2aa0baaSqaaiaaysW7caqGVbGaaeOyaiaa bohaaeaadaqadaqaaiaadkhaaiaawIcacaGLPaaaaaGccaaMb8UaaG ilaiaaysW7cuaH4oqCgaqcamaaBaaaleaacaWGPbGaamiDaaqabaqe euuDJXwAKbsr4rNCHbaceaGccqGF8iIocaWGobGaaGikaiqbeI7aXz aajaWaa0baaSqaaiaadMgacaWG0baabaWaaeWaaeaacaWGYbaacaGL OaGaayzkaaaaaOGaaGzaVlaaiYcacaaMe8Uaeq4SdC2aa0baaSqaai aadMgacaWG0baabaWaaeWaaeaacaWGYbaacaGLOaGaayzkaaaaaOGa eqiYdK3aaSbaaSqaaiaadMgacaWG0baabeaakiaaiMcacaaISaaaaa@6EDB@

where

θ ^ i t ( r ) = γ i t ( r ) θ ^ i t + ( 1 γ i t ( r ) ) x i t β u ( r ) , ( A .2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaDaaaleaacaWGPb GaamiDaaqaamaabmaabaGaamOCaaGaayjkaiaawMcaaaaakiaai2da cqaHZoWzdaqhaaWcbaGaamyAaiaadshaaeaadaqadaqaaiaadkhaai aawIcacaGLPaaaaaGccuaH4oqCgaqcamaaBaaaleaacaWGPbGaamiD aaqabaGccqGHRaWkdaqadaqaaiaaigdacqGHsislcqaHZoWzdaqhaa WcbaGaamyAaiaadshaaeaadaqadaqaaiaadkhaaiaawIcacaGLPaaa aaaakiaawIcacaGLPaaacaWH4bWaa0baaSqaaiaadMgacaWG0baaba qcLbwacWaGyBOmGikaaOGaaCOSdmaaDaaaleaacaWG1baabaWaaeWa aeaacaWGYbaacaGLOaGaayzkaaaaaOGaaGzaVlaacYcacaaMf8UaaG zbVlaaywW7caaMf8UaaGzbVlaacIcacaqGbbGaaeOlaiaabkdacaGG Paaaaa@6656@

with γ i t ( r ) = σ u 2 ( r ) / ( σ u 2 ( r ) + ψ i t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaHZoWzdaqhaaWcbaGaamyAaiaads haaeaadaqadaqaaiaadkhaaiaawIcacaGLPaaaaaGccaaI9aWaaSGb aeaacqaHdpWCdaqhaaWcbaGaamyDaaqaaiaaikdadaqadaqaaiaadk haaiaawIcacaGLPaaaaaaakeaadaqadaqaaiabeo8aZnaaDaaaleaa caWG1baabaGaaGOmamaabmaabaGaamOCaaGaayjkaiaawMcaaaaaki abgUcaRiabeI8a5naaBaaaleaacaWGPbGaamiDaaqabaaakiaawIca caGLPaaaaaGaaiOlaaaa@4BF6@

A.2  Model selection: The Chib estimator

The method proposed in Chib (1995) can be applied to perform model selection starting from the Gibbs sampler output. It is known that the posterior density can be written as the ratio of the product of the likelihood function and the priors divided by the marginal likelihood:

p ( U , ϕ lat , ϕ obs , Θ | Θ ^ ) = p ( U | ϕ lat ) π ( ϕ lat ) π ( ϕ obs ) p ( Θ | U , ϕ obs ) p ( Θ ^ | Θ ) p ( Θ ^ ) . ( A .3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaacaWHvbGaaGilai aaysW7iiWacqWFvpGzdaWgaaWcbaGaaGjbVlaabYgacaqGHbGaaeiD aaqabaGccaaMb8UaaGilaiaaysW7cqWFvpGzdaWgaaWcbaGaaGjbVl aab+gacaqGIbGaae4CaaqabaGccaaMb8UaaGilaiaaysW7caWHyoGa aGPaVpaaeeaabaGaaGPaVlqahI5agaqcaaGaay5bSdaacaGLOaGaay zkaaGaaGypamaalaaabaGaamiCamaabmaabaGaaCyvaiaaykW7daab baqaaiaaykW7cqWFvpGzdaWgaaWcbaGaaGjbVlaabYgacaqGHbGaae iDaaqabaaakiaawEa7aaGaayjkaiaawMcaaiabec8aWnaabmaabaGa e8x1dy2aaSbaaSqaaiaaysW7caqGSbGaaeyyaiaabshaaeqaaaGcca GLOaGaayzkaaGaeqiWda3aaeWaaeaacqWFvpGzdaWgaaWcbaGaaGjb Vlaab+gacaqGIbGaae4CaaqabaaakiaawIcacaGLPaaacaWGWbWaae WaaeaacaWHyoGaaGPaVpaaeeaabaGaaGPaVlaahwfacaaMb8Uaaiil aaGaay5bSdGaaGjbVlab=v9aMnaaBaaaleaacaaMe8Uaae4Baiaabk gacaqGZbaabeaaaOGaayjkaiaawMcaaiaadchadaqadaqaaiqahI5a gaqcaiaaykW7daabbaqaaiaaykW7caWHyoaacaGLhWoaaiaawIcaca GLPaaaaeaacaWGWbWaaeWaaeaaceWHyoGbaKaaaiaawIcacaGLPaaa aaGaaGOlaiaaywW7caaMf8UaaiikaiaabgeacaqGUaGaae4maiaacM caaaa@9C80@

Therefore, it is possible to write the marginal likelihood of the data Θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWHyoGbaKaaaaa@32A0@ as

p ( Θ ^ ) = p ( Θ ^ | Θ ) p ( U | ϕ lat ) π ( ϕ lat ) π ( ϕ obs ) p ( Θ | U , ϕ obs ) p ( U , ϕ lat , ϕ obs , Θ | Θ ^ ) , ( A .4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaceWHyoGbaKaaai aawIcacaGLPaaacaaI9aWaaSaaaeaacaWGWbWaaeWaaeaaceWHyoGb aKaacaaMc8+aaqqaaeaacaaMc8UaaCiMdaGaay5bSdaacaGLOaGaay zkaaGaamiCamaabmaabaGaaCyvaiaaykW7daabbaqaaiaaykW7iiWa cqWFvpGzdaWgaaWcbaGaaGjbVlaabYgacaqGHbGaaeiDaaqabaaaki aawEa7aaGaayjkaiaawMcaaiabec8aWnaabmaabaGae8x1dy2aaSba aSqaaiaaysW7caqGSbGaaeyyaiaabshaaeqaaaGccaGLOaGaayzkaa GaeqiWda3aaeWaaeaacqWFvpGzdaWgaaWcbaGaaGjbVlaab+gacaqG IbGaae4CaaqabaaakiaawIcacaGLPaaacaWGWbWaaeWaaeaacaWHyo GaaGPaVpaaeeaabaGaaGPaVlaahwfacaaISaaacaGLhWoacaaMe8Ua e8x1dy2aaSbaaSqaaiaaysW7caqGVbGaaeOyaiaabohaaeqaaaGcca GLOaGaayzkaaaabaGaamiCamaabmaabaGaaCyvaiaaiYcacaaMe8Ua e8x1dy2aaSbaaSqaaiaaysW7caqGSbGaaeyyaiaabshaaeqaaOGaaG zaVlaaiYcacaaMe8Uae8x1dy2aaSbaaSqaaiaaysW7caqGVbGaaeOy aiaabohaaeqaaOGaaGzaVlaaiYcacaaMe8UaaCiMdiaaykW7daabba qaaiaaykW7ceWHyoGbaKaaaiaawEa7aaGaayjkaiaawMcaaaaacaaI SaGaaGzbVlaaywW7caaMf8UaaiikaiaabgeacaqGUaGaaeinaiaacM caaaa@9C89@

for any U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHvbGaaiilaaaa@32FA@ ϕ lat , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaWgaaWcbaGaaGjbVl aabYgacaqGHbGaaeiDaaqabaGccaaMb8Uaaiilaaaa@3A03@ ϕ obs , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacqWFvpGzdaWgaaWcbaGaaGjbVl aab+gacaqGIbGaae4CaaqabaGccaaMb8Uaaiilaaaa@3A06@ Θ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHyoaaaa@3290@ and Θ ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWHyoGbaKaacaGGUaaaaa@3352@ We drop the dependence on k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbaaaa@325C@ for ease of notation. This is the model selection criterion used in Section 5. Then, choosing specific values of the latent variables and model parameters, denoted by U ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWHvbGbaebacaGGSaaaaa@3312@ ϕ ¯ lat , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacuWFvpGzgaqeamaaBaaaleaaca aMe8UaaeiBaiaabggacaqG0baabeaakiaaygW7caGGSaaaaa@3A1B@ ϕ ¯ obs , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaiiWacuWFvpGzgaqeamaaBaaaleaaca aMe8Uaae4BaiaabkgacaqGZbaabeaakiaaygW7caGGSaaaaa@3A1E@ and Θ ¯ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWHyoGbaebacaGGSaaaaa@3358@ we can estimate log p ( Θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaciGGSbGaai4BaiaacEgacaWGWbWaae WaaeaaceWHyoGbaKaaaiaawIcacaGLPaaaaaa@37EE@ through the following decomposition:

log p ( Θ ^ ) = log p ( Θ ^ | Θ ¯ ) + log p ( U ¯ | ϕ ¯ lat ) + log π ( ϕ ¯ lat ) + log π ( ϕ ¯ obs ) + log p ( Θ ¯ | U ¯ , ϕ ¯ obs ) log p ( U ¯ , ϕ ¯ lat , ϕ ¯ obs , Θ ¯ | Θ ^ ) . ( A .5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaafaqaaeGacaaabaGaciiBaiaac+gaca GGNbGaamiCamaabmaabaGabCiMdyaajaaacaGLOaGaayzkaaGaaGyp aaqaaiGacYgacaGGVbGaai4zaiaadchadaqadaqaaiqahI5agaqcai aaykW7daabbaqaaiaaykW7ceWHyoGbaebaaiaawEa7aaGaayjkaiaa wMcaaiabgUcaRiGacYgacaGGVbGaai4zaiaadchadaqadaqaaiqahw fagaqeaiaaykW7daabbaqaaiaaykW7iiWacuWFvpGzgaqeamaaBaaa leaacaaMe8UaaeiBaiaabggacaqG0baabeaaaOGaay5bSdaacaGLOa GaayzkaaGaey4kaSIaciiBaiaac+gacaGGNbGaeqiWda3aaeWaaeaa cuWFvpGzgaqeamaaBaaaleaacaaMe8UaaeiBaiaabggacaqG0baabe aaaOGaayjkaiaawMcaaiabgUcaRiGacYgacaGGVbGaai4zaiabec8a WnaabmaabaGaf8x1dyMbaebadaWgaaWcbaGaaGjbVlaab+gacaqGIb Gaae4CaaqabaaakiaawIcacaGLPaaaaeaaaeaacqGHRaWkciGGSbGa ai4BaiaacEgacaWGWbWaaeWaaeaaceWHyoGbaebacaaMc8+aaqqaae aacaaMc8UabCyvayaaraGaaiilaaGaay5bSdGaaGjbVlqb=v9aMzaa raWaaSbaaSqaaiaaysW7caqGVbGaaeOyaiaabohaaeqaaaGccaGLOa GaayzkaaGaeyOeI0IaciiBaiaac+gacaGGNbGaamiCamaabmaabaGa bCyvayaaraGaaGilaiaaysW7cuWFvpGzgaqeamaaBaaaleaacaaMe8 UaaeiBaiaabggacaqG0baabeaakiaaygW7caaISaGaaGjbVlqb=v9a MzaaraWaaSbaaSqaaiaaysW7caqGVbGaaeOyaiaabohaaeqaaOGaaG zaVlaaiYcacaaMe8UabCiMdyaaraGaaGPaVpaaeeaabaGaaGPaVlqa hI5agaqcaaGaay5bSdaacaGLOaGaayzkaaGaaGOlaiaaywW7caaMf8 UaaGzbVlaaywW7caaMf8UaaiikaiaabgeacaqGUaGaaeynaiaacMca aaaaaa@B8E6@

The use of the log transformation is motivated by numerical stability (Chib, 1995).

The first five terms at the right hand side of (A.5) can be computed directly from the assumed distributions of the parameters and the data. On the other hand obtaining the last component is more challanging. By the law of total probability, p ( U ¯ , ϕ ¯ lat , ϕ ¯ obs , Θ ¯ | Θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaceWHvbGbaebaca aISaGaaGjbVJGadiqb=v9aMzaaraWaaSbaaSqaaiaaysW7caqGSbGa aeyyaiaabshaaeqaaOGaaGzaVlaaiYcacaaMe8Uaf8x1dyMbaebada WgaaWcbaGaaGjbVlaab+gacaqGIbGaae4CaaqabaGccaaMb8UaaGil aiaaysW7ceWHyoGbaebacaaMc8+aaqqaaeaacaaMc8UabCiMdyaaja aacaGLhWoaaiaawIcacaGLPaaaaaa@52B7@ may be decomposed as

p ( U ¯ , ϕ ¯ lat , ϕ ¯ obs , Θ ¯ | Θ ^ ) = p ( U ¯ | ϕ ¯ lat , ϕ ¯ obs , Θ ¯ , Θ ^ ) p ( ϕ ¯ lat | ϕ ¯ obs , Θ ¯ , Θ ^ ) p ( ϕ ¯ obs | Θ ¯ , Θ ^ ) p ( Θ ¯ | Θ ^ ) . ( A .6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaceWHvbGbaebaca aISaGaaGjbVJGadiqb=v9aMzaaraWaaSbaaSqaaiaaysW7caqGSbGa aeyyaiaabshaaeqaaOGaaGzaVlaaiYcacaaMe8Uaf8x1dyMbaebada WgaaWcbaGaaGjbVlaab+gacaqGIbGaae4CaaqabaGccaaMb8UaaGil aiaaysW7ceWHyoGbaebacaaMc8+aaqqaaeaacaaMc8UabCiMdyaaja aacaGLhWoaaiaawIcacaGLPaaacaaI9aGaamiCamaabmaabaGabCyv ayaaraGaaGPaVpaaeeaabaGaaGPaVlqb=v9aMzaaraWaaSbaaSqaai aaysW7caqGSbGaaeyyaiaabshaaeqaaOGaaGzaVlaacYcaaiaawEa7 aiaaysW7cuWFvpGzgaqeamaaBaaaleaacaaMe8Uaae4Baiaabkgaca qGZbaabeaakiaaygW7caaISaGaaGjbVlqahI5agaqeaiaaiYcacaaM e8UabCiMdyaajaaacaGLOaGaayzkaaGaamiCamaabmaabaGaf8x1dy MbaebadaWgaaWcbaGaaGjbVlaabYgacaqGHbGaaeiDaaqabaGccaaM c8+aaqqaaeaacaaMc8Uaf8x1dyMbaebadaWgaaWcbaGaaGjbVlaab+ gacaqGIbGaae4CaaqabaGccaaMb8UaaiilaaGaay5bSdGaaGjbVlqa hI5agaqeaiaaiYcacaaMe8UabCiMdyaajaaacaGLOaGaayzkaaGaam iCamaabmaabaGaf8x1dyMbaebadaWgaaWcbaGaaGjbVlaab+gacaqG IbGaae4CaaqabaGccaaMc8+aaqqaaeaacaaMc8UabCiMdyaaraGaaG ilaaGaay5bSdGaaGjbVlqahI5agaqcaaGaayjkaiaawMcaaiaadcha daqadaqaaiqahI5agaqeaiaaykW7daabbaqaaiaaykW7ceWHyoGbaK aaaiaawEa7aaGaayjkaiaawMcaaiaai6cacaaMf8Uaaiikaiaabgea caqGUaGaaeOnaiaacMcaaaa@B2D5@

Following Chib (1995), we compute the first term of (A.6) following the Gibbs scheme outlined in Section A.1, whereas, the other three terms are estimated from the Gibbs output. In particular, we estimate

p ( ϕ ¯ lat | ϕ ¯ obs , Θ ¯ , Θ ^ ) = p ( ϕ ¯ lat | U ¯ , ϕ ¯ obs , Θ ¯ , Θ ^ ) p ( U ¯ | ϕ ¯ obs , Θ ¯ , Θ ^ ) d U ¯ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaiiWacuWFvpGzga qeamaaBaaaleaacaaMe8UaaeiBaiaabggacaqG0baabeaakiaaykW7 daabbaqaaiaaykW7cuWFvpGzgaqeamaaBaaaleaacaaMe8Uaae4Bai aabkgacaqGZbaabeaakiaaygW7caGGSaaacaGLhWoacaaMe8UabCiM dyaaraGaaGilaiaaysW7ceWHyoGbaKaaaiaawIcacaGLPaaacaaI9a Waa8qaaeqaleqabeqdcqGHRiI8aOGaamiCamaabmaabaGaf8x1dyMb aebadaWgaaWcbaGaaGjbVlaabYgacaqGHbGaaeiDaaqabaGccaaMc8 +aaqqaaeaacaaMc8UabCyvayaaraGaaGilaaGaay5bSdGaaGjbVlqb =v9aMzaaraWaaSbaaSqaaiaaysW7caqGVbGaaeOyaiaabohaaeqaaO GaaGzaVlaaiYcacaaMe8UabCiMdyaaraGaaGilaiaaysW7ceWHyoGb aKaaaiaawIcacaGLPaaacaWGWbWaaeWaaeaaceWHvbGbaebacaaMc8 +aaqqaaeaacaaMc8Uaf8x1dyMbaebadaWgaaWcbaGaaGjbVlaab+ga caqGIbGaae4CaaqabaGccaaMb8UaaGilaaGaay5bSdGaaGjbVlqahI 5agaqeaiaaiYcacaaMe8UabCiMdyaajaaacaGLOaGaayzkaaGaamiz aiqahwfagaqeaaaa@8956@

as R 1 r = 1 R p ( ϕ ¯ lat | U ( r ) , ϕ ¯ obs , Θ ¯ ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbWaaWbaaSqabeaacqGHsislca aIXaaaaOWaaabmaeqaleaacaWGYbGaaGypaiaaigdaaeaacaWGsbaa niabggHiLdGccaaMc8UaamiCamaabmaabaaccmGaf8x1dyMbaebada WgaaWcbaGaaGjbVlaabYgacaqGHbGaaeiDaaqabaGccaaMc8+aaqqa aeaacaaMc8UaaCyvamaaCaaaleqabaWaaeWaaeaacaWGYbaacaGLOa GaayzkaaaaaOGaaGzaVlaaiYcaaiaawEa7aiaaysW7cuWFvpGzgaqe amaaBaaaleaacaaMe8Uaae4BaiaabkgacaqGZbaabeaakiaaygW7ca aISaGaaGjbVlqahI5agaqeaaGaayjkaiaawMcaaiaaiYcaaaa@5C33@ based on R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbaaaa@3243@ draws from a reduced Gibbs sampling where U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWHvbaaaa@324A@ is not updated. In order to estimate

p ( ϕ ¯ obs | Θ ¯ , Θ ^ ) = p ( ϕ ¯ obs | U ¯ , ϕ ¯ lat , Θ ¯ , Θ ^ ) p ( U ¯ , ϕ ¯ lat | Θ ¯ , Θ ^ ) d U ¯ d ϕ ¯ lat , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaiiWacuWFvpGzga qeamaaBaaaleaacaaMe8Uaae4BaiaabkgacaqGZbaabeaakiaaykW7 daabbaqaaiaaykW7ceWHyoGbaebaaiaawEa7aiaaygW7caGGSaGaaG jbVlqahI5agaqcaaGaayjkaiaawMcaaiaai2dadaWdbaqabSqabeqa niabgUIiYdGccaWGWbWaaeWaaeaacuWFvpGzgaqeamaaBaaaleaaca aMe8Uaae4BaiaabkgacaqGZbaabeaakiaaykW7daabbaqaaiaaykW7 ceWHvbGbaebacaGGSaaacaGLhWoacaaMe8Uaf8x1dyMbaebadaWgaa WcbaGaaGjbVlaabYgacaqGHbGaaeiDaaqabaGccaaMb8Uaaiilaiaa ysW7ceWHyoGbaebacaaISaGaaGjbVlqahI5agaqcaaGaayjkaiaawM caaiaadchadaqadaqaaiqahwfagaqeaiaaygW7caaISaGaaGjbVlqb =v9aMzaaraWaaSbaaSqaaiaaysW7caqGSbGaaeyyaiaabshaaeqaaO GaaGPaVpaaeeaabaGaaGPaVlqahI5agaqeaiaaiYcacaaMe8UabCiM dyaajaaacaGLhWoaaiaawIcacaGLPaaacaWGKbGabCyvayaaraGaaG PaVlaadsgacuWFvpGzgaqeamaaBaaaleaacaaMe8UaaeiBaiaabgga caqG0baabeaakiaaygW7caaISaaaaa@8BB8@

we use R 1 r = 1 R p ( ϕ ¯ obs | U ( r , 1 ) , ϕ ¯ lat ( r , 1 ) , Θ ¯ ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbWaaWbaaSqabeaacqGHsislca aIXaaaaOWaaabmaeqaleaacaWGYbGaaGypaiaaigdaaeaacaWGsbaa niabggHiLdGccaaMc8UaamiCamaabmaabaaccmGaf8x1dyMbaebada WgaaWcbaGaaGjbVlaab+gacaqGIbGaae4CaaqabaGccaaMc8+aaqqa aeaacaaMc8UaaCyvamaaCaaaleqabaWaaeWaaeaacaWGYbGaaGzaVl aacYcacaaMc8UaaGymaaGaayjkaiaawMcaaaaakiaaygW7caGGSaaa caGLhWoacaaMe8Uaf8x1dyMbaebadaqhaaWcbaGaaGjbVlaabYgaca qGHbGaaeiDaaqaamaabmaabaGaamOCaiaaygW7caGGSaGaaGPaVlaa igdaaiaawIcacaGLPaaaaaGccaaMb8UaaGilaiaaysW7ceWHyoGbae baaiaawIcacaGLPaaacaGGUaaaaa@67AA@ Finally, to estimate

p ( Θ ¯ | Θ ^ ) = p ( Θ ¯ | U ¯ , ϕ ¯ lat , ϕ ¯ obs , Θ ^ ) p ( U ¯ , ϕ ¯ lat , ϕ ¯ obs | Θ ^ ) d U ¯ d ϕ ¯ lat d ϕ ¯ obs , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGWbWaaeWaaeaaceWHyoGbaebaca aMc8+aaqqaaeaacaaMc8UabCiMdyaajaaacaGLhWoaaiaawIcacaGL PaaacaaI9aWaa8qaaeqaleqabeqdcqGHRiI8aOGaamiCamaabmaaba GabCiMdyaaraGaaGPaVpaaeeaabaGaaGPaVlqahwfagaqeaiaaygW7 caGGSaaacaGLhWoacaaMe8occmGaf8x1dyMbaebadaWgaaWcbaGaaG jbVlaabYgacaqGHbGaaeiDaaqabaGccaaMb8UaaGilaiaaysW7cuWF vpGzgaqeamaaBaaaleaacaaMe8Uaae4BaiaabkgacaqGZbaabeaaki aaygW7caaISaGaaGjbVlqahI5agaqcaaGaayjkaiaawMcaaiaadcha daqadaqaaiqahwfagaqeaiaaiYcacaaMe8Uaf8x1dyMbaebadaWgaa WcbaGaaGjbVlaabYgacaqGHbGaaeiDaaqabaGccaaMb8UaaGilaiaa ysW7cuWFvpGzgaqeamaaBaaaleaacaaMe8Uaae4BaiaabkgacaqGZb aabeaakiaaykW7daabbaqaaiaaykW7ceWHyoGbaKaaaiaawEa7aaGa ayjkaiaawMcaaiaadsgaceWHvbGbaebacaaMc8Uaamizaiqb=v9aMz aaraWaaSbaaSqaaiaaysW7caqGSbGaaeyyaiaabshaaeqaaOGaaGPa VlaadsgacuWFvpGzgaqeamaaBaaaleaacaaMe8Uaae4Baiaabkgaca qGZbaabeaakiaaygW7caaISaaaaa@92AE@

we use R 1 r = 1 R p ( Θ ¯ | U ( r , 2 ) , ϕ lat ( r , 2 ) , ϕ obs ( r , 2 ) ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbWaaWbaaSqabeaacqGHsislca aIXaaaaOWaaabmaeqaleaacaWGYbGaaGypaiaaigdaaeaacaWGsbaa niabggHiLdGccaaMc8UaamiCamaabmaabaGabCiMdyaaraGaaGPaVp aaeeaabaGaaGPaVlaahwfadaahaaWcbeqaamaabmaabaGaamOCaiaa ygW7caGGSaGaaGjbVlaaikdaaiaawIcacaGLPaaaaaGccaaMb8Uaai ilaaGaay5bSdGaaGjbVJGadiab=v9aMnaaDaaaleaacaaMe8UaaeiB aiaabggacaqG0baabaWaaeWaaeaacaWGYbGaaGzaVlaacYcacaaMe8 UaaGOmaaGaayjkaiaawMcaaaaakiaaygW7caaISaGaaGjbVlab=v9a MnaaDaaaleaacaaMe8Uaae4BaiaabkgacaqGZbaabaWaaeWaaeaaca WGYbGaaGzaVlaacYcacaaMe8UaaGOmaaGaayjkaiaawMcaaaaaaOGa ayjkaiaawMcaaiaaiYcaaaa@6E88@ with R MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFj0xb9qqFj0db9qqvqFr0dXdHiVc=b YP0xb9peeu0xXdcrpe0db9Wqpepec9ar=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGsbaaaa@3243@ draws from a third reduced Gibbs sampling.

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