A. 'hearn and B. , A restricted maximum likelihood estimator for truncated height samples, Economics & Human Biology, vol.2, issue.1, p.519, 2004.

C. Ancey, C. Gervasoni, and M. Meunier, Computing extreme avalanches, Cold Regions Science and Technology, vol.39, issue.2, p.161180, 2004.
DOI : 10.1016/j.coldregions.2004.04.004

S. Banerjee, A. Gelfand, and B. Carlin, Hierarchical modeling and analysis for spatial data, 2003.

M. Barbolini and C. Keylock, A new method for avalanche hazard mapping using a combination of statistical and deterministic models, Natural Hazards and Earth System Science, vol.2, issue.3/4, p.239245, 2002.
URL : https://hal.archives-ouvertes.fr/hal-00298991

M. Barbolini, U. Gruber, C. Keylock, M. Naaim, and F. Savi, Application of statistical and hydraulic-continuum dense-snow avalanche models to ve real European sites, Cold Regions Science and Technology, vol.31, issue.2, p.133149, 2000.

P. Bartelt and V. Stockli, The inuence of tree and branch fracture, overturning and debris entrainment on snow avalanche ow, Annals of Glaciology, vol.32, issue.1, p.209216, 2001.

M. Beniston, H. Diaz, and R. Bradley, Climatic change at high elevation sites: an overview, Climatic Change, vol.36, issue.3-4, p.233251, 1997.
DOI : 10.1007/978-94-015-8905-5_1

N. Best, S. Richardson, and A. Thomson, A comparison of Bayesian spatial models for disease mapping, Statistical methods in medical research, vol.14, issue.1, p.3559, 2005.

J. Blanchet and A. Davison, Spatial modeling of extreme snow depth, The Annals of Applied Statistics, vol.5, issue.3, p.16991725, 2011.
DOI : 10.1214/11-aoas464
URL : https://doi.org/10.1214/11-aoas464

A. Lavigne,

S. Brooks and A. Gelman, Some Issues for Monitoring Convergence of Iterative Simulations, Proceedings of the Section on Statistical Computing, 1998.

B. Carlin and T. Louis, Bayes and empirical Bayes methods for data analysis, Statistics and Computing, vol.7, issue.2, p.153154, 1997.

N. Chopin, Fast simulation of truncated Gaussian distributions, Statistics and Computing, vol.21, issue.2, p.275288, 2011.
DOI : 10.1007/s11222-009-9168-1
URL : http://arxiv.org/pdf/1201.6140.pdf

E. W. Cope, Penalized likelihood estimators for truncated data, Journal of Statistical Planning and Inference, vol.141, issue.1, p.345358, 2011.
DOI : 10.1016/j.jspi.2010.06.010

C. Corona, J. Guiot, J. Edouard, F. Chalié, U. Büntgen et al., Millennium-long summer temperature variations in the European Alps as reconstructed from tree rings, Climate of the Past, vol.6, issue.3, p.379400, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00801942

N. Cressie and C. Wikle, Statistics for spatio-temporal data, 2011.

P. Diggle and P. J. Ribeiro, Model-based geostatistics, 2007.

Y. Durand, M. Laternser, G. Giraud, P. Etchevers, B. Lesare et al., Reanalysis of 44 Yr of Climate in the French Alps, Methodology, Model Validation, Climatology, and Trends for Air Temperature and Precipitation, vol.48, p.429, 1958.

N. Eckert, E. Parent, L. Belanger, and S. Garcia, Hierarchical Bayesian modelling for spatial analysis of the number of avalanche occurrences at the scale of the township, Cold Regions Science and Technology, vol.50, issue.1, p.97112, 2007.

N. Eckert, E. Parent, M. Naaim, and D. Richard, Bayesian stochastic modelling for avalanche predetermination: from a general system framework to return period computations, Stochastic Environmental Research and Risk Assessment, vol.22, issue.2, p.185206, 2008.
URL : https://hal.archives-ouvertes.fr/hal-01197576

N. Eckert, E. Parent, T. Faug, and M. Naaim, Bayesian optimal design of an avalanche dam using a multivariate numerical avalanche model, Stochastic Environmental Research and Risk Assessment, vol.23, issue.8, p.11231141, 2009.
URL : https://hal.archives-ouvertes.fr/hal-01197549

N. Eckert, E. Parent, R. Kies, and H. Baya, A spatio-temporal modelling framework for assessing the uctuations of avalanche occurrence resulting from climate change: application to 60 years of data in the northern French Alps, Climatic Change, vol.101, issue.3-4, p.515553, 2010.

N. Eckert, C. Keylock, H. Castebrunet, A. Lavigne, and M. Naaim, Temporal trends in avalanche activity in the French Alps and subregions: from occurrences and runout altitudes to unsteady return periods, Journal of Glaciology, vol.59, issue.213, p.93114, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01019099

M. Fuentes, Testing for separability of spatialtemporal covariance functions, Journal of statistical planning and inference, vol.136, issue.2, p.447466, 2006.

F. Garcia-papani, M. Uribe-opazo, V. Leiva, and R. Aykroyd, Birnbaum Saunders spatial modelling and diagnostics applied to agricultural engineering data, Stochastic Environmental Research and Risk Assessment, vol.120, 2016.

J. Gaume, N. Eckert, G. Chambon, M. Naaim, and L. Bel, Mapping extreme snowfalls in the French Alps using max-stable processes, Water Resources Research, vol.49, issue.7, p.10791098, 2013.
URL : https://hal.archives-ouvertes.fr/hal-01001316

A. Gelman, Analysis of variancewhy it is more important than ever, The Annals of Statistics, vol.33, issue.1, p.153, 2005.

A. Gelman, J. Carlin, H. Stern, and D. Rubin, Bayesian Data Analysis, 2004.

W. Gilks and P. Wild, Adaptive rejection sampling for Gibbs sampling, Applied Statistics, vol.41, issue.2, p.337348, 1992.

W. Gilks, N. Best, and K. Tan, Adaptive rejection Metropolis sampling within Gibbs sampling, Applied Statistics pp, p.455472, 1995.

W. Gilks, S. Richardson, and D. Spiegelhalter, Markov Chain Monte Carlo in Practice, 1995.

W. Griths, A Gibbs' sampler for the parameters of a truncated multivariate normal distribution, Contemporary issues in economics and econometrics: theory and application, p.7591, 2004.

B. Jamieson, S. Margreth, and A. Jones, Application and limitations of dynamic models for snow avalanche hazard mapping, Proceedings of the ISSW, p.730739, 2008.

C. Keylock, An alternative form for the statistical distribution of extreme avalanche runout distances, Cold regions science and technology, vol.42, issue.3, p.185, 2005.

A. Lavigne, L. Bel, E. Parent, and N. Eckert, A model for spatio-temporal clustering using multinomial probit regression: application to avalanche counts, Environmetrics, vol.23, p.522534, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01004120

A. Lavigne, N. Eckert, L. Bel, and E. Parent, Adding expert contributions to the spatiotemporal modelling of avalanche activity under dierent climatic inuences, Journal of the Royal Statistical Society: Series C (Applied Statistics, 2015.

V. Leiva, M. Ferreira, M. Gomes, and C. Lillo, Extreme value Birnbaum Saunders regression models applied to environmental data, Stochastic Environmental Research and Risk Assessment, vol.30, p.114, 2015.

K. Lied and S. Bakkehøi, Empirical calculations of snow-avalanche run-out distance based on topographic parameters, Journal of Glaciology, vol.26, p.165177, 1980.

B. Matérn, Meddelanden från Statens Skogsforskningsinstitut, vol.49, 1960.

D. Mcclung, Extreme avalanche runout: a comparison of empirical models, Canadian Geotechnical Journal, vol.38, issue.6, p.12541265, 2001.

D. Mcclung and K. Lied, Statistical and geometrical denition of snow avalanche runout, Cold Regions Science and Technology, vol.13, issue.2, p.107119, 1987.

M. Meunier and C. Ancey, Towards a conceptual approach to predetermining long-return-period avalanche run-out distances, Journal of Glaciology, vol.50, issue.169, p.268278, 2004.

F. Munoz, M. G. Pennino, D. Conesa, A. López-quílez, and J. M. Bellido, Estimation and prediction of the spatial occurrence of sh species using Bayesian latent Gaussian models, 2013.

A. Lavigne, , vol.27, p.11711180

M. Naaim, T. Faug, F. Naaim, and N. Eckert, Return period calculation and passive structure design at the Taconnaz avalanche path, Annals of Glaciology, vol.51, issue.54, p.8997, 2010.

M. Naaim, Y. Durand, N. Eckert, and G. Chambon, Dense avalanche friction coecients : inuence of physical properties of snow, Journal of Glaciology, vol.59, p.771782, 2013.

F. Pérez-cruz, Kullback-Leibler divergence estimation of continuous distributions, Information Theory, 2008.

A. Rabatel, A. Letréguilly, J. Dedieu, and N. Eckert, Changes in glacier equilibrium-line altitude (ELA) in the western Alps over the 1984-2010 period: evaluation by remote sensing and modeling of the morpho-topographic and climate controls, The Cryosphere, vol.7, p.14551471, 2013.

G. Rodriguez-yam, R. Davis, and L. Scharf, Ecient Gibbs sampling of truncated multivariate normal with application to constrained linear regression, 2004.

S. Sigurôsson, K. Jónasson, and P. Arnalds, Transferring avalanches between paths, Hestnes E (ed) Proceedings of the anniversary conference 25 years of snow avalanche research, p.259263, 1216.

M. Smith and D. Mcclung, Characteristics and prediction of high-frequency avalanche runout, Arctic and Alpine Research, vol.29, issue.3, p.352357, 1997.

P. Speckman and D. Sun, Bayesian nonparametric regression and autoregression priors, 2001.

M. L. Stein, Prediction and inference for truncated spatial data, Journal of Computational and Graphical Statistics, vol.1, issue.1, p.91110, 1992.

E. Vanem, A. Huseby, and B. Natvig, A Bayesian hierarchical spatio-temporal model for signicant wave height in the North Atlantic. Stochastic environmental research and risk assessment, vol.26, p.609632, 2012.

G. Wahba, Improper priors, spline smoothing and the problem of guarding against model errors in regression, Journal of the Royal Statistical Society Series B (Methodological), vol.40, issue.3, p.364372, 1978.

M. Zalina, M. Desa, V. Nguyen, and A. Kassim, Selecting a probability distribution for extreme rainfall series in Malaysia, Water Science & Technology, vol.45, issue.2, p.6368, 2002.
DOI : 10.2166/wst.2002.0028

X. Zhou, R. Giacometti, F. J. Fabozzi, and A. H. Tucker, Bayesian estimation of truncated data with applications to operational risk measurement, Quantitative Finance, vol.14, issue.5, p.862888, 2014.
DOI : 10.1080/14697688.2012.752103

, {1, · · · nc ? 1} Compute the k-dimension vectors u j = (u j 1

, Find the interval [a j , b j ] in which w j is drawn to satisfy the k constraints

, Sample w j in the normal truncated distribution N ((Lm 0 ) j , 1) with support

, Here, we use the method and the code proposed by Chopin, 2011.