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A unified view on Bayesian varying coefficient models. (English) Zbl 1434.62092
Summary: Varying coefficient models are useful in applications where the effect of the covariate might depend on some other covariate such as time or location. Various applications of these models often give rise to case-specific prior distributions for the parameter(s) describing how much the coefficients vary. In this work, we introduce a unified view of varying coefficients models, arguing for a way of specifying these prior distributions that are coherent across various applications, avoid overfitting and have a coherent interpretation. We do this by considering varying coefficients models as a flexible extension of the natural simpler model and capitalising on the recently proposed framework of penalized complexity (PC) priors. We illustrate our approach in two spatial examples where varying coefficient models are relevant.

62H11 Directional data; spatial statistics
62F15 Bayesian inference
Stem; spBayes
Full Text: DOI Euclid
[1] Banerjee, S., Carlin, B., and Gelfand, A. (2015)., Hierarchical Modeling and Analysis for Spatial Data, Second Edition. CRC Press/Chapman & Hall. Monographs on Statistics and Applied Probability. · Zbl 1358.62009
[2] Barndorff-Nielsen, O., and Schou, G. (1973). On the parametrization of autoregressive models by partial autocorrelations., Journal of Multivariate Analysis, 3:408-419. · Zbl 0275.62074
[3] Berger, J. O., and Yang, R. (1994). Noninformative priors and Bayesian testing for the AR(1) model., Econometric Theory, 10:461-482.
[4] Besag, J. (1974). Spatial interaction and the statistical analysis of lattice systems (with discussion)., Journal of the Royal Statistical Society Series B, 36(2):192-225. · Zbl 0327.60067
[5] Besag, J., York, J., and Mollie, A. (1991). Bayesian image restoration, with two applications in spatial statistics., Annals of the Institute of Statistical Mathematics, 43:1-21. · Zbl 0760.62029
[6] Biller, C. and Fahrmeir, L. (2001). Bayesian varying-coefficient models using adaptive regression splines., Statistical Modelling, 1(3):195-211. · Zbl 1104.62023
[7] Bitto, A. and Frühwirth-Schnatter, S. (2018). Achieving Shrinkage in a Time-Varying Parameter Model Framework., arXiv:1611.01310.
[8] Blangiardo, M. and Cameletti, M. (2017)., Spatial and Spatio-temporal Bayesian Models with R-INLA. Wiley. · Zbl 1318.62001
[9] Cai, Z. and Sun, Y. (2003). Local linear estimation for time-dependent coefficients in Cox’s regression models., Scandinavian Journal of Statistics, 30:93-11.
[10] Carvalho C., Polson, N., and Scott, J. (2010). The horseshoe estimator for sparse signals., Biometrika, 97:465-480. · Zbl 1406.62021
[11] Fahrmeir, L., Kneib, T., and Lang, S. (2004). Penalized structured additive regression for space-time data: a Bayesian perspective., Statistica Sinica, 14:715-745. · Zbl 1073.62025
[12] Fan, J. and Zhang, W. (1999). Statistical estimation in varying coefficient models., The Annals of Statistics, 27:1491-1518. · Zbl 0977.62039
[13] Ferguson, C., Bowman, A., Scott, E., and Carvalho, L. (2007). Model comparison for a complex ecological system., Journal of the Royal Statistical Society Series A, 170(3):691-711.
[14] Finazzi, F., Scott, M., and Fasso, A. (2013). A model-based framework for air quality indices and population risk evaluation, with an application to the analysis of Scottish air quality data., Journal of the Royal Statistical Society Series C, 62(2):287-308.
[15] Finley, A. (2011). Comparing spatially-varying coefficients models for analysis of ecological data with non-stationary and anisotropic residual dependence., Methods in Ecology and Evolution, 2:143-154.
[16] Franco-Villoria, M., Ventrucci, M., and Rue, H. (2019). Supplement A to “A unified view on Bayesian varying coefficient models”. DOI:, 10.1214/08-AOS99GSUPPA.
[17] Franco-Villoria, M., Ventrucci, M., and Rue, H. (2019). Supplement B to “A unified view on Bayesian varying coefficient models”. DOI:, 10.1214/08-AOS99GSUPPB.
[18] Frühwirth-Schnatter, S. and Wagner, H. (2010). Stochastic model specification search for Gaussian and partial non-Gaussian state space models., Journal of Econometrics, 154(1):85-100. · Zbl 1431.62373
[19] Frühwirth-Schnatter, S. and Wagner, H. (2011). Bayesian variable selection for random intercept modeling of Gaussian and non-Gaussian data. In, J. M. Bernardo, M. J. Bayarri, J. O. Berger, A. P. Dawid, D. Heckerman, A. F. M. Smith and M. West (Eds.), pages 165-200. Bayesian Statistics 9, Oxford.
[20] Fuglstad, G. A., Simpson, D., Lindgren, F., and Rue, H. (2018). Constructing priors that penalize the complexity of Gaussian random fields., Journal of the American Statistical Association. · Zbl 07095889
[21] Gamerman, D., Moreira, A. R., and Rue, H. (2003). Space-varying regression models: specifications and simulation., Computational Statistics & Data Analysis, 42(3):513-533. · Zbl 1429.62428
[22] Gelfand, A., Kim, J., Sirmans, C., and Banerjee, S. (2003). Spatial modeling with spatially varying coefficient processes., Journal of the American Statistical Association, 98(462):387-396. · Zbl 1041.62041
[23] Gelman, A. (2006). Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper)., Bayesian Analysis, 3:515-534. · Zbl 1331.62139
[24] Hastie, T. and Tibshirani, R. (1993). Varying-coefficient models., Journal of the Royal Statistical Society Series B, 55(4):757-796. · Zbl 0796.62060
[25] Hoover, D., Rice, J., and Wu, C. (1998). Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data., Biometrika, 85(4):809-822. · Zbl 0921.62045
[26] Kimeldorf, G. and Wahba, G. (1970). A correspondence between Bayesian estimation on stochastic processes and smoothing by splines., The Annals of Mathematical Statistics, 41(2):495-502. · Zbl 0193.45201
[27] Kowal, D. R., Matteson, D. S., and Ruppert, D. (2018). Dynamic Shrinkage Processes., arXiv:1707.00763. · Zbl 1428.62397
[28] Kullback, S. and Leibler, R. A. (1951). On information and sufficiency., The Annals of Mathematical Statistics, 22:79-86. · Zbl 0042.38403
[29] Laird, N. and Ware, J. (1982). Random-effects models for longitudinal data., Biometrics, 38(4):963-974. · Zbl 0512.62107
[30] Lindgren, F. and Rue, H. (2008). On the second-order random walk model for irregular locations., Scandinavian Journal of Statistics, 35(4):691-700. · Zbl 1199.60276
[31] Martins, T. G., Simpson, D., Lindgren, F., and Rue, H. (2013). Bayesian computing with INLA: New features., Computational Statistics & Data Analysis, 67(0):68-83. · Zbl 06970873
[32] Marx, B. (2010). P-spline varying coefficient models for complex data. In, T. Kneib and G. Tutz (Eds.). Statistical Modelling and Regression Structures, Physica-Verlag HD.
[33] Mu, J., Wang, G., and Wang, L. (2018). Estimation and inference in spatially varying coefficient models., Environmetrics, 29.
[34] Nelder, J. and Wedderburn, R. (1972). Generalized linear models., Journal of the Royal Statistical Society Series A, 135:370-384.
[35] Riebler, A., Sørbye, S. H., Simpson, D., and Rue, H. (2016). An intuitive Bayesian spatial model for disease mapping that accounts for scaling., Statistical Methods in Medical Research, 25(4):1145-1165. PMID: 27566770.
[36] Rue, H. and Held, L. (2005)., Gaussian Markov Random Fields. Chapman and Hall/CRC. · Zbl 1093.60003
[37] Scheipl, F. and Kneib, T. (2009). Locally adaptive Bayesian P-splines with a Normal-Exponential-Gamma prior., Computational Statistics & Data Analysis, 53:3533-3552. · Zbl 1453.62191
[38] Simpson, D., Rue, H., Riebler, A., Martins, T. G., and Sørbye, S. H. (2017). Penalising model component complexity: A principled, practical approach to constructing priors., Statistical Science, 32(1):1-28. · Zbl 1442.62060
[39] Sørbye, S. and Rue, H. (2014). Scaling intrinsic Gaussian Markov random field priors in spatial modelling., Spatial Statistics, 8:39-51.
[40] Sørbye, S. and Rue, H. (2017). Penalised complexity priors for stationary autoregressive processes., Journal of Time Series Analysis, 38:923-935. · Zbl 1416.62529
[41] Staubach, C., Schmid, V., Knorr-Held, L., and Ziller, M. (2002). A Bayesian model for spatial wildlife disease prevalence data., Preventive Veterinary Medicine, 56:75-87.
[42] Stein, M. (1999)., Interpolation of Spatial Data: Some Theory for Kriging. Springer-Verlag, New York. · Zbl 0924.62100
[43] Tian, L., Zucker, D., and Wei, L. (2005). On the Cox model with time-varying regression coefficients., Journal of the American Statistical Association, 100(469):172-183. · Zbl 1117.62435
[44] Waller, L., Zhu, L., Gotway, C., Gorman, D., and Gruenewald, P. (2007). Quantifying geographic variations in associations between alcohol distribution and violence: a comparison of geographically weighted regression and spatially varying coefficient models., Stochastic Environmental Research and Risk Assessment, 21:573-588.
[45] Warnes, J. and Ripley, B. (1987). Problems with likelihood estimation of covariance functions of spatial Gaussian processes., Biometrika, 74(3):640-642. · Zbl 0628.62087
[46] Yue, Y. R., Simpson, D., Lindgren, F. and Rue, H. (2017). Bayesian adaptive smoothing splines using stochastic differential equations., Bayesian Analysis, 2:397-424. · Zbl 1327.62234
[47] Zhang, H. (2004). Inconsistent estimation and asymptotically equal interpolations in model-based geostatistics., Journal of the American Statistical Association, 99(465):250-261. · Zbl 1089.62538
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