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Prior specification of neighbourhood and interaction structure in binary Markov random fields. (English) Zbl 06737694
Summary: We formulate a prior distribution for the energy function of stationary binary Markov random fields (MRFs) defined on a rectangular lattice. In the prior we assign distributions to all parts of the energy function. In particular we define priors for the neighbourhood structure of the MRF, what interactions to include in the model, and for potential values. We define a reversible jump Markov chain Monte Carlo (RJMCMC) procedure to simulate from the corresponding posterior distribution when conditioned to an observed scene. Thereby we are able to learn both the neighbourhood structure and the parametric form of the MRF from the observed scene. We circumvent evaluations of the intractable normalising constant of the MRF when running the RJMCMC algorithm by adopting a previously defined approximate auxiliary variable algorithm. We demonstrate the usefulness of our prior in two simulation examples and one real data example.

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[1] Arnesen, P; Tjelmeland, H, Fully Bayesian binary Markov random field models: prior specification and posterior simulation, Scand. J. Stat., 42, 967-987, (2015) · Zbl 1419.62270
[2] Augustin, NH; Mugglestone, MA; Buckland, ST, An autologistic model for the spatial distribution of wildlife, J. Appl. Ecol., 33, 339-347, (1996)
[3] Austad, H.M.: Approximations of binary Markov random fields, Ph.D. thesis, Norwegian University of Science and Technology. Thesis number 292:2011. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-14922 (2011) · Zbl 0928.60049
[4] Austad, H.M., Tjelmeland, H.: Approximate computations for binary Markov random fields and their use in Bayesian models. Technical report, Submitted (2016) · Zbl 06737711
[5] Besag, J, Spatial interaction and the statistical analysis of lattice systems, J. R. Stat. Soc. Ser. B, 36, 192-236, (1974) · Zbl 0327.60067
[6] Besag, J, Statistical analysis of non-lattice data, The Statistician, 24, 179-195, (1975)
[7] Besag, J, On the statistical analysis of dirty pictures, J. R. Stat. Soc. Ser. B, 48, 259-302, (1986) · Zbl 0609.62150
[8] Buckland, ST; Elston, DA, Empirical models for the spatial distribution of wildlife, J. Appl. Ecol., 30, 478-495, (1993)
[9] Caimo, A; Friel, N, Bayesian inference for exponential random graph models, Soc. Netw., 33, 41-55, (2011)
[10] Celeux, G; Hurn, MA; Robert, CP, Computational and inferential difficulties with mixture posterior distributions, J. Am. Stat. Assoc., 95, 957-970, (2000) · Zbl 0999.62020
[11] Clifford, P; Grimmett, G (ed.); Welsh, DJ (ed.), Markov random fields in statistics, (1990), New York · Zbl 0736.62081
[12] Cowles, MK; Carlin, BP, Markov chain Monte Carlo convergence diagnostics: a comparative review, J. Am. Stat. Assoc., 91, 883-904, (1996) · Zbl 0869.62066
[13] Cressie, N.A.: Statistics for Spatial Data, 2nd edn. Wiley, New York (1993) · Zbl 1347.62005
[14] Cressie, N; Davidson, JL, Image analysis with partially ordered Markov models, Comput. Stat. Data Anal., 29, 1-26, (1998) · Zbl 1042.62611
[15] Descombes, X., Mangin, J., Pechersky, E. and Sigelle, M.: Fine structures preserving Markov model for image processing. In: Proceedings of 9th SCIA 95, Uppsala, pp. 349-356 (1995) · Zbl 0982.91009
[16] Everitt, RG, Bayesian parameter estimation for latent Markov random fields and social networks, J. Comput. Graph. Stat., 21, 940-960, (2012)
[17] Friel, N; Rue, H, Recursive computing and simulation-free inference for general factorizable models, Biometrika, 94, 661-672, (2007) · Zbl 1135.62078
[18] Friel, N; Pettitt, AN; Reeves, R; Wit, E, Bayesian inference in hidden Markov random fields for binary data defined on large lattices, J. Comput. Graph. Stat., 18, 243-261, (2009)
[19] Geyer, CJ; Thompson, EA, Constrained Monte Carlo maximum likelihood for dependent data, J. R. Stat. Soc. Ser. B, 54, 657-699, (1992)
[20] Grabisch, M; Marichal, L-L; Roubens, M, Equivalent representation of set function, Math. Oper. Res., 25, 157-178, (2000) · Zbl 0982.91009
[21] Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading, MA (1988) · Zbl 0836.00001
[22] Hammer, P; Holzman, R, Approximations of pseudo-Boolean functions; application to game theory, Methods Models Oper. Res., 36, 3-21, (1992) · Zbl 0778.41009
[23] Heikkinen, J; Högmander, H, Fully Bayesian approach to image restoration with an application in biogeography, Appl. Stat., 43, 569-582, (1994) · Zbl 0825.62413
[24] Higdon, DM; Bowsher, JE; Johnsen, VE; Turkington, TG; Gilland, DR; Jaszczak, RJ, Fully Bayesian estimation of Gibbs hyperparameters for emission computed tomography data, IEEE Trans. Med. Imaging, 16, 516-526, (1997)
[25] Hurn, M., Husby, O. and Rue, H.: .A tutorial on image analysis. In: Møller J. (ed.), Spatial Statistics and Computational Methods. Lecture Notes in Statistics, vol. 173, pp. 87-141, Springer, New York (2003) · Zbl 1255.62185
[26] Kindermann, R., Snell, J.L.: Markov Random Fields and Their Applications. American Mathematical Society, Providence, Rhode Island (1980) · Zbl 1229.60003
[27] Lauritzen, S.L.: Graphical Models. Clarendon Press, Oxford (1996) · Zbl 0907.62001
[28] Lyne, AM; Girolami, M; Atchade, Y; Strathmann, H; Simpson, D, On russian roulette estimates for Bayesian inference with doubly-intractable likelihoods, Stat. Sci., 30, 443-467, (2015) · Zbl 1426.62092
[29] McGrory, CA; Pettitt, AN; Reeves, R; Griffin, M; Dwyer, M, Variational Bayes and the reduced dependence approximation for the autologistic model on an irregular grid with applications, J. Comput. Graph. Stat., 21, 781-796, (2012)
[30] Møller, J; Pettitt, AN; Reeves, R; Berthelsen, KK, An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants, Biometrika, 93, 451-458, (2006) · Zbl 1158.62020
[31] Murray, I., Ghahramani, Z. and MacKay, D.: MCMC for doubly-intractable distributions. In: Proceedings of the Twenty-Second Conference Annual Conference on Uncertainty in Artificial Intelligence (UAI-06), pp. 359-366, AUAI Press, Arlington, Virginia (2006) · Zbl 1419.62270
[32] Propp, JG; Wilson, DB, Exact sampling with coupled Markov chains and applications to statistical mechanics, Random Struct. Algorithms, 9, 223-252, (1996) · Zbl 0859.60067
[33] Qian, W; Titterington, DM, Multidimensional Markov chain models for image textures, J. R. Stat. Soc. Ser. B, 53, 661-674, (1991) · Zbl 0800.62512
[34] Tjelmeland, H; Austad, HM, Exact and approximate recursive calculations for binary Markov random fields defined on graphs, J. Comput. Graph. Stat., 21, 758-780, (2012)
[35] Tjelmeland, H; Besag, J, Markov random fields with higher order interactions, Scand. J. Stat., 25, 415-433, (1998) · Zbl 0928.60049
[36] Toftaker, H.: Modelling and parameter estimation for discrete random fields and spatial point processes, PhD thesis, Norwegian University of Science and Technology. Thesis number 200:2013. http://ntnu.diva-portal.org/smash/record.jsf?pid=diva2:662513 (2013)
[37] Walker, S, Posterior sampling when the normalizing constant is unknown, Commun. Stat. Simul. Comput., 40, 784-792, (2011) · Zbl 1216.62043
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