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A tutorial on bridge sampling. (English) Zbl 1402.62042

Summary: The marginal likelihood plays an important role in many areas of Bayesian statistics such as parameter estimation, model comparison, and model averaging. In most applications, however, the marginal likelihood is not analytically tractable and must be approximated using numerical methods. Here we provide a tutorial on bridge sampling [C.H. Bennett, “Efficient estimation of free energy differences from Monte Carlo data”, J. Comput. Phys. 22, 245–268 (1976; doi:10.1016/0021-9991(76)90078-4); X.-L. Meng and W. H. Wong, Stat. Sin. 6, No. 4, 831–860 (1996; Zbl 0857.62017)]), a reliable and relatively straightforward sampling method that allows researchers to obtain the marginal likelihood for models of varying complexity. First, we introduce bridge sampling and three related sampling methods using the beta-binomial model as a running example. We then apply bridge sampling to estimate the marginal likelihood for the Expectancy Valence (EV) model – a popular model for reinforcement learning. Our results indicate that bridge sampling provides accurate estimates for both a single participant and a hierarchical version of the EV model. We conclude that bridge sampling is an attractive method for mathematical psychologists who typically aim to approximate the marginal likelihood for a limited set of possibly high-dimensional models.

MSC:

62F15 Bayesian inference
62P15 Applications of statistics to psychology

Citations:

Zbl 0857.62017
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[1] Ahn, W.-Y.; Busemeyer, J. R.; Wagenmakers, E.-J.; Stout, J. C., Comparison of decision learning models using the generalization criterion method, Cognitive Science, 32, 1376-1402, (2008)
[2] Ahn, W.-Y.; Krawitz, A.; Kim, W.; Busemeyer, J. R.; Brown, J. W., A model-based fMRI analysis with hierarchical Bayesian parameter estimation, Journal of Neuroscience Psychology and Economics, 4, 95-110, (2011)
[3] Ando, T., Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models, Biometrika, 94, 443-458, (2007) · Zbl 1132.62005
[4] Andrews, M.; Baguley, T., Prior approval: the growth of Bayesian methods in psychology, The British Journal of Mathematical and Statistical Psychology, 66, 1-7, (2013)
[5] Bark, R.; Dieckmann, S.; Bogerts, B.; Northoff, G., Deficit in decision making in catatonic schizophrenia: an exploratory study, Psychiatry Research, 134, 131-141, (2005)
[6] Bayarri, M.; Benjamin, D. J.; Berger, J. O.; Sellke, T. M., Rejection odds and rejection ratios: A proposal for statistical practice in testing hypotheses, Journal of Mathematical Psychology, 72, 90-103, (2016) · Zbl 1357.62018
[7] Bechara, A.; Damasio, A. R.; Damasio, H.; Anderson, S. W., Insensitivity to future consequences following damage to human prefrontal cortex, Cognition, 50, 7-15, (1994)
[8] Bechara, A.; Damasio, H.; Damasio, A. R.; Lee, G. P., Different contributions of the human amygdala and ventromedial prefrontal cortex to decision-making, Journal of Neuroscience, 19, 5473-5481, (1999)
[9] Bechara, A.; Damasio, H.; Tranel, D.; Anderson, S. W., Dissociation of working memory from decision making within the human prefrontal cortex, Journal of Neuroscience, 18, 428-437, (1998)
[10] Bechara, A.; Damasio, H.; Tranel, D.; Damasio, A. R., Deciding advantageously before knowing the advantageous strategy, Science, 275, 1293-1295, (1997)
[11] Bechara, A.; Tranel, D.; Damasio, H., Characterization of the decision-making deficit of patients with ventromedial prefrontal cortex lesions, Brain, 123, 2189-2202, (2000)
[12] Bennett, C. H., Efficient estimation of free energy differences from Monte Carlo data, Journal of Computational Physics, 22, 245-268, (1976)
[13] Berger, J. O., Bayes factors, (Kotz, S.; Balakrishnan, N.; Read, C.; Vidakovic, B.; Johnson, N. L., Encyclopeida of statistical sciences, Vol. 1, (2006), Wiley Hoboken, NJ), 378-386
[14] Berger, J. O.; Molina, G., Posterior model probabilities via path-based pairwise priors, Statistica Neerlandica, 59, 3-15, (2005) · Zbl 1069.62021
[15] Blair, R. J.R.; Colledge, E.; Mitchell, D. G.V., Somatic markers and response reversal: Is there orbitofrontal cortex dysfunction in boys with psychopathic tendencies?, Journal of Abnormal Child Psychology, 29, 499-511, (2001)
[16] Brown, C. E., Coefficient of variation, (Applied multivariate statistics in geohydrology and related sciences, (1998), Springer), 155-157
[17] Busemeyer, J. R.; Stout, J. C., A contribution of cognitive decision models to clinical assessment: decomposing performance on the bechara gambling task, Psychological Assessment, 14, 253-262, (2002)
[18] Carlin, B. P.; Chib, S., Bayesian model choice via Markov chain Monte Carlo methods, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 473-484, (1995) · Zbl 0827.62027
[19] Cavedini, P.; Riboldi, G.; D’Annucci, A.; Belotti, P.; Cisima, M.; Bellodi, L., Decision-making heterogeneity in obsessive-compulsive disorder: ventromedial prefrontal cortex function predicts different treatment outcomes, Neuropsychologia, 40, 205-211, (2002)
[20] Cavedini, P.; Riboldi, G.; Keller, R.; D’Annucci, A.; Bellodi, L., Frontal lobe dysfunction in pathological gambling patients, Biological Psychiatry, 51, 334-341, (2002)
[21] Chen, M.-H.; Shao, Q.-M.; Ibrahim, J. G., Monte Carlo methods in Bayesian computation, (2012), Springer Science & Business Media
[22] Chib, S.; Jeliazkov, I., Marginal likelihood from the metropolis-Hastings output, Journal of the American Statistical Association, 96, 270-281, (2001) · Zbl 1015.62020
[23] Dai, J.; Kerestes, R.; Upton, D. J.; Busemeyer, J. R.; Stout, J. C., An improved cognitive model of the iowa and soochow gambling tasks with regard to model Fitting performance and tests of parameter consistency, Frontiers in Psychology, 6, 229, (2015)
[24] DiCiccio, T. J.; Kass, R. E.; Raftery, A.; Wasserman, L., Computing Bayes factors by combining simulation and asymptotic approximations, Journal of the American Statistical Association, 92, 903-915, (1997) · Zbl 1050.62520
[25] Dickey, J. M., The weighted likelihood ratio, linear hypotheses on normal location parameters, The Annals of Mathematical Statistics, 204-223, (1971) · Zbl 0274.62020
[26] Dickey, J. M.; Lientz, B., The weighted likelihood ratio, sharp hypotheses about chances, the order of a Markov chain, The Annals of Mathematical Statistics, 41, 214-226, (1970) · Zbl 0188.50102
[27] Didelot, X.; Everitt, R. G.; Johansen, A. M.; Lawson, D. J., Likelihood-free estimation of model evidence, Bayesian Analysis, 6, 1, 49-76, (2011) · Zbl 1330.62118
[28] Etz, A., & Wagenmakers, E.-J. (in press). J. B. S. Haldane’s contribution to the Bayes factor hypothesis test. Statist. Sci. · Zbl 1381.62009
[29] Frühwirth-Schnatter, S., Estimating marginal likelihoods for mixture and Markov switching models using bridge sampling techniques, The Econometrics Journal, 7, 143-167, (2004) · Zbl 1053.62087
[30] Gamerman, D.; Lopes, H. F., Markov chain Monte Carlo: Stochastic simulation for Bayesian inference, 237-286, (2006), CRC Press
[31] Gelfand, A. E.; Dey, D. K., Bayesian model choice: asymptotics and exact calculations, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 501-514, (1994) · Zbl 0800.62170
[32] Gelman, A.; Meng, X.-L., Simulating normalizing constants: from importance sampling to bridge sampling to path sampling, Statistical Science, 163-185, (1998) · Zbl 0966.65004
[33] Green, P. J., Reversible jump Markov chain Monte Carlo computation and Bayesian model determination, Biometrika, 82, 711-732, (1995) · Zbl 0861.62023
[34] Gronau, Q. F., Singmann, H., & Wagenmakers, E.-J. (2017). Bridgesampling: Bridge sampling for marginal likelihoods and Bayes factors. Retrieved from https://github.com/quentingronau/bridgesampling (R package version 0.2-2).
[35] Hammersley, J. M.; Handscomb, D. C., Monte Carlo methods, (1964), Methuen London
[36] Hoeting, J. A.; Madigan, D.; Raftery, A. E.; Volinsky, C. T., Bayesian model averaging: A tutorial, Statistical Science, 382-401, (1999) · Zbl 1059.62525
[37] Ionides, E. L., Truncated importance sampling, Journal of Computational and Graphical Statistics, 17, 295-311, (2008)
[38] Jeffreys, H., Theory of probability, (1961), Oxford University Press Oxford, England · Zbl 0116.34904
[39] Kass, R. E.; Raftery, A. E., Bayes factors, Journal of the American Statistical Association, 90, 773-795, (1995) · Zbl 0846.62028
[40] Lee, M. D., Three case studies in the Bayesian analysis of cognitive models, Psychonomic Bulletin & Review, 15, 1-15, (2008)
[41] Lewis, S. M.; Raftery, A. E., Estimating Bayes factors via posterior simulation with the Laplace-metropolis estimator, Journal of the American Statistical Association, 92, 648-655, (1997) · Zbl 0889.62018
[42] Lodewyckx, T.; Kim, W.; Lee, M. D.; Tuerlinckx, F.; Kuppens, P.; Wagenmakers, E.-J., A tutorial on Bayes factor estimation with the product space method, Journal of Mathematical Psychology, 55, 331-347, (2011) · Zbl 1225.62037
[43] Luce, R., Individual choice behavior, (1959), Wiley New York · Zbl 0093.31708
[44] Lunn, D. J.; Best, N.; Whittaker, J. C., Generic reversible jump MCMC using graphical models, Statistics and Computing, 19, 395-408, (2009)
[45] Lunn, D. J.; Thomas, A.; Best, N.; Spiegelhalter, D., Winbugs-a Bayesian modelling framework: concepts, structure, and extensibility, Statistics and Computing, 10, 325-337, (2000)
[46] Ly, A.; Verhagen, A.; Wagenmakers, E.-J., An evaluation of alternative methods for testing hypotheses, from the perspective of harold Jeffreys, Journal of Mathematical Psychology, 72, 43-55, (2016) · Zbl 1357.62118
[47] Ly, A.; Verhagen, J.; Wagenmakers, E.-J., Harold jeffreys’s default Bayes factor hypothesis tests: explanation, extension, and application in psychology, Journal of Mathematical Psychology, 72, 19-32, (2016) · Zbl 1357.62117
[48] Martino, D. J.; Bucay, D.; Butman, J. T.; Allegri, R. F., Neuropsychological frontal impairments and negative symptoms in schizophrenia, Psychiatry Research, 152, 121-128, (2007)
[49] Matzke, D.; Dolan, C. V.; Batchelder, W. H.; Wagenmakers, E.-J., Bayesian estimation of multinomial processing tree models with heterogeneity in participants and items, Psychometrika, 80, 205-235, (2015) · Zbl 1314.62275
[50] Matzke, D.; Wagenmakers, E.-J., Psychological interpretation of the ex-Gaussian and shifted Wald parameters: A diffusion model analysis, Psychonomic Bulletin & Review, 16, 798-817, (2009)
[51] Meng, X.-L.; Schilling, S., Warp bridge sampling, Journal of Computational and Graphical Statistics, 11, 552-586, (2002)
[52] Meng, X.-L.; Wong, W. H., Simulating ratios of normalizing constants via a simple identity: a theoretical exploration, Statistica Sinica, 831-860, (1996) · Zbl 0857.62017
[53] Mira, A.; Nicholls, G., Bridge estimation of the probability density at a point, Statistica Sinica, 14, 603-612, (2004) · Zbl 1045.62027
[54] Mulder, J.; Wagenmakers, E.-J., Editors’ introduction to the special issue Bayes factors for testing hypotheses in psychological research: practical relevance and new developments, Journal of Mathematical Psychology, 72, 1-5, (2016) · Zbl 1349.00253
[55] Myung, I. J.; Forster, M. R.; Browne, M. W., Guest editors’ introduction: special issue on model selection, Journal of Mathematical Psychology, 44, 1-2, (2000)
[56] Navarro, D. J.; Griffiths, T. L.; Steyvers, M.; Lee, M. D., Modeling individual differences using Dirichlet processes, Journal of Mathematical Psychology, 50, 101-122, (2006) · Zbl 1138.91594
[57] Neal, R. M., Annealed importance sampling, Statistics and Computing, 11, 125-139, (2001)
[58] Newton, M. A.; Raftery, A. E., Approximate Bayesian inference with the weighted likelihood bootstrap, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 3-48, (1994) · Zbl 0788.62026
[59] Ntzoufras, I., Bayesian model and variable evaluation, (Bayesian modeling using WinBUGS, (2009), John Wiley & Sons), 389-433
[60] Overstall, A. M.; Forster, J. J., Default Bayesian model determination methods for generalised linear mixed models, Computational Statistics & Data Analysis, 54, 3269-3288, (2010) · Zbl 1284.62462
[61] Owen, A.; Zhou, Y., Safe and effective importance sampling, Journal of the American Statistical Association, 95, 135-143, (2000) · Zbl 0998.65003
[62] Pajor, A., Estimating the marginal likelihood using the arithmetic mean identity, Bayesian Analysis, 1-27, (2016)
[63] Pitt, M. A.; Myung, I. J.; Zhang, S., Toward a method of selecting among computational models of cognition, Psychological Review, 109, 3, 472-491, (2002)
[64] Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. In Proceedings of the 3rd International workshop on distributed statistical computing, Vol. 124, (pp. 1-8).
[65] Plummer, M.; Best, N.; Cowles, K.; Vines, K., CODA: convergence diagnosis and output analysis for MCMC, R News, 6, 7-11, (2006)
[66] Poirier, D. J., The growth of Bayesian methods in statistics and economics Since 1970, Bayesian Analysis, 1, 969-979, (2006) · Zbl 1331.62493
[67] Raftery, A. E.; Banfield, J. D., Stopping the Gibbs sampler, the use of morphology, and other issues in spatial statistics (Bayesian image restoration, with two applications in spatial statistics)-(discussion), Annals of the Institute of Statistical Mathematics, 43, 32-43, (1991)
[68] Rescorla, R. A., & Wagner, A. R. (1972). A theory of Pavlovian conditioning: Variations in the effectiveness of reinforcement and nonreinforcement. In A.H. Black,& W.F. Prokasy, (Eds.), Classical conditioning II: Current research and theory, New York, (pp. 64-99).
[69] Robert, C., The expected demise of the Bayes factor, Journal of Mathematical Psychology, 72, 33-37, (2016) · Zbl 1357.62127
[70] Rouder, J. N.; Lu, J., An introduction to Bayesian hierarchical models with an application in the theory of signal detection, Psychonomic Bulletin & Review, 12, 573-604, (2005)
[71] Rouder, J. N.; Lu, J.; Morey, R. D.; Sun, D.; Speckman, P. L., A hierarchical process-dissociation model, Journal of Experimental Psychology: General, 137, 370-389, (2008)
[72] Rouder, J. N.; Lu, J.; Speckman, P.; Sun, D.; Jiang, Y., A hierarchical model for estimating response time distributions, Psychonomic Bulletin & Review, 12, 195-223, (2005)
[73] Rouder, J. N.; Lu, J.; Sun, D.; Speckman, P.; Morey, R.; Naveh-Benjamin, M., Signal detection models with random participant and item effects, Psychometrika, 72, 621-642, (2007) · Zbl 1291.62242
[74] Scheibehenne, B.; Pachur, T., Using Bayesian hierarchical parameter estimation to assess the generalizability of cognitive models of choice, Psychonomic Bulletin & Review, 22, 391-407, (2015)
[75] Schwarz, G., Estimating the dimension of a model, The Annals of Statistics, 6, 461-464, (1978) · Zbl 0379.62005
[76] Shiffrin, R. M.; Lee, M. D.; Kim, W.; Wagenmakers, E.-J., A survey of model evaluation approaches with a tutorial on hierarchical Bayesian methods, Cognitive Science, 32, 1248-1284, (2008)
[77] Sinharay, S.; Stern, H. S., An empirical comparison of methods for computing Bayes factors in generalized linear mixed models, Journal of Computational and Graphical Statistics, 14, 415-435, (2005)
[78] Spiegelhalter, D. J.; Best, N. G.; Carlin, B. P.; van der Linde, A., Bayesian measures of model complexity and fit, Journal of the Royal Statistical Society. Series B., 64, 583-639, (2002) · Zbl 1067.62010
[79] Stan Development Team. (2016). RStan: the R interface to Stan. Retrieved form http://mc-stan.org/ (R package version 2.14.1).
[80] Steingroever, H., Pachur, T., Šmíra, M., & Lee, M. D. (submitted for publication). Bayesian techniques for analyzing group differences in the Iowa gambling task: A case study of intuitive and deliberate decision makers.
[81] Steingroever, H.; Wetzels, R.; Horstmann, A.; Neumann, J.; Wagenmakers, E.-J., Performance of healthy participants on the iowa gambling task, Psychological Assessment, 25, 180-193, (2013)
[82] Steingroever, H.; Wetzels, R.; Wagenmakers, E.-J., A comparison of reinforcement-learning models for the iowa gambling task using parameter space partitioning, The Journal of Problem Solving, 5, (2013), Article 2
[83] Steingroever, H.; Wetzels, R.; Wagenmakers, E.-J., Validating the PVL-delta model for the iowa gambling task, Frontiers in Psychology, 4, 898, (2013)
[84] Steingroever, H.; Wetzels, R.; Wagenmakers, E.-J., Absolute performance of reinforcement-learning models for the iowa gambling task, Decision, 1, 161-183, (2014)
[85] Steingroever, H.; Wetzels, R.; Wagenmakers, E.-J., Bayes factors for reinforcement-learning models of the iowa gambling task, Decision, 3, 115-131, (2016)
[86] Stone, C. J.; Hansen, M. H.; Kooperberg, C.; Truong, Y. K., Polynomial splines and their tensor products in extended linear modeling: 1994 Wald memorial lecture, The Annals of Statistics, 25, 1371-1470, (1997) · Zbl 0924.62036
[87] Vandekerckhove, J.; Matzke, D.; Wagenmakers, E.-J., Model comparison and the principle of parsimony, (Busemeyer, J.; Townsend, J.; Wang, Z. J.; Eidels, A., Oxford handbook of computational and mathematical psychology, (2015), Oxford University Press Oxford)
[88] Vanpaemel, W., Prototypes, exemplars and the response scaling parameter: A Bayes factor perspective, Journal of Mathematical Psychology, 72, 183-190, (2016) · Zbl 1359.62507
[89] Verhagen, J.; Levy, R.; Millsap, R. E.; Fox, J.-P., Evaluating evidence for invariant items: A Bayes factor applied to testing measurement invariance in IRT models, Journal of Mathematical Psychology, 72, 171-182, (2015) · Zbl 1359.62508
[90] Wagenmakers, E.-J.; Lodewyckx, T.; Kuriyal, H.; Grasman, R., Bayesian hypothesis testing for psychologists: A tutorial on the savage-Dickey method, Cognitive Psychology, 60, 158-189, (2010)
[91] Wagenmakers, E.-J.; Waldorp, L., Editors’ introduction, Journal of Mathematical Psychology, 50, 99-100, (2006)
[92] Wang, L., & Meng, X.-L. (2016). Warp bridge sampling: The next generation. arXiv preprint arXiv:1609.07690.
[93] Wetzels, R.; Grasman, R. P.; Wagenmakers, E.-J., An encompassing prior generalization of the savage-Dickey density ratio, Computational Statistics & Data Analysis, 54, 2094-2102, (2010) · Zbl 1284.62135
[94] Wetzels, R.; Tutschkow, D.; Dolan, C.; van der Sluis, S.; Dutilh, G.; Wagenmakers, E.-J., A Bayesian test for the hot hand phenomenon, Journal of Mathematical Psychology, 72, 200-209, (2016) · Zbl 1357.62136
[95] Wetzels, R.; Vandekerckhove, J.; Tuerlinckx, F.; Wagenmakers, E.-J., Bayesian parameter estimation in the expectancy valence model of the iowa gambling task, Journal of Mathematical Psychology, 54, 14-27, (2010) · Zbl 1203.91255
[96] Worthy, D. A.; Maddox, W. T., A comparison model of reinforcement-learning and win-stay-lose-shift decision-making processes: A tribute to W. K. estes, Journal of Mathematical Psychology, 59, 41-49, (2014) · Zbl 1309.91124
[97] Worthy, D. A.; Pang, B.; Byrne, K. A., Decomposing the roles of perseveration and expected value representation in models of the iowa gambling task, Frontiers in Psychology, 4, (2013)
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