A tutorial on approximate Bayesian computation. (English) Zbl 1245.91084
Summary: This tutorial explains the foundation of approximate Bayesian computation (ABC), an approach to Bayesian inference that does not require the specification of a likelihood function, and hence that can be used to estimate posterior distributions of parameters for simulation-based models. We discuss briefly the philosophy of Bayesian inference and then present several algorithms for ABC. We then apply these algorithms in a number of examples. For most of these examples, the posterior distributions are known, and so we can compare the estimated posteriors derived from ABC to the true posteriors and verify that the algorithms recover the true posteriors accurately. We also consider a popular simulation-based model of recognition memory (REM) for which the true posteriors are unknown. We conclude with a number of recommendations for applying ABC methods to solve real-world problems.
References:
| [1] | Abelson, R. P., Statistics as principled argument (1995), Lawrence Erlbaum Associates: Lawrence Erlbaum Associates Hillsdale, NJ |
| [2] | Barthelme, S., & Chopin, N. (2011). ABC-EP: expectation propagation for likelihood-free Bayesian computation. In Proceedings of the 28th international conference on machine learning. Bellevue, WA; Barthelme, S., & Chopin, N. (2011). ABC-EP: expectation propagation for likelihood-free Bayesian computation. In Proceedings of the 28th international conference on machine learning. Bellevue, WA · Zbl 1367.62063 |
| [3] | Bazin, E.; Dawson, K. J.; Beaumont, M. A., Likelihood-free inference of population structure and local adaptation in a Bayesian hierarchical model, Genetics, 185, 587-602 (2010) |
| [4] | Beaumont, M. A., Approximate Bayesian computation in evolution and ecology, Annual Review of Ecology, Evolution, and Systematics, 41, 379-406 (2010) |
| [5] | Beaumont, M. A.; Cornuet, J.-M.; Marin, J.-M.; Robert, C. P., Adaptive approximate Bayesian computation, Biometrika, asp052, 1-8 (2009) |
| [6] | Beaumont, M. A.; Zhang, W.; Balding, D. J., Approximate Bayesian computation in population genetics, Genetics, 162, 2025-2035 (2002) |
| [7] | Blum, M. G.B.; François, O., Non-linear regression models for approximate Bayesian computation, Statistics and Computing, 20, 63-73 (2010) |
| [8] | Bortot, P.; Coles, S. G.; Sisson, S. A., Inference for stereological extremes, Journal of the American Statistical Association, 102, 84-92 (2007) · Zbl 1284.62795 |
| [9] | Brown, S.; Steyvers, M., Detecting and predicting changes, Cognitive Psychology, 58, 49-67 (2009) |
| [10] | Cappé, O.; Guillin, A.; Marin, J. M.; Robert, C. P., Population Monte Carlo, Journal of Computational and Graphical Statistics, 13, 907-929 (2004) |
| [11] | Christensen, R.; Johnson, W.; Branscum, A.; Hanson, T. E., Bayesian ideas and data analysis: an introduction for scientists and statisticians (2011), CRC Press, Taylor and Francis Group: CRC Press, Taylor and Francis Group Boca Ranton, FL |
| [12] | Criss, A. H.; McClelland, J. L., Differentiating the differentiation models: A comparison of the retrieving effectively from memory model (REM) and the subjective likelihood model (SLiM), Journal of Memory and Language, 55, 447-460 (2006) |
| [13] | Del Moral, P.; Doucet, A.; Jasra, A., Sequential Monte Carlo samplers, Journal of the Royal Statistical Society. Series B, 68, 411-432 (2006) · Zbl 1105.62034 |
| [14] | Dennis, S.; Lee, M.; Kinnell, A., Bayesian analysis of recognition memory: the case of the list-length effect, Journal of Mathematical Psychology, 59, 361-376 (2008) |
| [15] | Douc, R.; Guillin, A.; Marin, J.-M.; Robert, C., Convergence of adaptive mixtures of importance sampling schemes, Annals of Statistics, 35, 420-448 (2007) · Zbl 1132.60022 |
| [16] | Egan, J. P. (1958). Recognition memory and the operating characteristic. Tech. rep. AFCRC-TN-58-51; Egan, J. P. (1958). Recognition memory and the operating characteristic. Tech. rep. AFCRC-TN-58-51 |
| [17] | Farrell, S.; Ludwig, C. J.H., Bayesian and maximum likelihood estimation of hierarchical response time models, Psychonomic Bulletin and Review, 15, 1209-1217 (2008) |
| [18] | Gelman, A.; Carlin, J. B.; Stern, H. S.; Rubin, D. B., Bayesian data analysis (2004), Chapman and Hall: Chapman and Hall New York, NY · Zbl 1039.62018 |
| [19] | Glanzer, M.; Adams, J. K.; Iverson, G. J.; Kim, K., The regularities of recognition memory, Psychological Review, 100, 546-567 (1993) |
| [20] | Green, D. M.; Swets, J. A., Signal detection theory and psychophysics (1966), Wiley Press: Wiley Press New York |
| [21] | Heathcote, A.; Brown, S. D.; Mewhort, D. J.K., The power law repealed: the case for an exponential law of practice, Psychonomic Bulletin and Review, 7, 185-207 (2000) |
| [22] | Hickerson, M. J.; Meyer, C., Testing comparative phylogeographic models of marine vicariance and dispersal using a hierarchical Bayesian approach, BMC Evolutionary Biology, 8, 322 (2008) |
| [23] | Hickerson, M. J.; Stahl, E. A.; Lessios, H. A., Test for simultaneous divergence using approximate Bayesian computation, Evolution, 60, 2435-2453 (2006) |
| [24] | Kruschke, J. K., Doing Bayesian data analysis: a tutorial with R and BUGS (2011), Academic Press: Academic Press Burlington, MA · Zbl 1301.62001 |
| [25] | Kullback, S.; Keegel, J. C.; Kullback, J. H., (Topics in statistical information theory. Topics in statistical information theory, Lecture notes in statistics, Vol. 42 (1987), Springer-Verlag: Springer-Verlag New York) · Zbl 0632.62003 |
| [26] | Lee, M. D., A Bayesian analysis of retention functions, Journal of Mathematical Psychology, 48, 310-321 (2004) · Zbl 1118.91360 |
| [27] | Lee, M. D., Three case studies in the Bayesian analysis of cognitive models, Psychonomic Bulletin and Review, 15, 1-15 (2008) |
| [28] | Lee, M. D., Special issue on hierarchical Bayesian models, Journal of Mathematical Psychology, 55, 1-118 (2011) · Zbl 1208.91123 |
| [29] | Lee, M. D.; Fuss, I. G.; Navarro, D. J., A Bayesian approach to diffusion models of decision-making and response time, (Scholkopf, B.; Platt, J.; Hoffman, T., Advances in neural information processing (2006), MIT Press: MIT Press Cambridge, MA), 809-815 |
| [30] | Leuenberger, C.; Wegmann, D., Bayesian computation and model selection without likelihoods, Genetics, 184, 243-252 (2010) |
| [31] | Liu, C. C.; Aitkin, M., Bayes factors: prior sensitivity and model generalizability, Journal of Mathematical Psychology, 52, 362-375 (2008) · Zbl 1152.91771 |
| [32] | Malmberg, K. J.; Zeelenberg, R.; Shiffrin, R., Turning up the noise or turning down the volume? on the nature of the impairment of episodic recognition memory by Midazolam, Journal of Experimental Psychology: Learning, Memory, and Cognition, 30, 540-549 (2004) |
| [33] | Marjoram, P.; Molitor, J.; Plagnol, V.; Tavare, S., Markov chain Monte Carlo without likelihoods, Proceedings of the National Academy of Sciences of the United States, 100, 324-328 (2003) |
| [34] | Matzke, D.; Wagenmakers, E.-J., Psychological interpretation of the ex-Gaussian and shifted wald parameters: a diffusion model analysis, Psychonomic Bulletin and Review, 16, 798-817 (2009) |
| [35] | Montenegro, M., Myung, J. I., & Pitt, M. A. (2011). REM integral expressions. Unpublished Manuscript.; Montenegro, M., Myung, J. I., & Pitt, M. A. (2011). REM integral expressions. Unpublished Manuscript. |
| [36] | Myung, I. J., Tutorial on maximum likelihood estimation, Journal of Mathematical Psychology, 47, 90-100 (2003) · Zbl 1023.62112 |
| [37] | Myung, J. I.; Montenegro, M.; Pitt, M. A., Analytic expressions for the BCDMEM model of recognition memory, Journal of Mathematical Psychology, 51, 198-204 (2007) · Zbl 1160.91408 |
| [38] | Nilsson, H.; Rieskamp, J.; Wagenmakers, E.-J., Hierarchical Bayesian parameter estimation for cumulative prospect theory, Journal of Mathematical Psychology, 55, 84-93 (2011) · Zbl 1208.91038 |
| [39] | Nosofsky, R. M., Attention, similarity, and the identification-categorization relationship, Journal of Experimental Psychology: General, 115, 39-57 (1986) |
| [40] | Nosofsky, R. M.; Little, D. R.; Donkin, C.; Fific, M., Short-term memory scanning viewed as exemplar-based categorization, Psychological Review, 118, 280-315 (2011) |
| [41] | Oravecz, Z.; Tuerlinckx, F.; Vandekerckhove, J., A hierarchical latent stochastic differential equation model for affective dynamics, Psychological Methods, 16, 468-490 (2011) |
| [42] | O’Reilly, R. C., Generalization in interactive networks: the benefits of inhibitory competition and Hebbian learning, Neural Computation, 13, 1199-1242 (2001) · Zbl 0963.68643 |
| [43] | O’Reilly, R. C., Biologically based computational models of cortical cognition, Science, 314, 91-94 (2006) · Zbl 1226.92009 |
| [44] | (O’Reilly, R.; Munakata, Y., Computational explorations in cognitive neuroscience: understanding the mind by simulating the brain (2000), MIT Press: MIT Press Cambridge, MA) |
| [45] | Pritchard, J. K.; Seielstad, M. T.; Perez-Lezaun, A.; Feldman, M. W., Population growth of human \(Y\) chromosomes: a study of \(Y\) chromosome microsatellites, Molecular Biology and Evolution, 16, 1791-1798 (1999) |
| [46] | R Development Core Team (2008). R: a language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. ISBN: 3-900051-07-0. URL: http://www.R-project.org; R Development Core Team (2008). R: a language and environment for statistical computing. R Foundation for Statistical Computing. Vienna, Austria. ISBN: 3-900051-07-0. URL: http://www.R-project.org |
| [47] | Ratcliff, R., A theory of memory retrieval, Psychological Review, 85, 59-108 (1978) |
| [48] | Ratcliff, R.; McKoon, G.; Tindall, M. H., Empirical generality of data from recognition memory receiver-operating characteristic functions and implications for the global memory models, Journal of Experimental Psychology: Learning, Memory, and Cognition, 20, 763-785 (1994) |
| [49] | Ratcliff, R.; Sheu, C.-F.; Gronlund, S. D., Testing global memory models using ROC curves, Psychological Review, 99, 518-535 (1992) |
| [50] | Robert, C. P.; Casella, G., Monte Carlo statistical methods (2004), Springer: Springer New York, NY · Zbl 1096.62003 |
| [51] | Rouder, J. N.; Lu, J., An introduction to Bayesian hierarchical models with an application in the theory of signal detection, Psychonomic Bulletin and Review, 12, 573-604 (2005) |
| [52] | Rouder, J. N.; Speckman, P. L., An evaluation of the vincentizing method of forming group-level response time distributions, Psychonomic Bulletin and Review, 11, 419-427 (2004) |
| [53] | Rubin, D. C.; Wenzel, A. E., One hundred years of forgetting: a quantitative description of retention, Psychological Review, 4, 734-760 (1996) |
| [54] | 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) |
| [55] | Shiffrin, R. M.; Steyvers, M., A model for recognition memory: REM—retrieving effectively from memory, Psychonomic Bulletin and Review, 4, 145-166 (1997) |
| [56] | Sisson, S.; Fan, Y.; Tanaka, M. M., Sequential Monte Carlo without likelihoods, Proceedings of the National Academy of Sciences of the United States, 104, 1760-1765 (2007) · Zbl 1160.65005 |
| [57] | Sousa, V. C.; Fritz, M.; Beaumont, M. A.; Chikhi, L., Approximate Bayeisian computation without summary statistics: the case of admixture, Genetics, 181, 1507-1519 (2009) |
| [58] | Toni, T.; Welch, D.; Strelkowa, N.; Ipsen, A.; Stumpf, M. P., Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems, Journal of the Royal Society Interface, 6, 187-202 (2009) |
| [59] | Turner, B. M., Dennis, S., & Van Zandt, T. (2011). Bayesian analysis of memory models. Manuscript in preparation.; Turner, B. M., Dennis, S., & Van Zandt, T. (2011). Bayesian analysis of memory models. Manuscript in preparation. |
| [60] | Turner, B. M., & Sederberg, P. B. (2012). Approximate Bayesian computation with differential evolution. Manuscript (submitted for publication).; Turner, B. M., & Sederberg, P. B. (2012). Approximate Bayesian computation with differential evolution. Manuscript (submitted for publication). |
| [61] | Turner, B. M., & Van Zandt, T. (2012). Hierarchical approximate Bayesian computation. Manuscript (submitted for publication).; Turner, B. M., & Van Zandt, T. (2012). Hierarchical approximate Bayesian computation. Manuscript (submitted for publication). · Zbl 1288.62191 |
| [62] | Usher, M.; McClelland, J. L., On the time course of perceptual choice: the leaky competing accumulator model, Psychological Review, 108, 550-592 (2001) |
| [63] | Vandekerckhove, J.; Tuerlinckx, F.; Lee, M. D., Hierarchical diffusion models for two-choice response time, Psychological Methods, 16, 44-62 (2011) |
| [64] | Wagenmakers, E.-J.; van der Maas, H. J.L.; Grasman, R. P.P. P., An EZ-diffusion model for response time and accuracy, Psychonomic Bulletin and Review, 14, 3-22 (2007) |
| [65] | Wegmann, D.; Leuenberger, C.; Excoffier, L., Efficient approximate Bayesian computation coupled with Markov chain Monte Carlo without likelihood, Genetics, 182, 1207-1218 (2009) |
| [66] | 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 (2008) · Zbl 1203.91255 |
| [67] | Wilkinson, R. D. (2012). Approximate Bayesian computation (ABC) gives exact results under the assumption of model error. Manuscript (submitted for publication).; Wilkinson, R. D. (2012). Approximate Bayesian computation (ABC) gives exact results under the assumption of model error. Manuscript (submitted for publication). |
| [68] | Wixted, J. T., Analyzing the empirical course of forgetting, Journal of Experimental Psychology: Learning, Memory, and Cognition, 16, 927-935 (1990) |
| [69] | Wood, S. N., Statistical inference for noisy nonlinear ecological dynamic systems, Nature, 466, 1102-1104 (2010) |
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