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Hierarchical Approximate Bayesian Computation

Published online by Cambridge University Press:  01 January 2025

Brandon M. Turner  and
Trisha Van Zandt
Brandon M. Turner*
Affiliation:
Stanford University
Trisha Van Zandt
Affiliation:
The Ohio State University
*
Requests for reprints should be sent to Brandon M. Turner, Stanford University, Stanford, USA. E-mail: [email protected]
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Abstract

Approximate Bayesian computation (ABC) is a powerful technique for estimating the posterior distribution of a model’s parameters. It is especially important when the model to be fit has no explicit likelihood function, which happens for computational (or simulation-based) models such as those that are popular in cognitive neuroscience and other areas in psychology. However, ABC is usually applied only to models with few parameters. Extending ABC to hierarchical models has been difficult because high-dimensional hierarchical models add computational complexity that conventional ABC cannot accommodate. In this paper, we summarize some current approaches for performing hierarchical ABC and introduce a new algorithm called Gibbs ABC. This new algorithm incorporates well-known Bayesian techniques to improve the accuracy and efficiency of the ABC approach for estimation of hierarchical models. We then use the Gibbs ABC algorithm to estimate the parameters of two models of signal detection, one with and one without a tractable likelihood function.

Keywords

approximate Bayesian computationhierarchical Bayesian estimationsignal detection theorydynamic signal detection
Type
Original Paper
Information
Psychometrika , Volume 79 , Issue 2 , April 2014 , pp. 185 - 209
Copyright
Copyright © 2013 The Psychometric Society

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References

Bazin, E., Dawson, K.J., Beaumont, M.A. (2010). Likelihood-free inference of population structure and local adaptation in a Bayesian hierarchical model. Genetics, 185, 587–602CrossRefGoogle Scholar
Beaumont, M.A. (2010). Approximate Bayesian computation in evolution and ecology. Annual Review of Ecology, Evolution, and Systematics, 41, 379–406CrossRefGoogle Scholar
Beaumont, M.A., Cornuet, J.-M., Marin, J.-M., Robert, C.P. (2009). Adaptive approximate Bayesian computation. Biometrika, asp052, 1–8Google Scholar
Benjamin, A.S., Diaz, M., Wee, S. (2009). Signal detection with criterior noise: applications to recognition memory. Psychological Review, 116, 84–115CrossRefGoogle Scholar
Brown, S., Heathcote, A. (2005). A ballistic model of choice response time. Psychological Review, 112, 117–128CrossRefGoogle ScholarPubMed
Brown, S., Heathcote, A. (2008). The simplest complete model of choice reaction time: linear ballistic accumulation. Cognitive Psychology, 57, 153–178CrossRefGoogle ScholarPubMed
Brown, S., Steyvers, M. (2005). The dynamics of experimentally induced criterion shifts. Journal of Experimental Psychology. Learning, Memory, and Cognition, 31, 587–599CrossRefGoogle ScholarPubMed
Christensen, R., Johnson, W., Branscum, A., Hanson, T.E. (2011). Bayesian ideas and data analysis: an introduction for scientists and statisticians, Boca Raton: CRC Press, Taylor and Francis GroupGoogle Scholar
DeCarlo, L.T. (2012). On a signal detection approach to m-alternative forced choice with bias, with maximum likelihood and Bayesian approaches to estimation. Journal of Mathematical Psychology, 56, 196–207CrossRefGoogle Scholar
Dennis, S., Humphreys, M.S. (2001). A context noise model of episodic word recognition. Psychological Review, 108, 452–478CrossRefGoogle ScholarPubMed
Dorfman, D.D., Alf, E. (1969). Maximum-likelihood estimation of parameters of signal-detection theory and determination of confidence intervals—rating-method data. Journal of Mathematical Psychology, 6, 487–496CrossRefGoogle Scholar
Efron, B. (1986). Why isn’t everyone a Bayesian?. American Statistician, 40, 1–11CrossRefGoogle Scholar
Egan, J.P. (1958). Recognition memory and the operating characteristic (Tech. Rep. AFCRC-TN-58-51). Hearing and Communication Laboratory, Indiana University, Bloomington, Indiana. Google Scholar
Erev, I. (1998). Signal detection by human observers: a cutoff reinforcement learning model of categoriation decisions under uncertainty. Psychological Review, 105, 280–298CrossRefGoogle ScholarPubMed
Excoffer, L., Estoup, A., Cornuet, J.-M. (2005). Bayesian analysis of an admixture model with mutations and arbitrarily linked markers. Genetics, 169, 1727–1738CrossRefGoogle Scholar
Fearnhead, P., Prangle, D. (2012). Constructing summary statistics for approximate Bayesian computation: semi-automatic approximate Bayesian computation. Journal of the Royal Statistical Society. Series B, 74, 419–474CrossRefGoogle Scholar
Gelman, A., Carlin, J.B., Stern, H.S., Rubin, D.B. (2004). Bayesian data analysis, New York: Chapman and HallGoogle Scholar
Gilks, W.R., Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337–348CrossRefGoogle Scholar
Green, D.M., Swets, J.A. (1966). Signal detection theory and psychophysics, New York: WileyGoogle Scholar
Hickerson, M.J., Meyer, C. (2008). Testing comparative phylogeographic models of marine vicariance and dispersal using a hierarchical Bayesian approach. BMC Evolutionary Biology, 8, 322CrossRefGoogle ScholarPubMed
Hickerson, M.J., Stahl, E.A., Lessios, H.A. (2006). Test for simultaneous divergence using approximate Bayesian computation. Evolution, 60, 2435–2453CrossRefGoogle ScholarPubMed
Jilk, D.J., Lebiere, C., O’Reilly, R.C., Anderson, J.R. (2008). SAL: an explicitly pluralistic cognitive architecture. Journal of Experimental and Theoretical Artificial Intelligence, 20, 197–218CrossRefGoogle Scholar
Kubovy, M., Healy, A.F. (1977). The decision rule in probabilistic categorization: what it is and how it is learned. Journal of Experimental Psychology. General, 106, 427–446CrossRefGoogle Scholar
Lee, M.D. (2008). Three case studies in the Bayesian analysis of cognitive models. Psychonomic Bulletin & Review, 15, 1–15CrossRefGoogle ScholarPubMed
Lee, M. D., & Wagenmakers, E.-J. (2012). A course in Bayesian graphical modeling for cognitive science. Available from http://www.ejwagenmakers.com/BayesCourse/BayesBookWeb.pdf; last downloaded January 1, 2012. Google Scholar
Lunn, D., Thomas, A., Best, N., Spiegelhalter, D. (2000). WinBUGS—a Bayesian modelling framework: concepts, structure and extensibility. Statistics and Computing, 10, 325–337CrossRefGoogle Scholar
Macmillan, N.A., Creelman, C.D. (2005). Detection theory: a user’s guide, Mahwah: Lawrence Erlbaum AssociatesGoogle Scholar
Malmberg, K.J., Zeelenberg, R., Shiffrin, R. (2004). 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, 540–549CrossRefGoogle ScholarPubMed
Matzke, D., Wagenmakers, E.-J. (2009). Psychological interpretation of the ex-Gaussian and shifted Wald parameters: a diffusion model analysis. Psychonomic Bulletin & Review, 16, 798–817CrossRefGoogle ScholarPubMed
Mazurek, M.E., Roitman, J.D., Ditterich, J., Shadlen, M.N. (2003). A role for neural integrators in perceptual decision making. Cerebral Cortex, 13, 1257–1269CrossRefGoogle ScholarPubMed
McElree, B., Dosher, B.A. (1993). Serial retrieval processes in the recovery of order information. Journal of Experimental Psychology. General, 122, 291–315CrossRefGoogle Scholar
Mueller, S.T., Weidemann, C.T. (2008). Decision noise: an explanation for observed violations of signal detection theory. Psychonomic Bulletin & Review, 15, 465–494CrossRefGoogle ScholarPubMed
Myung, I.J. (2003). Tutorial on maximum likelihood estimation. Journal of Mathematical Psychology, 47, 90–100CrossRefGoogle Scholar
Nelder, J.A., Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308–313CrossRefGoogle Scholar
Nosofsky, R.M., Little, D.R., Donkin, C., Fific, M. (2011). Short-term memory scanning viewed as exemplar-based categorization. Psychological Review, 118, 280–315CrossRefGoogle ScholarPubMed
Nosofsky, R.M., Palmeri, T. (1997). Comparing exemplar-retrieval and decision-bound models of speeded perceptual classification. Perception and Psychophysics, 59, 1027–1048CrossRefGoogle ScholarPubMed
O’Reilly, R., Frank, M. (2006). Making working memory work: a computational model of learning in the prefrontal cortex and basal ganglia. Neural Computation, 18, 283–328CrossRefGoogle Scholar
Pritchard, J.K., Seielstad, M.T., Perez-Lezaun, A., Feldman, M.W. (1999). Population growth of human Y chromosomes: a study of Y chromosome microsatellites. Molecular Biology and Evolution, 16, 1791–1798CrossRefGoogle ScholarPubMed
Ratcliff, R. (1978). A theory of memory retrieval. Psychological Review, 85, 59–108CrossRefGoogle Scholar
Ratcliff, R., Rouder, J.N. (1998). Modeling response times for two-choice decisions. Psychological Science, 9, 347–356CrossRefGoogle Scholar
Ratcliff, R., Smith, P.L. (2004). A comparison of sequential sampling models for two-choice reaction time. Psychological Review, 111, 333–367CrossRefGoogle ScholarPubMed
Ratcliff, R., Starns, J. (2009). Modeling confidence and response time in recognition memory. Psychological Review, 116, 59–83CrossRefGoogle ScholarPubMed
Robert, C.P., Casella, G. (2004). Monte Carlo statistical methods, New York: SpringerCrossRefGoogle Scholar
Rouder, J.N., Lu, J. (2005). An introduction to Bayesian hierarchical models with an application in the theory of signal detection. Psychonomic Bulletin & Review, 12, 573–604CrossRefGoogle ScholarPubMed
Rouder, J.N., Lu, J., Speckman, P., Sun, D., Jiang, Y. (2005). A hierarchical model for estimating response time distributions. Psychonomic Bulletin & Review, 12, 195–223CrossRefGoogle ScholarPubMed
Rouder, J.N., Sun, D., Speckman, P., Lu, J., Zhou, D. (2003). A hierarchical Bayesian statistical framework for response time distributions. Psychometrika, 68, 589–606CrossRefGoogle Scholar
Shiffrin, R.M., Lee, M.D., Kim, W., Wagenmakers, E.-J. (2008). A survey of model evaluation approaches with a tutorial on hierarchical Bayesian methods. Cognitive Science, 32, 1248–1284CrossRefGoogle ScholarPubMed
Shiffrin, R.M., Steyvers, M. (1997). A model for recognition memory: REM—retrieving effectively from memory. Psychonomic Bulletin & Review, 4, 145–166CrossRefGoogle Scholar
Sisson, S., Fan, Y., Tanaka, M.M. (2007). Sequential Monte Carlo without likelihoods. Proceedings of the National Academy of Sciences of the United States of America, 104, 1760–1765CrossRefGoogle ScholarPubMed
Sousa, V.C., Fritz, M., Beaumont, M.A., Chikhi, L. (2009). Approximate Bayesian computation without summary statistics: the case of admixture. Genetics, 181, 1507–1519CrossRefGoogle ScholarPubMed
Toni, T., Welch, D., Strelkowa, N., Ipsen, A., Stumpf, M.P. (2009). Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. Journal of the Royal Society Interface, 6, 187–202CrossRefGoogle ScholarPubMed
Treisman, M., Williams, T. (1984). A theory of criterion setting with an application to sequential dependencies. Psychological Review, 91, 68–111CrossRefGoogle Scholar
Tsetsos, K., Usher, M., McClelland, J.L. (2011). Testing multi-alternative decision models with non-stationary evidence. Frontiers in Neuroscience, 5, 1–18CrossRefGoogle ScholarPubMed
Turner, B.M., Dennis, S., Van Zandt, T. (2013). Likelihood-free Bayesian analysis of memory models. Psychological Review, 120, 667–678CrossRefGoogle ScholarPubMed
Turner, B.M., Sederberg, P.B. (2012). Approximate Bayesian computation with differential evolution. Journal of Mathematical Psychology, 56, 375–385CrossRefGoogle Scholar
Turner, B.M., Van Zandt, T. (2012). A tutorial on approximate Bayesian computation. Journal of Mathematical Psychology, 56, 69–85CrossRefGoogle Scholar
Turner, B.M., Van Zandt, T., Brown, S.D. (2011). A dynamic, stimulus-driven model of signal detection. Psychological Review, 118, 583–613CrossRefGoogle ScholarPubMed
Usher, M., McClelland, J.L. (2001). On the time course of perceptual choice: the leaky competing accumulator model. Psychological Review, 108, 550–592CrossRefGoogle ScholarPubMed
Van Zandt, T. (2000). How to fit a response time distribution. Psychonomic Bulletin & Review, 7, 424–465CrossRefGoogle Scholar
Van Zandt, T., Colonius, H., Proctor, R.W. (2000). A comparison of two response time models applied to perceptual matching. Psychonomic Bulletin & Review, 7, 208–256CrossRefGoogle ScholarPubMed
Van Zandt, T., & Jones, M.R. (2011). Stimulus rhythm and choice performance. Unpublished manuscript. Google Scholar
Vandekerckhove, J., Tuerlinckx, F., Lee, M.D. (2011). Hierarchical diffusion models for two-choice response time. Psychological Methods, 16, 44–62CrossRefGoogle Scholar
Wald, A. (1947). Sequential analysis, New York: WileyGoogle Scholar
Wilkinson, R.D. (2011). Approximate Bayesian computation (ABC) gives exact results under the assumption of model error. Manuscript submitted for publication. Google Scholar