Bayesian model selection for group studies
Open accessAbstract
Bayesian model selection (BMS) is a powerful method for determining the most likely among a set of competing hypotheses about the mechanisms that generated observed data. BMS has recently found widespread application in neuroimaging, particularly in the context of dynamic causal modelling (DCM). However, so far, combining BMS results from several subjects has relied on simple (fixed effects) metrics, e.g. the group Bayes factor (GBF), that do not account for group heterogeneity or outliers. In this paper, we compare the GBF with two random effects methods for BMS at the between-subject or group level. These methods provide inference on model-space using a classical and Bayesian perspective respectively. First, a classical (frequentist) approach uses the log model evidence as a subject-specific summary statistic. This enables one to use analysis of variance to test for differences in log-evidences over models, relative to inter-subject differences. We then consider the same problem in Bayesian terms and describe a novel hierarchical model, which is optimised to furnish a probability density on the models themselves. This new variational Bayes method rests on treating the model as a random variable and estimating the parameters of a Dirichlet distribution which describes the probabilities for all models considered. These probabilities then define a multinomial distribution over model space, allowing one to compute how likely it is that a specific model generated the data of a randomly chosen subject as well as the exceedance probability of one model being more likely than any other model. Using empirical and synthetic data, we show that optimising a conditional density of the model probabilities, given the log-evidences for each model over subjects, is more informative and appropriate than both the GBF and frequentist tests of the log-evidences. In particular, we found that the hierarchical Bayesian approach is considerably more robust than either of the other approaches in the presence of outliers. We expect that this new random effects method will prove useful for a wide range of group studies, not only in the context of DCM, but also for other modelling endeavours, e.g. comparing different source reconstruction methods for EEG/MEG or selecting among competing computational models of learning and decision-making.
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Hidden individual differences in cognitive strategies can be detected by formulating alternative computational models and subjecting them to Bayesian model selection.
“Such hidden individual differences can be detected by formulating alternative computational models, each reflecting different cognitive strategies, and subjecting them to Bayesian model selection (62).”
From Can Bayesian Theories of Autism Spectrum Disorder Help Improve Clinical Practice? (2016) · cites this paper as [62]
The GBF approach implicitly assumes that every subject uses the same model, and is therefore a Fixed Effects analysis.
“As is made clear in [19] the GBF approach implicitly assumes that every subject uses the same model. It is therefore a Fixed Effects analysis.”
From Comparing families of dynamic causal models (2010) · cites this paper as [19]
The robustness and accuracy of the variational RFX method was verified via simulations using data from synthetic populations with known frequencies of competing models.
“The robustness and accuracy of this method was verified via simulations using data from synthetic populations with known frequencies of competing models [19].”
From Comparing families of dynamic causal models (2010) · cites this paper as [19]
The use of Gibbs sampling allows relaxation of the assumption made in VB that the posterior densities over m and α factorise.
“Additionally, the use of Gibbs sampling allows us to relax the assumption made in VB that the posterior densities over and factorise [19].”
From Comparing families of dynamic causal models (2010) · cites this paper as [19]
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Brodersen, K. H., Schofield, T. M., Leff, A. P., Ong, C. S., Lomakina, E. I., Buhmann, J. M., & Stephan, K. E. (2011) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1002079 · Cited by 165 · see citing works
Self-citationPharmacological Fingerprints of Contextual Uncertainty
Marshall, L., Mathys, C., Ruge, D., de Berker, A. O., Dayan, P., Stephan, K. E., & Bestmann, S. (2016) · PLOS Biology · doi:10.1371/journal.pbio.1002575 · Cited by 158 · see citing works
Self-citationSkvortsova, V., Palminteri, S., & Pessiglione, M. (2014) · The Journal of Neuroscience · doi:10.1523/jneurosci.1350-14.2014 · Cited by 177 · see citing works
Cho, Y. T., Fromm, S., Guyer, A. E., Detloff, A., Pine, D. S., Fudge, J. L., & Ernst, M. (2013) · NeuroImage · doi:10.1016/j.neuroimage.2012.10.013 · Cited by 173 · see citing works
The brain’s functional network architecture reveals human motives
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Modelling trial-by-trial changes in the mismatch negativity
Lieder, F., Daunizeau, J., Garrido, M. I., Friston, K. J., & Stephan, K. E. (2013) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1002911 · Cited by 168 · see citing works
Hauser, T. U., Iannaccone, R., Ball, J., Mathys, C., Brandeis, D., Walitza, S., & Brem, S. (2014) · JAMA Psychiatry · doi:10.1001/jamapsychiatry.2014.1093 · Cited by 168 · see citing works
Nicholson, A. A., Friston, K. J., Zeidman, P., Harricharan, S., McKinnon, M. C., Densmore, M., Neufeld, R. W. J., Théberge, J., Corrigan, F., Jetly, R., et al. (2017) · Human Brain Mapping · doi:10.1002/hbm.23748 · Cited by 159 · see citing works
Self-citationKlein-Flügge, M. C., Kennerley, S. W., Saraiva, A. C., Penny, W. D., & Bestmann, S. (2015) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1004116 · Cited by 163 · see citing works
TAPAS: An Open-Source Software Package for Translational Neuromodeling and Computational Psychiatry
Frässle, S., Aponte, E. A., Bollmann, S., Brodersen, K. H., Do, C. T., Harrison, O. K., Harrison, S. J., Heinzle, J., Iglesias, S., Kasper, L., et al. (2021) · Frontiers in Psychiatry · doi:10.3389/fpsyt.2021.680811 · Cited by 169 · see citing works
Self-citationAllefeld, C., Görgen, K., & Haynes, J.-D. (2016) · NeuroImage · doi:10.1016/j.neuroimage.2016.07.040 · Cited by 157 · see citing works
Spatial attention, precision, and Bayesian inference: a study of saccadic response speed
Vossel, S., Mathys, C., Daunizeau, J., Bauer, M., Driver, J., Friston, K. J., & Stephan, K. E. (2014) · Cerebral Cortex · doi:10.1093/cercor/bhs418 · Cited by 146 · see citing works
Self-citationBrain networks for confidence weighting and hierarchical inference during probabilistic learning
Meyniel, F., & Dehaene, S. (2017) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1615773114 · Cited by 138 · see citing works
Comparing Bayesian and non-Bayesian accounts of human confidence reports
Adler, W. T., & Ma, W. J. (2018) · PLoS Computational Biology · doi:10.1371/journal.pcbi.1006572 · Cited by 118 · see citing works
A hemodynamic model for layered BOLD signals
Heinzle, J., Koopmans, P. J., den Ouden, H. E. M., Raman, S., & Stephan, K. E. (2016) · NeuroImage · doi:10.1016/j.neuroimage.2015.10.025 · Cited by 144 · see citing works
Self-citationCompulsivity reveals a novel dissociation between action and confidence
Vaghi, M. M., Luyckx, F., Sule, A., Fineberg, N. A., Robbins, T. W., & De Martino, B. (2017) · Neuron · doi:10.1016/j.neuron.2017.09.006 · Cited by 145 · see citing works
Can Bayesian Theories of Autism Spectrum Disorder Help Improve Clinical Practice?
Haker, H., Schneebeli, M., & Stephan, K. E. (2016) · Frontiers in Psychiatry · doi:10.3389/fpsyt.2016.00107 · Cited by 130 · see citing works
Self-citationThe Control of Mimicry by Eye Contact Is Mediated by Medial Prefrontal Cortex
Wang, Y., Ramsey, R., & de C. Hamilton, A. F. (2011) · The Journal of Neuroscience · doi:10.1523/jneurosci.0845-11.2011 · Cited by 141 · see citing works
Encoding of Stimulus Probability in Macaque Inferior Temporal Cortex
Bell, A. H., Summerfield, C., Morin, E. L., Malecek, N. J., & Ungerleider, L. G. (2016) · Current Biology · doi:10.1016/j.cub.2016.07.007 · Cited by 118 · see citing works
Confidence controls perceptual evidence accumulation
Balsdon, T., Wyart, V., & Mamassian, P. (2020) · Nature Communications · doi:10.1038/s41467-020-15561-w · Cited by 137 · see citing works
A predictive coding account of bistable perception - a model-based fMRI study
Weilnhammer, V., Stuke, H., Hesselmann, G., Sterzer, P., & Schmack, K. (2017) · PLOS Computational Biology · doi:10.1371/journal.pcbi.1005536 · Cited by 120 · see citing works
Model averaging, optimal inference, and habit formation
FitzGerald, T. H. B., Dolan, R. J., & Friston, K. J. (2014) · Frontiers in Human Neuroscience · doi:10.3389/fnhum.2014.00457 · Cited by 121 · see citing works
Self-citationGeneralization of value in reinforcement learning by humans
Wimmer, G. E., Daw, N. D., & Shohamy, D. (2012) · European Journal of Neuroscience · doi:10.1111/j.1460-9568.2012.08017.x · Cited by 111 · see citing works
Computational nosology and precision psychiatry
Friston, K. J., Redish, A. D., & Gordon, J. A. (2017) · Computational Psychiatry · doi:10.1162/cpsy_a_00001 · Cited by 126 · see citing works
Self-citationChennu, S., Noreika, V., Gueorguiev, D., Shtyrov, Y., Bekinschtein, T. A., & Henson, R. (2016) · Journal of Neuroscience · doi:10.1523/jneurosci.1125-16.2016 · Cited by 107 · see citing works
Impaired top-down processes in schizophrenia: a DCM study of ERPs
Dima, D., Dietrich, D. E., Dillo, W., & Emrich, H. M. (2010) · NeuroImage · doi:10.1016/j.neuroimage.2009.12.086 · Cited by 122 · see citing works
Paret, C., Ruf, M., Gerchen, M. F., Kluetsch, R., Demirakca, T., Jungkunz, M., Bertsch, K., Schmahl, C., & Ende, G. (2016) · NeuroImage · doi:10.1016/j.neuroimage.2015.10.027 · Cited by 113 · see citing works
Cross, K. A., Torrisi, S., Reynolds Losin, E. A., & Iacoboni, M. (2013) · NeuroImage · doi:10.1016/j.neuroimage.2013.06.060 · Cited by 113 · see citing works
Effective connectivity analysis demonstrates involvement of premotor cortex during speech perception
Osnes, B., Hugdahl, K., & Specht, K. (2011) · NeuroImage · doi:10.1016/j.neuroimage.2010.09.078 · Cited by 112 · see citing works
Flexible combination of reward information across primates
Farashahi, S., Donahue, C. H., Hayden, B. Y., Lee, D., & Soltani, A. (2019) · Nature Human Behaviour · doi:10.1038/s41562-019-0714-3 · Cited by 112 · see citing works
Cooper, J. C., Dunne, S., Furey, T., & O'Doherty, J. P. (2012) · Journal of Cognitive Neuroscience · doi:10.1162/jocn_a_00114 · Cited by 101 · see citing works
An afferent white matter pathway from the pulvinar to the amygdala facilitates fear recognition
McFadyen, J., Mattingley, J. B., & Garrido, M. I. (2019) · eLife · doi:10.7554/elife.40766 · Cited by 115 · see citing works
Belief state representation in the dopamine system
Babayan, B. M., Uchida, N., & Gershman, S. J. (2018) · Nature Communications · doi:10.1038/s41467-018-04397-0 · Cited by 101 · see citing works
Autonomic and brain responses associated with empathy deficits in autism spectrum disorder
Gu, X., Eilam‐Stock, T., Zhou, T., Anagnostou, E., Kolevzon, A., Soorya, L., Hof, P. R., Friston, K. J., & Fan, J. (2015) · Human Brain Mapping · doi:10.1002/hbm.22840 · Cited by 110 · see citing works
Self-citationModeling ketamine effects on synaptic plasticity during the mismatch negativity
Schmidt, A., Diaconescu, A. O., Kometer, M., Friston, K. J., Stephan, K. E., & Vollenweider, F. X. (2013) · Cerebral Cortex · doi:10.1093/cercor/bhs238 · Cited by 109 · see citing works
Self-citationSchlösser, R. G. M., Wagner, G., Schachtzabel, C., Peikert, G., Koch, K., Reichenbach, J. R., & Sauer, H. (2010) · Human Brain Mapping · doi:10.1002/hbm.20980 · Cited by 111 · see citing works
Cortical coupling reflects Bayesian belief updating in the deployment of spatial attention
Vossel, S., Mathys, C., Stephan, K. E., & Friston, K. J. (2015) · The Journal of Neuroscience · doi:10.1523/jneurosci.1382-15.2015 · Cited by 99 · see citing works
Self-citationThe impact of evidence reliability on sensitivity and bias in decision confidence.
Boldt, A., de Gardelle, V., & Yeung, N. (2017) · Journal of Experimental Psychology: Human Perception and Performance · doi:10.1037/xhp0000404 · Cited by 110 · see citing works
Kraehenmann, R., Schmidt, A., Friston, K., Preller, K. H., Seifritz, E., & Vollenweider, F. X. (2016) · NeuroImage: Clinical · doi:10.1016/j.nicl.2015.08.009 · Cited by 104 · see citing works
Self-citationThe Tortoise and the Hare: Interactions between Reinforcement Learning and Working Memory
Collins, A. G. E. (2018) · Journal of Cognitive Neuroscience · doi:10.1162/jocn_a_01238 · Cited by 97 · see citing works
Evidence for surprise minimization over value maximization in choice behavior
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Self-citationNetwork activity underlying the illusory self‐attribution of a dummy arm
Limanowski, J., & Blankenburg, F. (2015) · Human Brain Mapping · doi:10.1002/hbm.22770 · Cited by 101 · see citing works
Moran, R. J., Jung, F., Kumagai, T., Endepols, H., Graf, R., Dolan, R. J., Friston, K. J., Stephan, K. E., & Tittgemeyer, M. (2011) · PLoS ONE · doi:10.1371/journal.pone.0022790 · Cited by 88 · see citing works
“fixed effects analysis of the models was performed using the group Bayes factor [47,50], with log-evidences averaged over hemi-”
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Schmidt, A., Smieskova, R., Aston, J., Simon, A., Allen, P., Fusar-Poli, P., McGuire, P. K., Riecher-Rössler, A., Stephan, K. E., & Borgwardt, S. (2013) · JAMA Psychiatry · doi:10.1001/jamapsychiatry.2013.117 · Cited by 93 · see citing works
Self-citationLong-lasting crossmodal cortical reorganization triggered by brief postnatal visual deprivation
Collignon, O., Dormal, G., de Heering, A., Lepore, F., Lewis, T. L., & Maurer, D. (2015) · Current Biology · doi:10.1016/j.cub.2015.07.036 · Cited by 94 · see citing works
Causal evidence for lateral prefrontal cortex dynamics supporting cognitive control
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Blunted medial prefrontal cortico-limbic reward-related effective connectivity and depression
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The hippocampus extrapolates beyond the view in scenes: an fMRI study of boundary extension
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Evidence for model-based computations in the human amygdala during Pavlovian conditioning
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Attractor-like dynamics in belief updating in schizophrenia
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Consistent spectral predictors for dynamic causal models of steady-state responses
Moran, R. J., Stephan, K. E., Dolan, R. J., & Friston, K. J. (2011) · NeuroImage · doi:10.1016/j.neuroimage.2011.01.012 · Cited by 75 · see citing works
Self-citationMulti-subject analyses with dynamic causal modeling
Kasess, C. H., Stephan, K. E., Weissenbacher, A., Pezawas, L., Moser, E., & Windischberger, C. (2010) · NeuroImage · doi:10.1016/j.neuroimage.2009.11.037 · Cited by 70 · see citing works
Self-citationAction-specific value signals in reward-related regions of the human brain
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Self-citationDiwadkar, V. A. (2012) · Archives of General Psychiatry · doi:10.1001/archgenpsychiatry.2011.1349 · Cited by 66 · see citing works
Crossmodal interactions in audiovisual emotion processing
Müller, V. I., Cieslik, E. C., Turetsky, B. I., & Eickhoff, S. B. (2012) · NeuroImage · doi:10.1016/j.neuroimage.2011.12.007 · Cited by 65 · see citing works
- Authors
- Stephan, K. E., Penny, W. D., Daunizeau, J., Moran, R. J., & Friston, K. J.
- Affiliations
- Wellcome Centre for Human Neuroimaging (GB)
- Corresponding author
- Stephan, Klaas Enno
- Venue
- NeuroImage
- Publisher
- Elsevier BV
- Published
- 2009-07
- Volume
- 46
- Issue
- 4
- Pages
- 1004-1017
- ISSN
- 1053-8119 (ISSN-L)
- PMC
- PMC2703732
- OpenAlex
- W2135173838
- Cited by
- 1,423
- References
- 49
- Citations by year
- 2026: 63 · 2025: 74 · 2024: 76 · 2023: 71 · 2022: 86 · 2021: 100 · 2020: 135 · 2019: 97 · 2018: 102 · 2017: 91 · 2016: 89 · 2015: 109 · 2014: 67 · 2013: 75 · 2012: 59
- FWCI
- 51.46
- Type
- journal article
- Language
- en
- Open access
- green
- www.zora.uzh.ch
- Licence
- Locations
- NeuroImage (publishedVersion)
- PubMed (publishedVersion)
- Zurich Open Repository and Archive (University of Zurich) (acceptedVersion, other-oa)
- PubMed Central (acceptedVersion)
- UCL Discovery (University College London) (submittedVersion)
- Explore Bristol Research (submittedVersion)
- UEA Digital Repository (University of East Anglia) (submittedVersion)
- Zurich Open Repository and Archive (University of Zurich) (submittedVersion, other-oa)
- Zurich Open Repository and Archive (University of Zurich)
- Funders
- Wellcome Trust (056750, 088130)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Life Sciences › Neuroscience › Cognitive Neuroscience › Functional Brain Connectivity Studies
- Keywords
- Bayesian model selection, group study, dynamic causal modeling, random effects, Bayesian hierarchical model, variational inference, outliers, exceedance probability, Dirichlet distribution, model comparison
- MeSH terms
- Algorithms, Bayes Theorem, Brain, Cognition, Diagnostic Imaging, Humans, Models, Neurological
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
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