Definition: Group Bayes factor

Index: The Book of Statistical Proofs ▷ Model Selection ▷ Bayesian model selection ▷ Bayes factor ▷ Group Bayes factor

Definition: Let $Y = \left\lbrace y_1, \ldots, y_N \right\rbrace$ be a set of $N$ data sets (forming a “group”) and consider two generate models $m_1$ and $m_2$ with model evidences (i.e. marginal likelihoods) $p(y_i \vert m)$ where $i = 1,\ldots,N$ and $m \in \left\lbrace m_1, m_2 \right\rbrace$. Then, the group Bayes factor for $Y$ in favor of $m_1$ over $m_2$ is defined as:

\[\label{eq:GBF} \mathrm{GBF}_{12} = \prod_{i=1}^N \frac{p(y_i \vert m_1)}{p(y_i \vert m_2)} \; .\]

 

Sources:

• Stephan KE, Penny WD, Daunizeau J, Moran RJ, Friston KJ (2009): "Bayesian model selection for group studies"; in: NeuroImage, vol. 46, pp. 1004–1017, eq. 2; URL: https://www.sciencedirect.com/science/article/abs/pii/S1053811909002638; DOI: 10.1016/j.neuroimage.2009.03.025.
• Penny WD, Stephan KE, Daunizeau J, Rosa MJ, Friston KJ, Schofield TM, Leff AP (2010): "Comparing Families of Dynamic Causal Models"; in: PLoS Computational Biology, vol. 6, iss. 3, art. e1000709, eqs. 6/8; URL: https://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1000709; DOI: 10.1371/journal.pcbi.1000709.
• Soch J, Allefeld C (2018): "MACS – a new SPM toolbox for model assessment, comparison and selection"; in: Journal of Neuroscience Methods, vol. 306, pp. 19-31, eq. 20; URL: https://www.sciencedirect.com/science/article/pii/S0165027018301468; DOI: 10.1016/j.jneumeth.2018.05.017.

Metadata: ID: D233 | shortcut: gbf | author: JoramSoch | date: 2026-06-04, 17:23.