Proof: Log Bayes factor in terms of log model evidences
Index: The Book of Statistical Proofs ▷ Model Selection ▷ Bayesian model selection ▷ Bayes factor ▷ Calculation from log model evidences
Metadata: ID: P64 | shortcut: lbf-lme | author: JoramSoch | date: 2020-02-27, 20:51.
Theorem: Let $m_1$ and $m_2$ be two statistical models with log model evidences $\mathrm{LME}(m_1)$ and $\mathrm{LME}(m_2)$. Then, the log Bayes factor in favor of model $m_1$ and against model $m_2$ is the difference of the log model evidences:
\[\label{eq:LBF-LME} \mathrm{LBF}_{12} = \mathrm{LME}(m_1) - \mathrm{LME}(m_2) \; .\]Proof: The Bayes factor is defined as the ratio of the model evidences of $m_1$ and $m_2$
\[\label{eq:BF} \mathrm{BF}_{12} = \frac{p(y|m_1)}{p(y|m_2)}\]and the log Bayes factor is defined as the logarithm of the Bayes factor
\[\label{eq:LBF-s1} \mathrm{LBF}_{12} = \log \mathrm{BF}_{12} \; .\]Thus, the log Bayes factor can be expressed as
\[\label{eq:LBF-s2} \mathrm{LBF}_{12} = \log \frac{p(y|m_1)}{p(y|m_2)} \; .\]and, with the definition of the log model evidence
\[\label{eq:LME} \mathrm{LME}(m) = \log p(y|m)\]and resolving the logarithm, we finally have:
\[\label{eq:LBF-LME-qed} \begin{split} \mathrm{LBF}_{12} &= \log p(y|m_1) - \log p(y|m_2) \\ &= \mathrm{LME}(m_1) - \mathrm{LME}(m_2) \; . \end{split}\]∎
Sources: - 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. 18; URL: https://www.sciencedirect.com/science/article/pii/S0165027018301468; DOI: 10.1016/j.jneumeth.2018.05.017.
Metadata: ID: P64 | shortcut: lbf-lme | author: JoramSoch | date: 2020-02-27, 20:51.