Index: The Book of Statistical ProofsModel Selection ▷ Bayesian model selection ▷ Log Bayes factor ▷ Calculation from log model evidences

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

$\label{eq:LBF} \mathrm{LBF}_{12} = \log \mathrm{BF}_{12} = \log \frac{p(y|m_1)}{p(y|m_2)} \; .$

With the definition of the log model evidence

$\label{eq:LME} \mathrm{LME}(m) = \log p(y|m)$

the log Bayes factor can be expressed as:

Resolving the logarithm and applying the definition of the log model evidence, we finally have:

$\label{eq:LBF-LME-qed} \begin{split} \mathrm{LBF}_{12} &= \log p(y|m_1) - \log p(y|m_2) \\ &= \mathrm{LME}(m_1) - \mathrm{LME}(m_2) \; . \end{split}$
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Metadata: ID: P64 | shortcut: lbf-lme | author: JoramSoch | date: 2020-02-27, 20:51.