Index: The Book of Statistical Proofs ▷ Model Selection ▷ Bayesian model selection ▷ Posterior model probability ▷ Calculation from Bayes factors

Theorem: Let $m_0, m_1, \ldots, m_M$ be $M+1$ statistical models with model evidences $p(y \vert m_0), p(y \vert m_1), \ldots, p(y \vert m_M)$. Then, the posterior model probabilities of the models $m_1, \ldots, m_M$ are given by

\[\label{eq:PMP-BF} p(m_i|y) = \frac{\mathrm{BF}_{i,0} \cdot \alpha_i}{\sum_{j=1}^{M} \mathrm{BF}_{j,0} \cdot \alpha_j}, \quad i = 1,\ldots,M \; ,\]

where $\mathrm{BF}_{i,0}$ is the Bayes factor comparing model $m_i$ with $m_0$ and $\alpha_i$ is the prior odds ratio of model $m_i$ against $m_0$.

Proof: Define the Bayes factor for $m_i$

\[\label{eq:BF-i0} \mathrm{BF}_{i,0} = \frac{p(y|m_i)}{p(y|m_0)}\]

and prior odds ratio of $m_i$ against $m_0$

\[\label{eq:prior-i0} \alpha_i = \frac{p(m_i)}{p(m_0)} \; .\]

The posterior model probability of $m_i$ is given by

\[\label{eq:PMP-s1} p(m_i|y) = \frac{p(y|m_i) \cdot p(m_i)}{\sum_{j=1}^{M} p(y|m_j) \cdot p(m_j)} \; .\]

Now applying \eqref{eq:BF-i0} and \eqref{eq:prior-i0} to \eqref{eq:PMP-s1}, we have

\[\label{eq:PMP-s2} \begin{split} p(m_i|y) &= \frac{ \mathrm{BF}_{i,0} \, p(y|m_0) \cdot \alpha_i \, p(m_0)}{\sum_{j=1}^{M} \mathrm{BF}_{j,0} \, p(y|m_0) \cdot \alpha_j \, p(m_0)} \\ &= \frac{\left[ p(y|m_0) \, p(m_0) \right] \mathrm{BF}_{i,0} \cdot \alpha_i}{\left[ p(y|m_0) \, p(m_0) \right] \sum_{j=1}^{M} \mathrm{BF}_{j,0} \cdot \alpha_j} \; , \end{split}\]

such that

\[\label{eq:PMP-BF-qed} p(m_i|y)= \frac{\mathrm{BF}_{i,0} \cdot \alpha_i}{\sum_{j=1}^{M} \mathrm{BF}_{j,0} \cdot \alpha_j} \; .\]
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Metadata: ID: P74 | shortcut: pmp-bf | author: JoramSoch | date: 2020-03-03, 13:13.