Index: The Book of Statistical ProofsModel Selection ▷ Bayesian model selection ▷ Log family evidence ▷ Derivation

Theorem: Let $f$ be a family of $M$ generative models $m_1, \ldots, m_M$ with model evidences $p(y \vert m_1), \ldots, p(y \vert m_M)$. Then, the log family evidence

\[\label{eq:LFE-term} \mathrm{LFE}(f) = \log p(y|f)\]

can be expressed as

\[\label{eq:LFE-marg} \mathrm{LFE}(f) = \log \sum_{i=1}^M p(y|m_i) \, p(m_i|f)\]

where $p(m_i \vert f)$ are the within-family prior model probabilities.

Proof: We will assume “prior addivivity”

\[\label{eq:fam-prior} p(f) = \sum_{i=1}^M p(m_i)\]

and “posterior additivity” for family probabilities:

\[\label{eq:fam-post} p(f|y) = \sum_{i=1}^M p(m_i|y)\]

Bayes’ theorem for the family evidence gives

\[\label{eq:fe-bayes-th} p(y|f) = \frac{p(f|y) \, p(y)}{p(f)} \; .\]

Applying \eqref{eq:fam-prior} and \eqref{eq:fam-post}, we have

\[\label{eq:fe-me} p(y|f) = \frac{\sum_{i=1}^M p(m_i|y) \, p(y)}{\sum_{i=1}^M p(m_i)} \; .\]

Bayes’ theorem for the model evidence gives

\[\label{eq:me-bayes-th} p(y|m_i) = \frac{p(m_i|y) \, p(y)}{p(m_i)}\]

which can be rearranged into

\[\label{eq:me-bayes-th-dev} p(m_i|y) \, p(y) = p(y|m_i) \, p(m_i) \; .\]

Plugging \eqref{eq:me-bayes-th-dev} into \eqref{eq:fe-me}, we have

\[\label{eq:fe-marg-qed} \begin{split} p(y|f) &= \frac{\sum_{i=1}^M p(y|m_i) \, p(m_i)}{\sum_{i=1}^M p(m_i)} \\ &= \sum_{i=1}^M p(y|m_i) \cdot \frac{p(m_i)}{\sum_{i=1}^M p(m_i)} \\ &= \sum_{i=1}^M p(y|m_i) \cdot \frac{p(m_i,f)}{p(f)} \\ &= \sum_{i=1}^M p(y|m_i) \cdot p(m_i|f) \; . \end{split}\]

Equation \eqref{eq:LFE-marg} follows by logarithmizing both sides of \eqref{eq:fe-marg-qed}.


Metadata: ID: P132 | shortcut: lfe-der | author: JoramSoch | date: 2020-07-13, 22:58.