Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Count data ▷ General contingency table ▷ Log Bayes factor

Theorem: Consider a $k \times l$ contingency table, denote observed counts as

\[\label{eq:y} \left\lbrace y_{ij} \, | \, i \in \left\lbrace 1,\ldots,k \right\rbrace, \, j \in \left\lbrace 1,\ldots,l \right\rbrace \right\rbrace\]

and assume that these counts come a multinomial distribution with unknown cell probabilities:

\[\label{eq:y-p} y = \left[ y_{11}, \ldots, y_{kl} \right] \sim \mathrm{Mult}(n; \left[ p_{11}, \ldots, p_{kl} \right]) \; .\]

Let $m_0$ be a null model which assumes statistical independence of $A$ and $B$, i.e. cell probabilities

\[\label{eq:m0-p} m_0: \quad p_{ij} = r_i s_j, \quad i = 1,\ldots,k, \quad j = 1,\ldots,l\]

and Dirichlet prior distributions over the model parameters $r$ and $s$:

\[\label{eq:m0-rs} \begin{split} r &\sim \mathrm{Dir}(\alpha_0) \\ s &\sim \mathrm{Dir}(\beta_0) \; . \end{split}\]

Let $m_1$ be an alternative model which allows for arbitrary parametrization of cell probabilities

\[\label{eq:m1-p} m_1: \quad p_{ij} = r_{ij}, \quad i = 1,\ldots,k, \quad j = 1,\ldots,l\]

and Dirichlet prior distributions over the model parameters $r$:

\[\label{eq:m1-r} \mathrm{vec}(r) \sim \mathrm{Dir}\left( \mathrm{vec}\left(\gamma^{(0)}\right) \right) \; .\]

Then, the log Bayes factor in favor of $m_1$ against $m_0$ is

\[\label{eq:ct-lbf} \begin{split} \mathrm{LBF}_{10} &= \log \Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right) - \log \Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right) \\ &+ \sum_{i=1}^k \sum_{j=1}^l \log \Gamma\left( \gamma_{ij}^{(n)} \right) - \sum_{i=1}^k \sum_{j=1}^l \log \Gamma\left( \gamma_{ij}^{(0)} \right) \\ &- \log \Gamma\left( \sum_{i=1}^k \alpha_{0i} \right) + \log \Gamma\left( \sum_{i=1}^k \alpha_{ni} \right) - \sum_{i=1}^k \log \Gamma\left( \alpha_{ni} \right) + \sum_{i=1}^k \log \Gamma\left( \alpha_{0i} \right) \\ &- \log \Gamma\left( \sum_{j=1}^l \beta_{0j} \right) + \log \Gamma\left( \sum_{j=1}^l \beta_{nj} \right) - \sum_{j=1}^l \log \Gamma\left( \beta_{nj} \right) + \sum_{j=1}^l \log \Gamma\left( \beta_{0j} \right) \; . \end{split}\]

where $\Gamma(x)$ is the gamma function; $\alpha_n$ and $\beta_n$ are the posterior hyperparameters of $m_0$; and $\gamma_{ij}^{(n)}$ are the posterior hyperparameters of $m_1$.

Proof: The log Bayes factor is equal to the difference of two log model evidences:

\[\label{eq:lbf-lme} \mathrm{LBF}_{12} = \mathrm{LME}(m_1) - \mathrm{LME}(m_2) \; .\]

The log model evidence of $m_0$ is

\[\label{eq:m0-lme} \begin{split} \mathrm{LME}(m_0) &= \log \Gamma(n+1) - \sum_{i=1}^k \sum_{j=1}^l \log \Gamma(y_{ij}+1) \\ &+ \log \Gamma\left( \sum_{i=1}^k \alpha_{0i} \right) - \log \Gamma\left( \sum_{i=1}^k \alpha_{ni} \right) + \sum_{i=1}^k \log \Gamma\left( \alpha_{ni} \right) - \sum_{i=1}^k \log \Gamma\left( \alpha_{0i} \right) \\ &+ \log \Gamma\left( \sum_{j=1}^l \beta_{0j} \right) - \log \Gamma\left( \sum_{j=1}^l \beta_{nj} \right) + \sum_{j=1}^l \log \Gamma\left( \beta_{nj} \right) - \sum_{j=1}^l \log \Gamma\left( \beta_{0j} \right) \end{split}\]

and the log model evidence of $m_1$ is

\[\label{eq:m1-lme} \begin{split} \mathrm{LME}(m_1) &= \log \Gamma(n+1) - \sum_{i=1}^k \sum_{j=1}^l \log \Gamma(y_{ij}+1) \\ &+ \log \Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right) - \log \Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right) \\ &+ \sum_{i=1}^k \sum_{j=1}^l \log \Gamma\left( \gamma_{ij}^{(n)} \right) - \sum_{i=1}^k \sum_{j=1}^l \log \Gamma\left( \gamma_{ij}^{(0)} \right) \; . \end{split}\]

Subtracting the two LMEs from each other, the LBF emerges as

\[\label{eq:ct-lbf-qed} \begin{split} \mathrm{LBF}_{10} &= \log \Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right) - \log \Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right) \\ &+ \sum_{i=1}^k \sum_{j=1}^l \log \Gamma\left( \gamma_{ij}^{(n)} \right) - \sum_{i=1}^k \sum_{j=1}^l \log \Gamma\left( \gamma_{ij}^{(0)} \right) \\ &- \log \Gamma\left( \sum_{i=1}^k \alpha_{0i} \right) + \log \Gamma\left( \sum_{i=1}^k \alpha_{ni} \right) - \sum_{i=1}^k \log \Gamma\left( \alpha_{ni} \right) + \sum_{i=1}^k \log \Gamma\left( \alpha_{0i} \right) \\ &- \log \Gamma\left( \sum_{j=1}^l \beta_{0j} \right) + \log \Gamma\left( \sum_{j=1}^l \beta_{nj} \right) - \sum_{j=1}^l \log \Gamma\left( \beta_{nj} \right) + \sum_{j=1}^l \log \Gamma\left( \beta_{0j} \right) \; . \end{split}\]
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Metadata: ID: P554 | shortcut: ct-lbf | author: JoramSoch | date: 2026-09-18, 11:57.