Index: The Book of Statistical Proofs ▷ Statistical Models ▷ Count data ▷ General contingency table ▷ Log model evidences

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 model evidence of $m_0$ is

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

where

\[\label{eq:m0-post-par} \alpha_{ni} = \alpha_{0i} + \sum_{j=1}^l y_{ij}, \quad \beta_{nj} = \beta_{0j} + \sum_{i=1}^k y_{ij}, \quad i = 1,\ldots,k, \quad j = 1,\ldots,l\]

and the log model evidence of $m_1$ is

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

where

\[\label{eq:m1-post-par} \gamma_{ij}^{(n)} = \gamma_{ij}^{(0)} + y_{ij}, \quad i = 1,\ldots,k, \quad j = 1,\ldots,l\]

and $\Gamma(x)$ denotes the gamma function.

Proof: The likelihood function for observed counts $y$, given cell probabilities $p$ is given by the probability mass function of the multinomial distribution:

\[\label{eq:p-y-p} \begin{split} \mathrm{p}(y|p) &= \mathrm{Mult}(n; \left[ p_{11}, \ldots, p_{kl} \right]) \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \prod_{i=1}^k \prod_{j=1}^l {p_{ij}}^{y_{ij}} \; . \end{split}\]

The null model has $(k-1) + (l-1)$ free parameters ($r_1,\ldots,r_{k-1}$ plus $s_1,\ldots,s_{l-1}$) while the alternative has $k \cdot l - 1$ free parameters ($r_1,\ldots,r_{k,l-1}$).

1) For the null model $m_0$, with \eqref{eq:m0-p}, the likelihood function \eqref{eq:p-y-p} becomes:

\[\label{eq:m0-p-y-rs} p(y|r,s) = {n \choose {y_{11}, \ldots, y_{kl}}} \prod_{i=1}^k \prod_{j=1}^l (r_i s_j)^{y_{ij}} \; .\]

From \eqref{eq:m0-rs}, the prior densities for $r$ and $s$ are:

\[\label{eq:m0-p-rs} \begin{split} p(r) &= \mathrm{Dir}(r; \alpha_0) = \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \prod_{i=1}^k {r_i}^{\alpha_{0i}-1} \\ p(s) &= \mathrm{Dir}(s; \beta_0) = \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\prod_{j=1}^l \Gamma(\beta_{0j})} \prod_{j=1}^l {s_j}^{\beta_{0j}-1} \; . \end{split}\]

Thus, combining \eqref{eq:m0-p-y-rs} and \eqref{eq:m0-p-rs}, the joint likelihood function is:

\[\label{eq:m0-p-yrs-s1} \begin{split} p(y,r,s) &= p(y|r,s) \cdot p(r) \cdot p(s) \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \prod_{i=1}^k \prod_{j=1}^l (r_i s_j)^{y_{ij}} \cdot \\ &\hphantom{=} \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \prod_{i=1}^k {r_i}^{\alpha_{0i}-1} \cdot \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\prod_{j=1}^l \Gamma(\beta_{0j})} \prod_{j=1}^l {s_j}^{\beta_{0j}-1} \; . \end{split}\]

Collecting identical variables, we obtain:

\[\label{eq:m0-p-yrs-s2} \begin{split} p(y,r,s) &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \, \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\prod_{j=1}^l \Gamma(\beta_{0j})} \cdot \\ &\phantom{=} \prod_{i=1}^k {r_i}^{\alpha_{0i} + \sum_{j=1}^l y_{ij} - 1} \cdot \prod_{j=1}^l {s_j}^{\beta_{0j} + \sum_{i=1}^k y_{ij} - 1} \\ &\overset{\eqref{eq:m0-post-par}}{=} {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\Gamma\left( \sum_{i=1}^k \alpha_{ni} \right)} \, \frac{\prod_{i=1}^k \Gamma(\alpha_{ni})}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \, \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\Gamma\left( \sum_{j=1}^l \beta_{nj} \right)} \, \frac{\prod_{j=1}^l \Gamma(\beta_{nj})}{\prod_{j=1}^l \Gamma(\beta_{0j})} \cdot \\ &\phantom{=} \frac{\Gamma\left( \sum_{i=1}^k \alpha_{ni} \right)}{\prod_{i=1}^k \Gamma(\alpha_{ni})} \prod_{i=1}^k {r_i}^{\alpha_{ni}-1} \cdot \frac{\Gamma\left( \sum_{j=1}^l \beta_{nj} \right)}{\prod_{j=1}^l \Gamma(\beta_{nj})} \prod_{j=1}^l {s_j}^{\beta_{nj}-1} \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\Gamma\left( \sum_{i=1}^k \alpha_{ni} \right)} \, \frac{\prod_{i=1}^k \Gamma(\alpha_{ni})}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \, \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\Gamma\left( \sum_{j=1}^l \beta_{nj} \right)} \, \frac{\prod_{j=1}^l \Gamma(\beta_{nj})}{\prod_{j=1}^l \Gamma(\beta_{0j})} \cdot \\ &\phantom{=} \mathrm{Dir}(r; \alpha_0) \cdot \mathrm{Dir}(s; \beta_0) \; . \end{split}\]

With that, we can integrate out $r$ and $s$:

\[\label{eq:m0-p-y-m} \begin{split} p(y|m_0) &= \iint p(y,r,s) \, \mathrm{d}r \, \mathrm{d}s \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\Gamma\left( \sum_{i=1}^k \alpha_{ni} \right)} \, \frac{\prod_{i=1}^k \Gamma(\alpha_{ni})}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \, \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\Gamma\left( \sum_{j=1}^l \beta_{nj} \right)} \, \frac{\prod_{j=1}^l \Gamma(\beta_{nj})}{\prod_{j=1}^l \Gamma(\beta_{0j})} \cdot \\ &\phantom{=} \iint \mathrm{Dir}(r; \alpha_0) \cdot \mathrm{Dir}(s; \beta_0) \, \mathrm{d}r \, \mathrm{d}s \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\Gamma\left( \sum_{i=1}^k \alpha_{ni} \right)} \, \frac{\prod_{i=1}^k \Gamma(\alpha_{ni})}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \, \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\Gamma\left( \sum_{j=1}^l \beta_{nj} \right)} \, \frac{\prod_{j=1}^l \Gamma(\beta_{nj})}{\prod_{j=1}^l \Gamma(\beta_{0j})} \\ &= \frac{n!}{\prod_{i=1}^k \prod_{j=1}^l y_{ij}!} \, \frac{\Gamma\left( \sum_{i=1}^k \alpha_{0i} \right)}{\Gamma\left( \sum_{i=1}^k \alpha_{ni} \right)} \, \frac{\prod_{i=1}^k \Gamma(\alpha_{ni})}{\prod_{i=1}^k \Gamma(\alpha_{0i})} \, \frac{\Gamma\left( \sum_{j=1}^l \beta_{0j} \right)}{\Gamma\left( \sum_{j=1}^l \beta_{nj} \right)} \, \frac{\prod_{j=1}^l \Gamma(\beta_{nj})}{\prod_{j=1}^l \Gamma(\beta_{0j})} \; . \end{split}\]

Finally, logarithmizing both sides and applying the relation $n! = \Gamma(n+1)$, we obtain:

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

2) For the alternative $m_1$, with \eqref{eq:m1-p}, the likelihood function \eqref{eq:p-y-p} becomes:

\[\label{eq:m1-p-y-r} p(y|r) = {n \choose {y_{11}, \ldots, y_{kl}}} \prod_{i=1}^k \prod_{j=1}^l {r_{ij}}^{y_{ij}} \; .\]

From \eqref{eq:m1-r}, the prior density for $r$ ps:

\[\label{eq:m1-p-r} p(\mathrm{vec}(r)) = \mathrm{Dir}\left(\mathrm{vec}(r); \mathrm{vec}\left(\gamma^{(0)}\right)\right) = \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \prod_{i=1}^k \prod_{j=1}^l {r_{ij}}^{\gamma_{ij}^{(0)}-1} \; .\]

Thus, combining \eqref{eq:m1-p-r} and \eqref{eq:m1-p-r}, the joint likelihood function is:

\[\label{eq:m1-p-yr-s1} \begin{split} p(y,r) &= p(y|r) \cdot p(r) \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \prod_{i=1}^k \prod_{j=1}^l {r_{ij}}^{y_{ij}} \cdot \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \prod_{i=1}^k \prod_{j=1}^l {r_{ij}}^{\gamma_{ij}^{(0)}-1} \; . \end{split}\]

Collecting identical variables, we obtain:

\[\label{eq:m1-p-yr-s2} \begin{split} p(y,r) &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \cdot \prod_{i=1}^k \prod_{j=1}^l {r_{ij}}^{\gamma_{ij}^{(0)}+y_{ij}-1} \\ &\overset{\eqref{eq:m1-post-par}}{=} {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right)} \, \frac{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(n)}\right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \cdot \\ &\hphantom{=} \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(n)}\right)} \prod_{i=1}^k \prod_{j=1}^l {r_{ij}}^{\gamma_{ij}^{(n)}-1} \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right)} \, \frac{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(n)}\right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \cdot \\ &\hphantom{=} \mathrm{Dir}\left(\mathrm{vec}(r); \mathrm{vec}\left(\gamma^{(n)}\right)\right) \; . \end{split}\]

With that, we can integrate out $r$:

\[\label{eq:m1-p-y-m} \begin{split} p(y|m_1) &= \int p(y,r) \, \mathrm{d}r \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right)} \, \frac{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(n)}\right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \cdot \\ &\hphantom{=} \int \mathrm{Dir}\left(\mathrm{vec}(r); \mathrm{vec}\left(\gamma^{(n)}\right)\right) \, \mathrm{d}r \\ &= {n \choose {y_{11}, \ldots, y_{kl}}} \, \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right)} \, \frac{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(n)}\right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \\ &= \frac{n!}{\prod_{i=1}^k \prod_{j=1}^l y_{ij}!} \, \frac{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(0)} \right)}{\Gamma\left( \sum_{i=1}^k \sum_{j=1}^l \gamma_{ij}^{(n)} \right)} \, \frac{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(n)}\right)}{\prod_{i=1}^k \prod_{j=1}^l \Gamma\left(\gamma_{ij}^{(0)}\right)} \; . \end{split}\]

Finally, logarithmizing both sides and applying the relation $n! = \Gamma(n+1)$, we obtain:

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

This completes the proof.

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Metadata: ID: P553 | shortcut: ct-lme | author: JoramSoch | date: 2026-09-18, 11:29.