Index: The Book of Statistical ProofsStatistical Models ▷ Categorical data ▷ Multinomial observations ▷ Log model evidence

Theorem: Let $y = [y_1, \ldots, y_k]$ be the number of observations in $k$ categories resulting from $n$ independent trials with unknown category probabilities $p = [p_1, \ldots, p_k]$, such that $y$ follows a multinomial distribution:

\[\label{eq:Mult} y \sim \mathrm{Mult}(n,p) \; .\]

Moreover, assume a Dirichlet prior distribution over the model parameter $p$:

\[\label{eq:Mult-prior} \mathrm{p}(p) = \mathrm{Dir}(p; \alpha_0) \; .\]

Then, the log model evidence for this model is

\[\label{eq:Mult-LME} \begin{split} \log \mathrm{p}(y|m) &= \log \Gamma(n+1) - \sum_{j=1}^{k} \log \Gamma(k_j+1) \\ &+ \log \Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right) - \log \Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right) \\ &+ \sum_{j=1}^k \log \Gamma(\alpha_{nj}) - \sum_{j=1}^k \log \Gamma(\alpha_{0j}) \; . \end{split}\]

and the posterior hyperparameters are given by

\[\label{eq:Mult-post-par} \alpha_{nj} = \alpha_{0j} + y_j, \; j = 1,\ldots,k \; .\]

Proof: With the probability mass function of the multinomial distribution, the likelihood function implied by \eqref{eq:Mult} is given by

\[\label{eq:Mult-LF} \mathrm{p}(y|p) = {n \choose {y_1, \ldots, y_k}} \prod_{j=1}^{k} {p_j}^{y_j} \; .\]

Combining the likelihood function \eqref{eq:Mult-LF} with the prior distribution \eqref{eq:Mult-prior}, the joint likelihood of the model is given by

\[\label{eq:Mult-JL} \begin{split} \mathrm{p}(y,p) &= \mathrm{p}(y|p) \, \mathrm{p}(p) \\ &= {n \choose {y_1, \ldots, y_k}} \prod_{j=1}^{k} {p_j}^{y_j} \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \prod_{j=1}^{k} {p_j}^{\alpha_{0j}-1} \\ &= {n \choose {y_1, \ldots, y_k}} \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \prod_{j=1}^{k} {p_j}^{\alpha_{0j}+y_j-1} \; . \end{split}\]

Note that the model evidence is the marginal density of the joint likelihood:

\[\label{eq:Mult-ME-s1} \mathrm{p}(y) = \int \mathrm{p}(y,p) \, \mathrm{d}p \; .\]

Setting $\alpha_{nj} = \alpha_{0j} + y_j$, the joint likelihood can also be written as

\[\label{eq:Mult-JL-s2} \mathrm{p}(y,p) = {n \choose {y_1, \ldots, y_k}} \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \, \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})} {\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \, \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)}{\prod_{j=1}^k \Gamma(\alpha_{nj})} \prod_{j=1}^{k} {p_j}^{\alpha_{nj}-1} \; .\]

Using the probability density function of the Dirichlet distribution, $p$ can now be integrated out easily

\[\label{eq:Mult-ME-s2} \begin{split} \mathrm{p}(y) &= \int {n \choose {y_1, \ldots, y_k}} \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \, \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \, \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)}{\prod_{j=1}^k \Gamma(\alpha_{nj})} \prod_{j=1}^{k} {p_j}^{\alpha_{nj}-1} \, \mathrm{d}p \\ &= {n \choose {y_1, \ldots, y_k}} \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \, \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \int \mathrm{Dir}(p; \alpha_n) \, \mathrm{d}p \\ &= {n \choose {y_1, \ldots, y_k}} \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \, \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \; , \end{split}\]

such that the log model evidence (LME) is shown to be

\[\label{eq:Mult-LME-s1} \begin{split} \log \mathrm{p}(y|m) = \log {n \choose {y_1, \ldots, y_k}} &+ \log \Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right) - \log \Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right) \\ &+ \sum_{j=1}^k \log \Gamma(\alpha_{nj}) - \sum_{j=1}^k \log \Gamma(\alpha_{0j}) \; . \end{split}\]

With the definition of the multinomial coefficient

\[\label{eq:mult-coeff} {n \choose {k_1, \ldots, k_m}} = \frac{n!}{k_1! \cdot \ldots \cdot k_m!}\]

and the definition of the gamma function

\[\label{eq:gam-fct} \Gamma(n) = (n-1)! \; ,\]

the LME finally becomes

\[\label{eq:Mult-LME-s2} \begin{split} \log \mathrm{p}(y|m) &= \log \Gamma(n+1) - \sum_{j=1}^{k} \log \Gamma(k_j+1) \\ &+ \log \Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right) - \log \Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right) \\ &+ \sum_{j=1}^k \log \Gamma(\alpha_{nj}) - \sum_{j=1}^k \log \Gamma(\alpha_{0j}) \; . \end{split}\]

Metadata: ID: P81 | shortcut: mult-lme | author: JoramSoch | date: 2020-03-11, 15:17.