Index: The Book of Statistical ProofsStatistical Models ▷ Count data ▷ Multinomial observations ▷ Posterior probability

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 two statistical models, one assuming that each $p_j$ is $1/k$ (null model), the other imposing a Dirichlet distribution as the prior distribution on the model parameters $p_1, \ldots, p_k$ (alternative):

$\label{eq:Mult-m01} \begin{split} m_0&: \; y \sim \mathrm{Mult}(n,p), \; p = [1/k, \ldots, 1/k] \\ m_1&: \; y \sim \mathrm{Mult}(n,p), \; p \sim \mathrm{Dir}(\alpha_0) \; . \end{split}$

Then, the posterior probability of the alternative model is given by

$\label{eq:Mult-PP1} p(m_1|y) = \frac{1}{1 + k^{-n} \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{0j})}{\prod_{j=1}^k \Gamma(\alpha_{nj})}}$

where $\Gamma(x)$ is the gamma function and $\alpha_n$ are the posterior hyperparameters for multinomial observations which are functions of the numbers of observations $y_1, \ldots, y_k$.

$\label{eq:PP-LBF} p(m_1|y) = \frac{\exp(\mathrm{LBF}_{12})}{\exp(\mathrm{LBF}_{12}) + 1} \; .$ $\label{eq:Mult-LBF10} \begin{split} \mathrm{LBF}_{10} &= \log \Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right) - \log \Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right) \\ &+ \sum_{j=1}^k \log \Gamma(\alpha_{nj}) - \sum_{j=1}^k \log \Gamma(\alpha_{0j}) - n \log \left( \frac{1}{k} \right) \end{split}$

and the corresponding Bayes factor, i.e. exponentiated log Bayes factor, is equal to

$\label{eq:Mult-BF10} \mathrm{BF}_{10} = \exp(\mathrm{LBF}_{10}) = k^n \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \; .$

Thus, the posterior probability of the alternative, assuming a prior distribution over the probabilities $p_1, \ldots, p_k$, compared to the null model, assuming fixed probabilities $p = [1/k, \ldots, 1/k]$, follows as

$\label{eq:Mult-PP1-qed} \begin{split} p(m_1|y) &\overset{\eqref{eq:PP-LBF}}{=} \frac{\exp(\mathrm{LBF}_{10})}{\exp(\mathrm{LBF}_{10}) + 1} \\ &\overset{\eqref{eq:Mult-BF10}}{=} \frac{k^n \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\prod_{j=1}^k \Gamma(\alpha_{0j})}}{k^n \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\prod_{j=1}^k \Gamma(\alpha_{0j})} + 1} \\ &= \frac{k^n \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\prod_{j=1}^k \Gamma(\alpha_{0j})}}{k^n \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{nj})}{\prod_{j=1}^k \Gamma(\alpha_{0j})} \left( 1 + k^{-n} \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{0j})}{\prod_{j=1}^k \Gamma(\alpha_{nj})} \right)} \\ &= \frac{1}{1 + k^{-n} \cdot \frac{\Gamma \left( \sum_{j=1}^{k} \alpha_{nj} \right)}{\Gamma \left( \sum_{j=1}^{k} \alpha_{0j} \right)} \cdot \frac{\prod_{j=1}^k \Gamma(\alpha_{0j})}{\prod_{j=1}^k \Gamma(\alpha_{nj})}} \end{split}$ $\label{eq:Mult-post-par} \alpha_n = \alpha_0 + y, \quad \text{i.e.} \quad \alpha_{nj} = \alpha_{0j} + y_j$

with the numbers of observations $y_1, \ldots, y_k$.

Sources:

Metadata: ID: P388 | shortcut: mult-pp | author: JoramSoch | date: 2022-12-02, 18:03.