Index: The Book of Statistical Proofs ▷ Model Selection ▷ Bayesian model selection ▷ Random-effects Bayesian model selection ▷ Posterior frequencies

Theorem: Consider model evidences $p(y_i \vert m)$ from $K$ models describing $N$ data sets $y = \left\lbrace y_1, \ldots, y_N \right\rbrace$ and assume that those data follow the group-level model specified by random-effects Bayesian model selection (RFX BMS):

\[\label{eq:bms-rfx} \begin{split} p(y_i|m_i = e_j) \quad &\mathrm{for} \quad i = 1,\ldots,N \quad \mathrm{and} \quad j = 1,\ldots,K \\ m_i \sim \mathrm{Cat}(r) \quad &\mathrm{for} \quad i = 1,\ldots,N \\ r \sim \mathrm{Dir}(\alpha) \; . \quad & \end{split}\]

Furthermore, let $p(r \vert y)$ be a Dirichlet posterior density over model frequencies, e.g. obtained via variational Bayes for RFX BMS, parametrized by concentration parameters $\alpha_n$:

\[\label{eq:bms-rfx-post} p(r|y) = \mathrm{Dir}(r; \alpha_n) \; .\]

Then, the posterior expected frequency, likeliest frequency and exceedance probability of the $j$-th model are given by:

\[\label{eq:bms-rfx-freq-post} \begin{split} \mathrm{EF}_j &= \frac{\alpha_{nj}}{\sum_{k=1}^K \alpha_{nk}} \\ \mathrm{LF}_j &= \frac{\alpha_{nj} - 1}{\sum_{k=1}^K \alpha_{nk} - K} \\ \mathrm{EP}_j &= \int_0^\infty \prod_{i \neq j} \left( \frac{\gamma(\alpha_{ni},q_j)}{\Gamma(\alpha_{ni})} \right) \, \frac{q_j^{\alpha_{nj}-1} \exp[-q_j]}{\Gamma(\alpha_{nj})} \, \mathrm{d}q_j \; . \end{split}\]

Proof: Let $\alpha_n$ be posterior concentration parameters specifying a Dirichlet probability distribution over possible combinations of model frequencies $r$:

\[\label{eq:bms-rfx-post-dir} r|y \sim \mathrm{Dir}(\alpha_n) \quad \mathrm{where} \quad 0 \leq r_j \leq 1 \quad \mathrm{for} \quad j = 1,\ldots,K \quad \mathrm{and} \quad \sum_{j=1}^K r_j = 1 \; .\]

1) The mean of the Dirichlet distribution is:

\[\label{eq:dir-mean} X \sim \mathrm{Dir}(\alpha) \quad \Rightarrow \quad \mathrm{E}(X_j) = \frac{\alpha_j}{\sum_{k=1}^K \alpha_k} \; .\]

Thus, the expected frequency according to the posterior distribution is:

\[\label{eq:bms-rfx-freq-exp} \mathrm{EF}_j = \frac{\alpha_{nj}}{\sum_{k=1}^K \alpha_{nk}} \quad \mathrm{for} \quad j = 1,\ldots,K \; .\]

2) The mode of the Dirichlet distribution is:

\[\label{eq:dir-mode} X \sim \mathrm{Dir}(\alpha) \quad \Rightarrow \quad \mathrm{mode}(X_j) = \frac{\alpha_j - 1}{\sum_{k=1}^K \alpha_k - K} \; .\]

Thus, the likeliest frequency according to posterior distribution is:

\[\label{eq:bms-rfx-freq-lik} \mathrm{LF}_j = \frac{\alpha_{nj} - 1}{\sum_{k=1}^K \alpha_{nk} - K} \quad \mathrm{for} \quad j = 1,\ldots,K \; .\]

3) The exceedance probabilities for the Dirichlet distribution are given by

\[\label{eq:dir-prob-exc} \begin{split} X \sim \mathrm{Dir}(\alpha) \quad \Rightarrow \quad \varphi(X_j) &= \mathrm{Pr}\left( \forall i \in \left\lbrace 1, \ldots, n | i \neq j \right\rbrace: \, X_j > X_i \right) \\ &= \int_0^\infty \prod_{i \neq j} \left( \frac{\gamma(\alpha_i,q_j)}{\Gamma(\alpha_i)} \right) \, \frac{q_j^{\alpha_j-1} \exp[-q_j]}{\Gamma(\alpha_j)} \, \mathrm{d}q_j \end{split}\]

where $\Gamma(x)$ is the gamma function and $\gamma(s,x)$ is the lower incomplete gamma function. Thus, the probability that, according to the posterior distribution, $r_j$ is higher than all other model frequencies $r_i$, $i \neq j$, is equal to:

\[\label{eq:bms-rfx-prob-exc} \mathrm{EP}_j = \int_0^\infty \prod_{i \neq j} \left( \frac{\gamma(\alpha_{ni},q_j)}{\Gamma(\alpha_{ni})} \right) \, \frac{q_j^{\alpha_{nj}-1} \exp[-q_j]}{\Gamma(\alpha_{nj})} \, \mathrm{d}q_j \quad \mathrm{for} \quad j = 1,\ldots,K \; .\]
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Metadata: ID: P545 | shortcut: bmsrfx-freqpost | author: JoramSoch | date: 2026-06-29, 17:36.