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

Definition: Let $\mathcal{M}$ be a set of $K$ statistical models describing a group of $N$ independent data sets $y = \left\lbrace y_1, \ldots, y_N \right\rbrace$ with model evidences $p(y_i \vert m)$ for $i = 1,\ldots,N$ and $m \in \mathcal{M}$.

Assume that $m_i \in \left\lbrace 0, 1 \right\rbrace^K$ is a discrete random vector specifying the generating model of $y_i$ and let $m_i = e_j$ denote the situation that the $j$-th model generated the $i$-th data set. Then, the model evidences $p(y_i \vert m)$ constitute a probability distribution specifying the how likely it is to observe a specific data set given a specific model:

\[\label{eq:p-y-m} p(y_i|m_i = e_j) \quad \mathrm{for} \quad i = 1,\ldots,N \quad \mathrm{and} \quad j = 1,\ldots,K \; .\]

Moreover, let the generating model for each data set come from a categorical distribution with unknown category probabilities, or model frequencies $r = [r_1, \ldots, r_K] \in [0, 1]^K$:

\[\label{eq:p-m-r} m_i \sim \mathrm{Cat}(r) \quad \mathrm{for} \quad i = 1,\ldots,N \; .\]

Finally, let the prior distribution over model frequencies be a Dirichlet distribution with known concentration $\alpha = [\alpha_1, \ldots, \alpha_K] \in \mathbb{R}^K$:

\[\label{eq:p-r-a} r \sim \mathrm{Dir}(\alpha) \; .\]

Taken together, the probability distributions specified by \eqref{eq:p-y-m} and \eqref{eq:p-m-r} and \eqref{eq:p-r-a}, treating generative model as a random effect, are called random-effects Bayesian model selection (RFX BMS).

 
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Metadata: ID: D235 | shortcut: bmsrfx | author: JoramSoch | date: 2026-06-24, 11:43.