Finite-model posterior probabilities and Bayes-factor odds

Form the measurable disjoint union $E={⨆}_{i\in I}(\{i\}\times {\mathrm{\Theta }}_i)$ and put $\mathrm{\Pi }(A)={\sum }_i{w}_i{\mathrm{\Pi }}_i(A_i)$, where $A_i=\{\theta :(i,\theta )\in A\}$. This is a probability measure because each ${\mathrm{\Pi }}_i$ is a probability measure and the finite weights sum to one. On $(i,\theta )$ use the observation kernel $Q_i(\theta ,\cdot )$ with density $\mathrm{\ell }((i,\theta ),y)=f_i(\theta ,y)$. Each section is a probability kernel, and finite piecewise measurability plus the stipulated joint measurability makes $\mathrm{\ell }$ a specified jointly measurable density; [F5] warns that separate density versions alone would not suffice. Its prior-predictive density is $m(y)={\sum }_i{w}_i{m}_i(y)$. The given finite-positive hypotheses make $m(y)$ finite and positive, so [F2] confirms its likelihood-weighted prior has a probability normalization.

Apply [F3] to $(E,\mathrm{\Pi })$ and this dominated observation kernel. Define its global posterior kernel $K(y,\cdot )$ by the normalized likelihood-weighted prior on $G=\{y:0<m(y)<\mathrm{\infty }\}$ and by the fixed prior $\mathrm{\Pi }$ on $G^c$, exactly as in [F3]. The set $G$ is measurable, and [F3] states that $G^c$ has predictive probability zero. At the specified fiber, where every model evidence is finite and positive, this posterior kernel assigns the tagged slice $\{i\}\times {\mathrm{\Theta }}_i$ mass ${\int }_{\{i\}\times {\mathrm{\Theta }}_i}\mathrm{\ell }((l,\theta ),y)\mathrm{\Pi }(dl,d\theta )\mathrm{/}m(y)=w_i{\int }_{{\mathrm{\Theta }}_i}{f}_i(\theta ,y){\mathrm{\Pi }}_i(d\theta )\mathrm{/}({\sum }_l{w}_l{m}_l(y))=w_i{m}_i(y)\mathrm{/}({\sum }_l{w}_l{m}_l(y))$. The model-index map $(i,\theta )\mapsto i$ is measurable, so the image of this regular conditional law is the stated posterior model probability.

For any measurable data event $B$ with positive predictive probability, the global kernel in step 2.1 has the event identity ${\int }_{B}K(y,\{i\}\times {\mathrm{\Theta }}_i)m(y)d\mu (y)=w_i{\int }_B{m}_i(y)d\mu (y)$ by [F3]. The integral on the left means integration against the predictive measure $md\mu$, so no product with an infinite density on an exceptional fiber is evaluated pointwise. Dividing by the event probability agrees with [F4]. This event identity uses the global kernel, whereas the displayed odds claim only concerns the given finite-positive fiber. At that fiber, divide the positive finite probabilities from step 2.1 for $j$ and $i$; [F1] gives $(w_j\mathrm{/}{w}_i)(m_j\mathrm{/}{m}_i)=(w_j\mathrm{/}{w}_i)B{F}_{ji}$.