Marginal and conditional densities

Although the posterior density given by (1) provides all that is needed for inference about ${\mbox{\boldmath$\theta$}}$, there may be particular interest in a subset of ${\mbox{\boldmath$\theta$}},  {\mbox{\boldmath$\theta$}}_I$ of dimension² $t$. where

$${\mbox{\boldmath$\theta$}}_I = (\theta_{i_1}, \theta_{i_2}\ldots, \theta_{i_t})^T$$ (6)

and, if ${\mbox{\boldmath$\theta$}}$ is of dimension $k$,

$$I=(i_1, i_2,\ldots ,i_t)\subset(1,2,\ldots ,k).$$ (7)

Denoting the complement of ${\mbox{\boldmath$\theta$}}_I$ with respect to ${\mbox{\boldmath$\theta$}}$ as ${\mbox{\boldmath$\theta$}}_I^c$, then the Bayesian paradigm gives inference for ${\mbox{\boldmath$\theta$}}_I$ as the marginal posterior density

$$p({\mbox{\boldmath$\theta$}}_I\vert {\mbox{\boldmath$x$}})=... ...vert {\mbox{\boldmath$x$}})d{\mbox{\boldmath$\theta$}}_I^c,$$ (8)

where $\Theta_{k-t}$ is the parameter space supporting ${\mbox{\boldmath$\theta$}}_I^c$, i.e. the appropriate region of integration for the subset ${\mbox{\boldmath$\theta$}}_I^c$ of ${\mbox{\boldmath$\theta$}}$.

In a similar way, inference about ${\mbox{\boldmath$\theta$}}_I$, when ${\mbox{\boldmath$\theta$}}_I^c$ are known, is given by the conditional posterior density

$$p({\mbox{\boldmath$\theta$}}_I\vert{\mbox{\boldmath$\theta$... ...p({\mbox{\boldmath$\theta$}}_I^c\vert {\mbox{\boldmath$x$}})}$$ (9)