Predictive densities

Similarly, inference about future observations ${\mbox{\boldmath$y$}}$, having observed ${\mbox{\boldmath$x$}}$ is given by the posterior predictive density

$$p({\mbox{\boldmath$y$}}\vert{\mbox{\boldmath$x$}} )=\int_{{... ...a$}}\vert{\mbox{\boldmath$x$}}) d{\mbox{\boldmath$\theta$}}$$ (10)

where $p({\mbox{\boldmath$\theta$}}\vert{\mbox{\boldmath$x$}})$ is the full posterior distribution. Note that the predictive density $p({\mbox{\boldmath$y$}}\vert{\mbox{\boldmath$x$}})$ provides inference about ${\mbox{\boldmath$y$}}$ conditional only on the observed data ${\mbox{\boldmath$x$}}$, without reference to any specific value of ${\mbox{\boldmath$\theta$}}$. This is in contrast to an estimative approach which might give $p({\mbox{\boldmath$y$}}\vert{\mbox{\boldmath$\hat \theta$}})$ where ${\mbox{\boldmath$\hat \theta$}}$ is a point estimate (e.g. maximum likelihood).