\begin{frame}
\begin{itemize}
\item
Time effects can also be incorporated in this framework, to allow for temporal 
evolution over time.

\item
There is a variety of Bayesian spatio-temporal models that may be used in ecological studies,
but here we introduce relatively simple but practically useful extension to the proposed framework.

\item
Specifically, we introduce both purely temporal effects and trends as well as an interaction effect
between space and time.

\end{itemize}
\end{frame}


\begin{frame}
So the all-singing, all-dancing `full monty' Bayesian model is:
\begin{eqnarray*}
y_{it}|\phi_i,\nu_i,\delta_t &\sim& \mbox{Poisson}(\mu_i)=\mbox{Poisson}(e_{it} \rho_{it})\\
\log\mu_{it}&=&\log e_{it} + \log \rho_{it} = \log e_{it} + \beta_0+\sum_{j=1}^p\beta_jx_{ij} + 
\gamma t + \delta_t +\phi_i+\nu_i + \eta_{it}\\
\delta_t &\sim&\mbox{Normal}(\delta_{t-1},\sigma^2_\delta) \\
\eta_{it}&\sim$ \mbox{Normal}(\alpha_i\eta_{it-1},\sigma^2_\eta) \\
\alpha_i&\sim&\mbox{Normal}(0,\sigma^2_\alpha)\\
\phi_i&\sim&\mbox{Normal}(0,\sigma^2_\phi)\\
\nu_i|\nu_{j\neq i} &\sim& \mbox{N}\left(\frac{\sum_{j\neq i} w_{ij}\nu_j}
{\sum_{j\neq i} w_{ij}},\frac{\sigma^2_\nu}{\sum_{j\neq i} w_{ij}}\right)\\
\beta_0&\sim&\mbox{U}(-\infty,+\infty)\\
\beta_j&\sim&\mbox{N}(0,a);\quad j=1,\ldots,p\\
\gamma&\sim&\mbox{N}(0,a)\\
1/\sigma^2_\phi&\sim&\mbox{Gamma}(b,c)\\
1/\sigma^2_\nu&\sim&\mbox{Gamma}(d,e)\\
1/\sigma^2_\delta&\sim&\mbox{Gamma}(f,g)\\
1/\sigma^2_\delta&\sim&\mbox{Gamma}(h,k)\\
\end{eqnarray*}
(where $a,b,c,d,e,f,g,h,k$ are suitably chosen fixed constants).\\
Note that $\alpha_i$ could also be spatially structured.
\end{frame}
