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\section*{A time series with seasonal component: the drivers data}
This example is  taken from \cite[Sec 4.4.2]{book80}. 

    The data consist in monthly counts of car drivers in Great Britain
    killed or seriously injured in car accidents from January 1969 to
    December 1984. The time series has $n_d=192$ data points and
    exhibits a strong seasonal pattern. One of our goals is to predict
    the pattern of counts in the 12 month following the last
    observation.

    We assume the squared root of the counts $y_t$ to be conditionally
    independent Gaussian random variables:
    \[
    y_t|\eta_t,\lambda_y\sim\mathcal{N}(\eta_t,1/\lambda_y),\quad
    t=0,\dots,n_d-1
    \]
    The conditional mean $\eta_t$ is then a sum of a smooth trend and
    a seasonal component:
    \begin{equation}
        \eta_t=\text{season}_t+\text{trend}_t,\quad t=0,\dots,n_{\eta}-1
    \end{equation}
    We assume the vector
    $\textbf{season}=(\text{season}_0,\dots,\text{season}_{n_{\eta}-1})$
    to follow the seasonal model in (3.58) of \cite{book80}, with
    length 12 and unknown precision $\lambda_{\textbf{season}}$, and
    the vector
    $\textbf{trend}=(\text{trend}_0,\dots,\text{trend}_{n_{\eta}-1})$
    to follow a RW2 with unknown precision $\lambda_{\textbf{trend}}$.

    Note that we have that $n_{\eta}=n_d+12=204$, since no
    observations $y_t$ are available for
    $t=n_d,n_d+1,\dots,n_d+11$. For prediction we are interested in
    the posterior marginals of $(\eta_{n_d},\dots,\eta_{n_d+11})$.

    There are three hyperparameters in the model
    $\mm{\theta}=(\log\lambda_y,\log\lambda_{\textbf{season}},\log\lambda_{\textbf{trend}})$
    for which we choose the following prior distributions:
    \[
    \begin{array}{lcl}
        \lambda_y&\sim&\text{LogGamma}(4,4)\\
        \lambda_{\textbf{season}}&\sim&\text{LogGamma}(1,0.1)\\
        \lambda_{\textbf{trend}}&\sim&\text{LogGamma}(1,0.0005)\\
    \end{array}
    \]
    See \cite{book80} for more details.

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