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\section*{Tokyo rainfall data}
 The number of occurrences of rainfall over 1 mm in the Tokyo area
    for each calendar year during two years (1983-84) are
    registered. It is of interest to estimate the underlying
    probability $p_t$ of rainfall for calendar day $t$ which is,
    apriori, assumed to change gradually over time. The likelihood
    model is binomial
    \[y_t|\eta_t\sim\text{Bin}(n_t,p_t)\] with logit link function
    \[p_t=\frac{\exp(\eta_t)}{1+\exp(\eta_t)}.\] The model for the
    latent variables can be written as
    \[\eta_t=f(t)\]
    where $t$ is the observed time whose effect is modelled as a
    smooth function $f(\cdot)$. Following \cite{book80}, the random
    vector $\mm{f}=\{f_0,\dots,f_{365}\}$ is assumed to have a {\it
        circular} random walk of order 2 (RW2) prior with unknown
    precision $\lambda_f$.

    There is only one hyperparameter $\mm{\theta}=(\log\lambda_f)$
    which we assign a $\text{LogGamma}(a,b)$ prior distribution with
    $a=1$ and $b=0.0001$. The LogGamma distribution is defined such
    that if $X\sim\text{LogGamma}(a,b)$, the
    $Y=\exp(X)\sim\text{Gamma}(a,b)$ with $\text{E}(Y)=a/b$ and
    $\text{Var}(Y)=a/b^2$.

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