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\begin{document}


%\vspace*{1cm}


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\def\mm#1{\ensuremath{\boldsymbol{#1}}} % version: amsmath

\begin{poster}

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\begin{minipage}[c][16cm][c]{0.85\textwidth}
  \begin{center}
    {\bf \Enormous  Approximate Inference for
        Hierarchical Gaussian Markov Random Field Models}\\[10mm]

    {\Huge Sara Martino and H\aa vard Rue\\[7.5mm]
        Norwegian University of Science and Technology, Trondheim, Norway}
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\vspace*{1.5cm}

%%%%%%%%%%%%%%%%%%%%%
%%% Content
%%%%%%%%%%%%%%%%%%%%%
\begin{multicols}{3}
    \vspace{1cm}
    \pbox{\columnwidth-\columnsep}{5.5cm}{linewidth=2mm,framearc=0.1,linecolor=whiteblue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em,shadowsize=10pt}{
        \Large{{\bf Abstract:} Approximate deterministic inference is
            produced for posterior marginals for hierarchical GMRF
            models without doing any simulation at all.}
    }
    \vspace{1cm}
    \begin{center}{\huge{\bf Theory}}
    \end{center}
    \vspace{1cm}
    \Large{\bf Hierarchical GMRF}
    
    \setlength{\columnseprule}{0pt}
    \begin{multicols}{2}
        A hierarchical GMRF is a model characterised by several stages of
        observables and parameters, typically:
        \begin{center}
            \small
            \psfrag{theta}{$\mm{\theta}$}
            \psfrag{xn}{$x_n$}
            \psfrag{x1}{$x_1$}
            \psfrag{x2}{$x_2$}
            \psfrag{yn}{$y_n$}
            \psfrag{y1}{$y_1$}
            \psfrag{y2}{$y_2$}
            \epsfig{file=plots/graph.eps,width=8cm,angle=0}
            \newline
            \normalsize{Fig 1: graph}
        \end{center}
        
    \end{multicols}
    \begin{itemize}
    \item {\bf Distribution for the data} \[\pi(\mm{y}\mid
        x)=\prod \pi(y_i\mid x_i)\]
    \item {\bf Distribution for latent variables (GMRF)}
        \[
        \pi(\mm{x}\mid\mm{\theta})\propto
        \left(-\frac{1}{2}\mm{x}^T\mm{Q}\mm{x}\right)
        \]
        where $\mm{Q}$ is typically a sparse matrix
    \item {\bf Distribution for hyperparameters} $\pi(\mm{\theta})$
    \end{itemize}
    The posterior is then
    {\large
        \begin{equation}\label{eq22}%
            \pi(\mm{x}, \mm{\theta}\mid \mm{y})\propto
            \exp\left(-\frac{1}{2}\mm{x}^T\mm{Q}\mm{x}+\sum \log\pi(y_i|x_i)\right)\pi(\mm{\theta})  
        \end{equation}
    }
    \pbox{\columnwidth-\columnsep}{7.2cm}{linewidth=2mm,framearc=0.1,linecolor=whiteblue,fillstyle=gradient,gradangle=0,gradbegin=whiteblue,gradend=whiteblue,gradmidpoint=1.0,framesep=1em,shadowsize=10pt}{
        {\bf Goal of the analysis:} posteriors for the latent field
        \[
        \pi(x_i \mid \mm{y}) = \int_{\mm{\theta}} \int_{\mm{x}_{-i}}
        \pi(\mm{x}, \mm{\theta} \mid \mm{y}) \; d\mm{x}_{-i}\;
        d\mm{\theta}
        \]
        and for hyperparameters $\pi(\mm{\theta}\mid \mm{y})$.
    }
    \vspace{1cm}
    
    Analytical integration $\pi(\mm{x},\mm{\theta}\mid \mm{y})$ is
    usually not possible,
    MCMC it  the common answer to inferential
    problems, {\bf BUT} MCMC can take unreasonable long time to obtain precise
    estimate.
    \vspace{1cm}
    
    \pbox{\columnwidth-\columnsep}{5.2cm}{linewidth=2mm,framearc=0.1,linecolor=NTNUblue,fillstyle=gradient,gradangle=0,gradbegin=white,gradend=white,gradmidpoint=1.0,framesep=1em,shadowsize=10pt}{
        We propose  a {\bf deterministic approximate} alternative, computed almost
        instant compared to the time any MCMC needs to reach a decent
        accuracy [1].%~\cite{tech78}
    }
    \vspace{1cm}
    
    \Large{\bf GMRF-approximation}
    \vspace{1cm}
    
    Build a Gaussian approximation to the full conditional for $\mm{x}$
    {\large
        \begin{equation}\label{eq23}%
            \widetilde{\pi}_G(\mm{x}\mid \mm{\theta}, \mm{y}) \propto
            \exp\left(-\frac{1}{2}(\mm{x}-\mm{\mu})^T(\mm{Q}+\mbox{diag}(\mm{c}))
              (\mm{x}-\mm{\mu})\right) 
        \end{equation}
    }
    by matching the mode and curvature at the mode.
    \begin{itemize}
    \item{\bf Approximate posterior for} $\mm{\theta}$
        \[
        {\pi}(\mm{\theta} \mid \mm{y}) =
        \frac{\pi(\mm{x}, \mm{\theta}\mid
            \mm{y})}{\tilde{{\pi}}_G(\mm{x}\mid \mm{\theta}, \mm{y})}.
        \]
        rhs evaluated at the modal value of $\mm{x}$ for each
        $\mm{\theta}$. This is the Laplace approximation.
    \item{\bf Approximate posterior for }$x_i$
        \[
        \widehat{\widetilde{\pi}}(x_i \mid \mm{y}) =
        \sum_j \widetilde{\pi}_G(x_i \mid \mm{\theta}_j, \mm{y})\;
        \widetilde{\pi}(\mm{\theta}_j \mid \mm{y})\;
        \Delta_j
        \]
    \end{itemize}
    
All computations are feasible and fast due to the sparseness of
the precision matrix $\mm{Q}$ 

\vspace{1cm}
\begin{center}{\huge{\bf Examples}}
\end{center}
The plots show the comparison between the GMRF approximation
({\color{red}{\bf red}}) and
the density estimation obtained from a very long MCMC run ({\bf black})
\vspace{0.7cm}
\begin{center}{\Large{\bf Time series}}
\end{center}
Number of occurrences of rainfall over 1mm in the Tokyo area for each
day during 1983-84
\vspace{0.5cm}

\pbox{\columnwidth-\columnsep}{6.5cm}{linewidth=2mm,framearc=0.1,linecolor=whiteblue,fillstyle=gradient,gradangle=0,gradbegin=whiteblue,gradend=whiteblue,gradmidpoint=1.0,framesep=1em,shadowsize=10pt}
{{\large
        {\bf The model:}
        \begin{eqnarray*}
        y_i \mid x_i &\sim &
        \text{Bin}\{n_i,p_i\}; i=1,\dots,366\\
        p_i&=&\exp(x_i)/(1+\exp(x_i)); \\
        \mm{x}\mid \kappa &&\text{circular RW2 with precision $\kappa\mm{Q}$ }\\
        \kappa&\sim &\text{Gamma}(a,b)
        \end{eqnarray*}
    }}  
\vspace{1cm}

\begin{center}
    \begin{tabular}{cccc}
        %\myfig[0]{plots/tokyo/kappa_approx.eps}{0.3}
        \epsfig{file=plots/tokyo/kappa_approx.eps,width=6.6cm,angle=0} 
        &
        \epsfig{file=plots/tokyo/kappa10.x340.eps,width=6.6cm,angle=0}
        &
        \epsfig{file=plots/tokyo/kappa10000.x340.eps,width=6.6cm,angle=0}
        &
        \epsfig{file=plots/tokyo/gaussia.x341.eps,width=6.6cm,angle=0}\\
        \small{(a) $\pi(\kappa\mid\mm{y})$}&\small{(b)
            $\pi(x_i\mid\mm{y},\kappa=10)$} &\small{(c)
            $\pi(x_i\mid\mm{y},\kappa=1000)$}&\small{(d) $\pi(x_i\mid\mm{y})$}\\
        \multicolumn{4}{c}
        {\normalsize{Fig 2: posteriors for the Tokyo example}}
    \end{tabular}
\end{center}

\begin{center}{\Large{\bf Model for disease mapping} [2]}%\cite{art149}}
\end{center}
\pbox{\columnwidth-\columnsep}{13.8cm}{linewidth=2mm,framearc=0.1,linecolor=whiteblue,fillstyle=gradient,gradangle=0,gradbegin=whiteblue,gradend=whiteblue,gradmidpoint=1.0,framesep=1em,shadowsize=10pt}
{{\large
        {\bf The model:}
        \[ y_i \mid \eta_i \sim \text{Pois}(E_{i} \exp(\eta_i)),:i=1,\ldots,n\]
        where $E_i$ is the expected number of cases in region $i$, and  
        $\eta_i=u_i+v_i$ is the relative risk.
        
        Spatial component
        \[   \pi(\mm{u}\mid \kappa_u)\propto
        \kappa_u^{(n-1)/2} \exp(-\frac{\kappa_u}{2}\sum_{i\sim j}(u_i-u_j)^2)
        \]
        Random component $\mm{v}\sim N(0,\kappa_v^{-1}\mm{I})$

        Gamma priors for the hyperparameters $(\kappa_u,\kappa_v)$
    }}  
\vspace{1cm}

The results are presented for two data sets
\pbox{\columnwidth-\columnsep}{6.8cm}{linewidth=2mm,framearc=0.1,linecolor=whiteblue,fillstyle=gradient,gradangle=0,gradbegin=whiteblue,gradend=whiteblue,gradmidpoint=1.0,framesep=1em,shadowsize=10pt}
{
    {\large
        \begin{itemize}
        \item {\bf Easy case} Oral cavity cancer mortality for males
            in Germany (1986-1990). It's considered ``easy'' since the
            observed counts are high
        \item {\bf Hard case} Insulin dependent Diabetes Mellitus in
        Sardinia. This is a sparse data set, which makes the Poisson
        distribution different from a Gaussian.
    \end{itemize}
}
}  
\vspace{1cm}

{\bf Approximating $\pi(\theta\mid\mm{y})$}
\begin{center}
    \begin{tabular}{ccc}
        \large{{\bf Easy case}}&&\\
        \epsfig{file=plots/germany/joint_log_kappalog_tau.eps,width=7cm,angle=0} 
        &
        \epsfig{file=plots/germany/marginal_kappa.eps,width=7cm,angle=0}
        &
        \epsfig{file=plots/germany/marginal_tau.eps,width=7cm,angle=0}
        \\
        \small{a) $\pi(\kappa_v,\kappa_u\mid\mm{y})$} &  \small{b)
            $\pi(\kappa_u\mid\mm{y})$} &  \small{c)
            $\pi(\kappa_u\mid\mm{y})$}\\
        \multicolumn{3}{c}
        {\normalsize{Fig 3: posteriors for the precision parameters}}
    \end{tabular}
\end{center}
\begin{center}
    \begin{tabular}{ccc}
        \large{{\bf Hard case}}&&\\
        \epsfig{file=plots/sardinia/approx_joint_post.eps,width=7cm,angle=0} 
        &
        \epsfig{file=plots/sardinia/marginal_logkappa.eps,width=7cm,angle=0}
        &
        \epsfig{file=plots/sardinia/marginal_logtau.eps,width=7cm,angle=0} \\
        \small{(a) $\pi(\kappa_v,\kappa_u\mid\mm{y})$} &  \small{(b)
            $\pi(\kappa_u\mid\mm{y})$} &  \small{(c)
            $\pi(\kappa_u\mid\mm{y})$}\\
        \multicolumn{3}{c}
        {\normalsize{Fig 4: posteriors for the precision parameters}}
    \end{tabular}
\end{center}
\vspace{1cm}
{\bf Approximating $\pi(x_i\mid \theta,\mm{y})$}
\begin{center}
    \begin{tabular}{cccc}
        \large{{\bf Easy case}}&&&\\
        \epsfig{file=plots/germany/joint_log_kappalog_tau_1.eps,width=6.4cm,angle=0} 
        &
        \epsfig{file=plots/germany/x157_1.eps,width=6.5cm,angle=0}
        &
        \epsfig{file=plots/germany/joint_log_kappalog_tau_4.eps,width=6.4cm,angle=0}
        &
        \epsfig{file=plots/germany/x157_4.eps,width=6.4cm,angle=0} \\
        & (a) \small{$\pi(u_i|\mm{\theta},\mm{y})$}&
        &(b) \small{$\pi(u_i|\mm{\theta},\mm{y})$}\\
        \multicolumn{4}{c}
        {\normalsize{Fig 5: posteriors  spatial
                component for fixed hyperparameter}}
    \end{tabular}
\end{center}


\begin{center}
    \begin{tabular}{cccc}
        \large{{\bf Hard case}}&&&\\
        \epsfig{file=plots/sardinia/approx_joint_post_1.eps,width=6.4cm,angle=0} 
        &
        \epsfig{file=plots/sardinia/x171_1.eps,width=6.4cm,angle=0}
        &
        \epsfig{file=plots/sardinia/approx_joint_post_5.eps,width=6.4cm,angle=0} 
        &
        \epsfig{file=plots/sardinia/x171_5.eps,width=6.4cm,angle=0} \\
        & (a) \small{$\pi(u_i|\mm{\theta},\mm{y})$}&
        &(b) \small{$\pi(u_i|\mm{\theta},\mm{y})$}\\
        \multicolumn{4}{c}
        {\normalsize{Fig 6: posteriors  spatial
                component for fixed hyperparameter}}
    \end{tabular}
\end{center}
{\bf Approximating $\pi(x_i\mid\mm{y})$}
\begin{center}
    \begin{tabular}{cc}
        \large{{\bf Easy case}}& \large{{\bf Hard case}}\\
        \epsfig{file=plots/germany/x157_integ.eps,width=7.4cm,angle=0} 
        &
        \epsfig{file=plots/sardinia/x171_integ.eps,width=7.4cm,angle=0} \\
          \multicolumn{2}{c}
        {\normalsize{Fig 7: posteriors for one node in the spatial
                component}}
        \end{tabular}
\end{center}
\vspace{1cm}
\begin{center}{\huge{\bf New improved approximation}}
\end{center}
\vspace{1cm}
At the cost of some more computations it is possible to go beyond the
Gaussian approximation, thus getting more precise estimates in those
cases where the Gaussian approximation is not enough, for example
those in Figure 2(b) and Figure 6(b). The figure below shows the
improved approximation ({\color{green}{\bf green}}) compared to the
Gaussian approximation ({\color{red}{\bf red}}) and the MCMC based
estimate ({\bf black}).

\begin{center}
    \begin{tabular}{cc}
        \large{{\bf }}& \large{{\bf }}\\
        \epsfig{file=plots/new_approx/tokyo.eps,width=9.4cm,angle=0} 
        &
        \epsfig{file=plots/new_approx/sardinia.eps,width=9.4cm,angle=0} \\
        \multicolumn{2}{c}
        {\normalsize{Fig 8: improved approximation}}
        \end{tabular}
\end{center}

\small{
    \bibliographystyle{unsrt}
    \bibliography{mybib}
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}
\end{multicols}

\end{poster}

\end{document}

