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%25/05/10 21.30 SG: added last DAG and references
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\begin{document}
\large\sffamily

  \postertitle{0.96\textwidth}
              {figures/shield.eps}
              {Building genuine beliefs into a prior distribution for the covariance matrix}
              {Sarah E.\ Germain, Richard J.\ Boys \& Malcolm Farrow}
              {School of Mathematics and Statistics, University of Newcastle
	        upon Tyne, U.K.}
              {E-mail: s.e.germain@ncl.ac.uk \quad richard.boys@ncl.ac.uk \quad malcolm.farrow@ncl.ac.uk}
              {whiteblue}
              {dblue}

  \begin{multicols}{3}
    \heading{0.82\columnwidth}{whiteblue}{black}{1: The Problem}

\begin{description}
\item[Introduction]: Analyses of multidimensional data often use the dependence structure of the multivariate normal (MVN) distribution to build relationships between variables. Constructing a prior for the covariance matrix,
$ \bs{\Sigma}$, of a MVN distribution is difficult because of the large number of parameters it can contain and the constraint that the matrix be positive definite. Tackling this problem is especially challenging if the prior is intended to convey substantive initial information.
\item[Examples]: Prior beliefs we might hold in
\begin{itemize}
\item time series problems: \textit{covariances between pairs of time points at the same lag are likely to be similar}.
\item spatial problems: \textit{covariances between pairs of points at similar distances apart are likely to be similar}.
\end{itemize}
\textcolor{dred}{Note}: we may want to express belief about ``similarity'' through choices for $\Covar(\Sigma_{jk},\Sigma_{\ell m})$ as well as $\Exp(\bs{\Sigma})$.
%\item Denoting the covariance matrix by $\bs{\Sigma}=(\Sigma_{jk})$, we can reduce the dimensionality of the problem, as well as the complexity of the positive definiteness condition, by assuming some parametric form for the covariance matrix, say $\bs{\Sigma}=\bs{\Sigma}(\bs{\theta})$, then placing an appropriate prior on the parameter vector $\bs{\theta}$ e.g. for example 1, a natural choice might be the AR(1) structure, $\Sigma_{jk}=\sigma^2 \rho^{[j-k]}/(1-\rho^2)$, choosing a prior for $\bs{\theta}=(\rho,\sigma^2)$ that ensures $|\rho| < 1$ and $\sigma^2>0$.
%\item \bs{However} imposing structure is restrictive because the covariance matrix is constrained to take the assumed form in the posterior, irrespective of the extent to which the data might contradict its suitability. We might, however, like our prior for $\bs{\Sigma}$ to support belief in a parametric form without making it a structural assumption. e.g. for example 1, we may want to base our prior mean for $\bs{\Sigma}$ on an AR(1) structure, and discourage deviations from it by making covariances $\Sigma_{jk}$ on the same diagonal highly correlated \textit{a priori}.
\item[Inadequacy of the inverse Wishart prior]: The inverse Wishart distribution is commonly chosen as a prior for $\bs{\Sigma}$ because it is conjugate and therefore convenient. However, it has just $n(n+1)/2+1$ hyperparameters, and so, once $\Exp(\bs{\Sigma})$ has been chosen, there is only one hyperparameter to set all the prior variances and covariances, $\Covar(\Sigma_{jk},\Sigma_{\ell m})$.
\item[Illustrative example]: For $n=5$ sites in Yorkshire, UK, (see Figure~\ref{fig:Yorkshire_map}) over $T$ years, let $Y_{tj}$ denote the log January rainfall total in year $t$ at site $j$. Suppose we model
\begin{equation*}
\textcolor{dred}{\bs{Y}_t=(Y_{t1},\ldots,Y_{tn})^{\prime} \mid \bs{\mu},\bs{\Sigma} \overset{\text{\emph{iid}}}{\sim} \text{N}_n(\bs{\mu},\bs{\Sigma})}
\end{equation*}
and assume $\pi(\bs{\mu},\bs{\Sigma})=\pi(\bs{\mu})\pi(\bs{\Sigma})$. We want to build a prior, $\pi(\bs{\Sigma})$, that expresses genuine initial beliefs.
\item[Notation]: Write $\bs{R}=\bs{Y}-\bs{\mu}$ where $\bs{Y} \mid \bs{\mu},\bs{\Sigma} \sim \text{N}_n(\bs{\mu},\bs{\Sigma})$.
\end{description}

 \heading{0.82\columnwidth}{whiteblue}{black}{2: Some priors proposed by other authors}

Priors on various reparameterisations of the covariance matrix have been proposed. Generally, $\bs{\Sigma}$ is transformed to a new set of parameters that belong to a less constrained sample space.
\begin{description}
\item[1. Spectral decomposition \cite{DK99} or matrix logarithm \cite{LH92}]:
\begin{itemize}
\item[$\textcolor{dred}{\CheckedBox}$] can lead to parameters in an unconstrained space 
\item[$\textcolor{dblue}{\XBox}$] the new parameters lack a natural interpretation
\end{itemize}
\item[2. Standard deviation / correlation decomposition \cite{BMM00}]:
\begin{itemize}
\item[$\textcolor{dred}{\CheckedBox}$] the new parameters have a clear statistical meaning 
\item[$\textcolor{dblue}{\XBox}$] the constraints on the set of correlation matrices make it difficult to elicit a prior and to sample correlation matrices during MCMC
\end{itemize}
\item[3. Modified Cholesky (MC) decomposition \cite{DP02}]:\\

%\psboxfill{\includegraphics{figures/figDP.eps}}
\vspace{3cm}
\hspace{1.25cm}
%\psboxfill{\includegraphics{figures/yorkshire_map_plain.eps}}
\begin{pspicture}(13.0,5.0)
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\captionof{figure}{Example DAG, MC decomposition (or remove $E_j$\\
nodes and make $R_j$ nodes non--deterministic, e.g. Figure~\ref{fig:Yorkshire_map})}% makes a caption for non-floats
\label{fig:DP}
\vspace{-0.15cm}
\[ \textcolor{dblue}{R_{j} = \sum_{k<j} \phi_{jk}R_{k} + E_{j}, \quad \Var(E_j)=\sigma_j^2} \]

Let $\bs{\Sigma}^{-1}=\mathbf{T}^{\prime}\mathbf{D}^{-1}\mathbf{T}$ where $\mathbf{D}$ is a diagonal matrix with $(j,j)$--th entry $\sigma_j^2>0$ and $\mathbf{T}$ is a unit lower triangular matrix with $(j,k)$--th entry $-\phi_{jk} \in \mathbb{R}$. Let $\bs{\phi}_j=(\phi_{j1},\ldots,\phi_{j,j-1})^{\prime}$ and $\bs{R}_{1:j-1}=(R_1,\ldots,R_{j-1})^{\prime}$. Now write
\vspace{-0.5cm}
\begin{gather*}
\textcolor{dred}{p(\bs{R}) = p(R_1) \prod_{j=2}^n p(R_j \mid \bs{R}_{1:j-1})} \; \text{where} \; \textcolor{dred}{R_1 \sim \text{N}(0,\sigma_1^2)}\\
%\end{equation*}
%\begin{equation*}
\text{and} \quad \textcolor{dred}{R_j \mid \bs{R}_{1:j-1} \sim \text{N}(\bs{\phi}_j^{\prime} \bs{R}_{1:j-1},\sigma_j^2)} \quad \text{for $j=2,\ldots,n$.}
\end{gather*}
%\vspace{-1cm}
The $\phi_{jk}$ are interpreted as coefficients in the regressions of $R_j$ on $\bs{R}_{1:j-1}$ and the $\sigma_j^2$ as conditional variances. Independent MVN and inverse gamma priors are assigned to $\bs{\phi}=(\bs{\phi}_2^{\prime},\ldots,\bs{\phi}_n^{\prime})^{\prime}$ and the $\sigma_j^2$
\begin{itemize}
\item[$\textcolor{dred}{\CheckedBox}$] leads to parameters in an unconstrained space 
\item[$\textcolor{dred}{\CheckedBox}$] the prior is semi--conjugate
\item[$\textcolor{dred}{\CheckedBox}$] the new parameters have a regression interpretation
\item[$\textcolor{dblue}{\XBox}$] an order must be imposed amongst $R_1,\ldots,R_n$
\end{itemize}
\item[4. Generalised inverse Wishart prior \cite{BLZ94}]: similar to \textbf{3.} but conjugate, necessitating \textit{a priori} dependence between the coefficients and conditional variances.
\end{description}

%\begin{figure}
%\vspace{-4cm}
%\begin{center}
%\includegraphics{figures/figDP.eps}
%\vspace{-4cm}
%\caption{\textcolor{mygreen}{Example DAG, MC decomposition}}
%\label{fig:DP}
%\end{center}
%\end{figure}

%\begin{figure*}[ht]
%\resizebox{0.4\textwidth}{!}{\includegraphics{figures/yorkshire_map.eps}}
%\end{figure*}

    \heading{0.82\columnwidth}{whiteblue}{black}{3: Uncertainty factors}

\vspace{5.5cm}
\hspace{4.3cm}
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\captionof{figure}{Example DAG, Uncertainty factors. (Different structures\\
\centerline{could be chosen).}}% makes a caption for non-floats
\label{fig:UF}
\vspace{-0.35cm}
\[ \textcolor{dblue}{{\rm E.g.,}\ {\rm for}\ {\rm DAG}\ {\rm above},\ \ R_{1} = \lambda_{112}U_{12} + \lambda_{113} U_{13} + E_{1}} \]

\begin{itemize}
\item Develop from \cite{Far03} to allow learning covariance matrix.
\item Unknown $ \Var(E_{j}) $ (as in MC decomposition) and
\end{itemize}
\renewcommand{\arraystretch}{0.1}
\begin{tabular}{m{12.7cm}!{\vrule width 2pt}m{12.7cm}}
\multicolumn{1}{c!{\vrule width 2pt}}{\textcolor{dred}{EITHER (A)}} &\multicolumn{1}{c}{\textcolor{dred}{OR (B)}}\\
\begin{itemize}
\item[-] Specify arc coefficients, $ \lambda_{ijk}= \pm 1. $
\item[-] Unknown $ \Var(U_{jk}). $
\end{itemize}
&
\begin{itemize}
\item[-] Specify $ \Var(U_{jk})=1. $
\item[-] Unknown arc coefficients, $ \lambda_{ijk} $, have MVN prior.
\end{itemize}
\end{tabular}
\begin{itemize}
\item Unknown variances have, e.g., MV lognormal prior.
\item 2nd--level uncertainty factor structure for unknowns.
\item Choices A or B place some restrictions on the possible $ \bs{\Sigma}$, but are still much more flexible than typical ``parametric'' specifications.
\end{itemize}

    \heading{0.82\columnwidth}{whiteblue}{black}{4: The MC decomposition approach}

The main problem with the MC decomposition (see Section 2) is that an order must be imposed amongst $Y_1,\ldots,Y_n$. In spatial problems, there is no natural ordering of the sites. This makes elicitation difficult. To choose the ordering, $N_1,\ldots,N_n$, and prior we propose:
\begin{itemize}
\item Distinguish between ``strong'' arcs and ``weak'' arcs.
\item Draw strong arcs: subjectively chosen to represent the relationships we expect.
\item Add weak arcs: until DAG is maximally connected.
\end{itemize}

\begin{description}
\item[Full algorithm]: 
\begin{enumerate}
\item[\textcolor{dred}{1.}] Select node $N_1$, chosen to be ``typical'' and ``central''.
\item[\textcolor{dred}{2.}] Make all other nodes children of $N_1$.
\item[\textcolor{dred}{3.}] Add other arcs where required, e.g. between neighbours, subject to not having directed cycles. For these arcs, we should be prepared to express prior beliefs about their coefficients and about the conditional variances of the nodes, given their parents.
\item[\textcolor{dred}{4.}] Express our prior beliefs about the coefficients on the strong arcs and about the conditional variances of the nodes, given their parents.
\item[\textcolor{dred}{5.}] Add the weak arcs to complete the DAG as follows. For $j$ in $2,\ldots,n-1$
\begin{enumerate}
\item[\textcolor{dred}{(a)}] There must be at least one node with just $j-1$ parents. Choose one such node to be $N_j$.
\item[\textcolor{dred}{(b)}] Make $N_j$ a parent of the remaining $N_{j+1},\ldots,N_n$.
\end{enumerate}
The coefficients on the weak arcs are given zero prior means and small prior variances. We do not change our beliefs about the conditional variances of the nodes given their parents.
\end{enumerate}

\item[Illustrative example]: For the Yorkshire rainfall problem, Figure~\ref{fig:Yorkshire_map} illustrates our choice of strong arcs (\textcolor{red}{\ding{222}}), weak arcs (\textcolor{red!50!white}{\ding{221}}) and the order. 

\vspace{6.5cm}

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        \ncline[arrowscale=2,linewidth=2.25pt,linecolor=red]{->}{N1}{N4}
        \ncline[arrowscale=2,linewidth=2.25pt,linecolor=red]{->}{N1}{N5}
        \ncline[arrowscale=2,linewidth=2.25pt,linecolor=red]{->}{N2}{N3}
        \ncline[arrowscale=2,linewidth=2.25pt,linecolor=red]{->}{N4}{N5}
        \ncline[arrowscale=2,linewidth=1.5pt,linecolor=red!50!white]{->}{N2}{N4}
        \ncline[arrowscale=2,linewidth=1.5pt,linecolor=red!50!white]{->}{N2}{N5}
        \ncline[arrowscale=2,linewidth=1.5pt,linecolor=red!50!white]{->}{N3}{N4}
        \ncline[arrowscale=2,linewidth=1.5pt,linecolor=red!50!white]{->}{N3}{N5}
        \uput[-90](13.7,2){\footnotesize{Lincolnshire}}
        \uput[-90](2.9,1.6){\footnotesize{Mersey}}
        \uput[-90](5.5,2.2){\footnotesize{G. Mancs}}
        \uput[-90](4,9.2){\footnotesize{Cumbria}}
        \uput[-90](9.7,9.9){\footnotesize{Teesside}}
        \uput[-90](5.0,4.8){\footnotesize{Lancs}}
      }
\end{pspicture}
\captionof{figure}{Yorkshire map and overlaid DAG, MC decomposition}% makes a caption for non-floats
\label{fig:Yorkshire_map}

\item[Remaining problem]: choosing the prior covariances between the slope coefficients, $\phi_{jk}$. There is no straightforward relationship between the $\phi_{jk}$ and the covariances, $\Sigma_{jk}$, e.g. omitting a site would change the coefficients but not the covariances between remaining sites. This makes it difficult to express beliefs about $\Covar(\Sigma_{jk},\Sigma_{\ell m})$ through our choices $\Covar(\phi_{jk},\phi_{\ell m})$.
\end{description}
    
 \heading{0.82\columnwidth}{whiteblue}{black}{5: Alternative decomposition}

\vspace{3.15cm}
\hspace{1.25cm}
%\psboxfill{\includegraphics{figures/yorkshire_map_plain.eps}}
\begin{pspicture}(13.0,5.0)
      \scalebox{1.6}{
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        %\rput(5,70){\pnode{S0}}
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        %\ncline[arrowscale=2,linewidth=1.5pt,linecolor=red!50!white]{->}{N2}{N4}
      }
\end{pspicture}
\captionof{figure}{Example DAG, Alternative decomposition}% makes a caption for non-floats
\label{fig:MOD}

%\begin{figure}
%\vspace{-4cm}
%\begin{center}
%\includegraphics[angle=270]{figures/figMOD.eps}
%\vspace{-4cm}
%\caption{\textcolor{mygreen}{Example DAG, Modified structure}}
%\label{fig:MOD}
%\end{center}
%\end{figure}

\vspace{-1cm}
\[ \textcolor{dblue}{R_{i} = \sum_{j=1}^{i} \lambda_{ij} U_{j},\ \ \ \Var(U_{j}) = 1,}\]
\vspace{-1cm}
\[ \textcolor{dblue}{\Covar(R_{i},R_{k}) = \sum_{j=1}^{k} \lambda_{ij} \lambda_{jk}\ \ (k \leq i)} \]

\begin{itemize}
\item Unknown arc coefficients have MVN prior.
\item Compare with Figures \ref{fig:DP} and \ref{fig:UF} above.
\item Figure \ref{fig:MOD} combines ideas from both.
\item Avoids restrictions of uncertainty factor approach.
\item Leads to simpler expressions for variances and covariances than the MC decomposition approach.
\item Again, start with ``strong'' arcs, e.g. just $ \lambda_{i1}. $
\end{itemize}

   \heading{0.91\columnwidth}{whiteblue}{black}{6: References}
\vspace{-5cm}
\normalsize
\bibliographystyle{siam}
\renewcommand\refname{} 
\bibliography{referencesP}


\end{multicols}

\end{document}