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\title{{\bf Bayesian Posterior Comprehension via Message from Monte Carlo}}

\author{Leigh J. Fitzgibbon, David L. Dowe and Lloyd Allison\\
School of Computer Science and Software Engineering\\
Monash University, Clayton, VIC 3800, Australia\\
{\tt \{leighf,dld,lloyd\}@bruce.csse.monash.edu.au} 
}
\date{}



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\begin{figure}\begin{algorithm}{Pseudo-code for optimising the MMLD message length in the unimodal likelihood case.  The algorithm is given as pseudo-code only and in practice extra care would be taken in its implementation to avoid numerical problems.}sample from the posterior distribution $S = \{ \theta_t : t = 1,...,N \}$\  \\
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$\displaystyle J = \left[   \begin{array}{ccc}   \sum_{\theta \in Q} f(x|\theta)^{-1} \left( \frac{\partial^2 KL(\theta,\hat{\theta})}{\partial \hat{\vartheta_1} \partial \hat{\vartheta_1}} \Bigl\lvert_{\hat{\theta}^{(k)}} \right) & \cdots & \sum_{\theta \in Q} f(x|\theta)^{-1} \left( \frac{\partial^2 KL(\theta,\hat{\theta})}{\partial \hat{\vartheta_1} \partial \hat{\vartheta_d}} \Bigl\lvert_{\hat{\theta}^{(k)}} \right) \\\vdots & \ddots & \vdots \\\sum_{\theta \in Q} f(x|\theta)^{-1} \left( \frac{\partial^2 KL(\theta,\hat{\theta})}{\partial \hat{\vartheta_d} \partial \hat{\vartheta_1}} \Bigl\lvert_{\hat{\theta}^{(k)}} \right) & \cdots & \sum_{\theta \in Q} f(x|\theta)^{-1} \left( \frac{\partial^2 KL(\theta,\hat{\theta})}{\partial \hat{\vartheta_d} \partial \hat{\vartheta_d}} \Bigl\lvert_{\hat{\theta}^{(k)}} \right)  \end{array} \right]$%
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$\displaystyle d\hat{\theta}^{(k)} = \left[     \begin{array}{c}   d\hat{\vartheta_0}^{(k)} \\\vdots \\d\hat{\vartheta_d}^{(k)} \\\end{array}  \right] ,\,\, r = \left[     \begin{array}{c}   \sum_{\theta \in Q} f(x|\theta)^{-1} \left( \frac{\partial KL(\theta,\hat{\theta})}{\partial \hat{\vartheta_1}} \Bigl\lvert_{\hat{\theta}^{(k)}} \right) \\\vdots \\\sum_{\theta \in Q} f(x|\theta)^{-1} \left( \frac{\partial KL(\theta,\hat{\theta})}{\partial \hat{\vartheta_d}} \Bigl\lvert_{\hat{\theta}^{(k)}} \right) \\\end{array}  \right]$%
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$ J$%
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$\displaystyle p(y|\psi) = a(y) e^{\psi \bullet y - b(\psi)}$%
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$\displaystyle b(\psi) = \ln \int_y e^{\psi \bullet y} a(y) dy$%
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$\displaystyle KL(\psi,\hat{\psi})$%
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$\displaystyle -H(\psi) - E_{\psi}\left[ \log a(y) \right] + b(\hat{\psi}) - \hat{\psi} \bullet E_{\psi} y$%
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{\newpage\clearpage
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$\displaystyle -b(\psi) + b(\hat{\psi}) + (\psi - \hat{\psi}) \bullet E_{\psi} y$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2111}%
$\displaystyle -b(\psi) + b(\hat{\psi}) + (\psi - \hat{\psi}) \bullet \nabla b(\psi)$%
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$ i^{\mbox{th}}$%
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{\newpage\clearpage
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$ \pi_k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2118}%
$ k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2120}%
$ x_{ik}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2122}%
$ k-x_{ik}$%
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{\newpage\clearpage
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$\displaystyle \pi_{ik} = \kappa^{x_{ik}} \times \upsilon^{k-x_{ik}}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2126}%
$\displaystyle \log(\pi_{ik}) = \alpha x_{ik} + \beta (k-x_{ik})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2128}%
$ \alpha=\log(\kappa)$%
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\lthtmlinlinemathA{tex2html_wrap_inline2130}%
$ \beta=\log(\upsilon)$%
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$ \pi_{k}=\pi_{k-1} \kappa$%
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\lthtmlinlinemathA{tex2html_wrap_inline2134}%
$ k-1$%
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$ \pi_{k}=\pi_{k-1} \upsilon$%
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$ \kappa$%
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\lthtmlinlinemathA{tex2html_wrap_inline2142}%
$ \upsilon$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2145}%
$\displaystyle \theta$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2149}%
$\displaystyle (\alpha,\beta)$%
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$\displaystyle \psi(\theta,x_{ik})$%
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$\displaystyle \log \frac{\pi_{ik}}{1 - \pi_{ik}}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2157}%
$\displaystyle a(y)$%
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$\displaystyle 1$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2163}%
$\displaystyle b(\psi(\theta,x_{ik}))$%
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{\newpage\clearpage
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$\displaystyle \log (1+e^\pi_{ik})$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2169}%
$\displaystyle \frac{\partial b(\psi(\theta,x_{ik}))}{\partial \psi}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2173}%
$\displaystyle \frac{e^{\psi(\theta,x_{ik})}}{1+e^{\psi(\theta,x_{ik})}}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2175}%
$\displaystyle \frac{\partial \psi(\theta,x_{ik})}{\partial \alpha}$%
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$\displaystyle x_{ik} (1 - \pi_{ik})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2181}%
$\displaystyle \frac{\partial \psi(\theta,x_{ik})}{\partial \beta}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2185}%
$\displaystyle (k-x_{ik}) (1 - \pi_{ik})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2187}%
$\displaystyle \frac{\partial b(\psi(\theta,x_{ik}))}{\partial \alpha}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2191}%
$\displaystyle \frac{e^{\psi(\theta,x_{ik})} x_{ik} (1 - \pi_{ik})}{1+e^{\psi(\theta,x_{ik})}}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay2193}%
$\displaystyle \frac{\partial b(\psi(\theta,x_{ik}))}{\partial \beta}$%
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{\newpage\clearpage
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$\displaystyle \frac{e^{\psi(\theta,x_{ik})} (j-x_{ik}) (1 - \pi_{ik})}{1+e^{\psi(\theta,x_{ik})}}$%
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{\newpage\clearpage
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$ \kappa=0.788$%
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$ \upsilon=0.924$%
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$\displaystyle \varepsilon = \{ ((\kappa=0.788,\upsilon=0.924),1) \}$%
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{\newpage\clearpage
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\includegraphics[width=0.6\textwidth]{/home/leighf/static/Paper-PostComp/dogs1}%
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$ x_i$%
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{\newpage\clearpage
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$ \pi_i$%
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{\newpage\clearpage
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$\displaystyle \pi_i$%
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$\displaystyle = \frac{e^{\alpha + \beta x_i}}{1 + \alpha + \beta x_i}$%
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$\displaystyle = \Phi (\alpha + \beta x_i)$%
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{\newpage\clearpage
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$\displaystyle = 1 - e^{- e^{\alpha + \beta x_i}}$%
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$\displaystyle \varepsilon = \{ (\hat{\theta}_{logit},0.03), (\hat{\theta}_{probit},0.07),(\hat{\theta}_{cloglog},0.9) \}$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay2260}%
$\displaystyle KL(\theta,\hat{\theta}) \le \frac{\sum_{\theta' \in Q} f(x|\theta')^{-1} KL(\theta',\hat{\theta}) }{\sum_{\theta' \in Q} f(x|\theta')^{-1}} + 1 - err(\theta)$%
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$ err(.)$%
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$ KL(.|.)$%
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$ U = \{ Q_1,...,Q_K \}$%
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\begin{figure}\begin{algorithm}{Pseudo-code for optimising the MMLD message length in the variable dimension case: Second phase.}changed \= true \\
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changed \= false\\
for each $Q$\  in $U$\  do
\begin{Block}
find the point estimate for $Q$\  and store it in $\hat{Q}$\end{Block}\\
for each $Q$\  in $U$\  do
\begin{Block}
for each $\theta$\  in $Q$\  do
\begin{Block}
for each $Q'$\  in $U - \{ Q \}$\  do
\begin{Block}
if $KL(\theta,\hat{Q'}) < KL(\theta,\hat{Q})$\  and \\
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   $KL(\theta,\hat{Q'}) \le \frac{\sum_{\theta' \in Q'} KL(\theta',\hat{Q'}) f(x|\theta')^{-1} }{\sum_{\theta' \in Q'} f(x|\theta')^{-1}} + 1 - err(\theta)$\  then
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move $\theta$\  from $Q$\  to $Q'$\   \\
changed \= true
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$ \lambda = 0.5$%
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{\newpage\clearpage
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\includegraphics[width=0.7\textwidth]{/home/leighf/static/Paper-PostComp/blocks}%
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\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-noise}} \\
\multicolumn{2}{c}{\mbox{\bf The data sample from the blocks test function}} \\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-map}} \\
\multicolumn{2}{c}{\mbox{\bf Maximum posterior estimate}} \\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc1}} \\
\multicolumn{2}{c}{\mbox{\bf Region 1 point estimate ($w_1 = 0.42$)}} \\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc2}} \\
\multicolumn{2}{c}{\mbox{\bf Region 2 point estimate ($w_2 = 0.30$)}} \\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc3}} \\
\multicolumn{2}{c}{\mbox{\bf Region 3 point estimate ($w_3 = 0.13$)}} \\
\multicolumn{2}{c}{\includegraphics[width=0.7\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mean}} \\
\multicolumn{2}{c}{\mbox{\bf Posterior mean (point-wise) estimate.}} \\
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\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc4} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc5} \\
\mbox{\bf Region 4 point estimate ($w_4 = 0.06$)} & \mbox{\bf Region 5 point estimate ($w_5 = 0.06$)} \\
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc6} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc7} \\
\mbox{\bf Region 6 point estimate ($w_6 = 0.02$)} & \mbox{\bf Region 7 point estimate ($w_7 = 0.01$)} \\
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc8} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/small-blocks-mmc9} \\
\mbox{\bf Region 8 point estimate ($w < 0.01$)} & \mbox{\bf Region 9 point estimate ($w < 0.01$)} \\
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\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc4} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc5} \\
\mbox{\bf Region 4 point estimate ($w_4 = 0.14$)} & \mbox{\bf Region 5 point estimate ($w_5 = 0.12$)} \\
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc6} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc7} \\
\mbox{\bf Region 6 point estimate ($w_6 = 0.09$)} & \mbox{\bf Region 7 point estimate ($w_7 = 0.08$)} \\
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc8} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc9} \\
\mbox{\bf Region 8 point estimate ($w_8 = 0.07$)} & \mbox{\bf Region 9 point estimate ($w_9 = 0.03$)} \\
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc10} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc11} \\
\mbox{\bf Region 10 point estimate ($w_{10} = 0.01$)} & \mbox{\bf Region 11 point estimate ($w_{11} < 0.01$)} \\
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc12} &
\includegraphics[width=0.45\textwidth]{/home/leighf/static/Paper-PostComp/blocks-mmc13} \\
\mbox{\bf Region 12 point estimate ($w_{12} < 0.01$)} & \mbox{\bf Region 13 point estimate ($w_{13} < 0.01$)} \\
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2329}%
$ [0,1]$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2343}%
$ w_1 = 0.42$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline2345}%
$ w_7 = 0.01$%
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\stepcounter{section}
\stepcounter{subsection}
\stepcounter{subsection}
\stepcounter{section}

\end{document}
