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Markus Kaiser 2 years ago
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  1. 37
      ecml-poster/poster.tex

37
ecml-poster/poster.tex

@ -155,8 +155,43 @@
\begin{block}{Scalable Inference}
\begin{itemize}
\item AMO-GP connects multiple deep GPs via a shared layer which is a multi-output GP
\smallskip
\smallskip
\item Application to non-linear time series alignment with very noisy observations
\smallskip
\item Variational distribution
\begin{align}
\Variat*{\mat{F}, \mat{\alpha}, \mat{U}}
&= \Variat*{\mat{\alpha}, \Set*{\mat{F^{\pix{k}}}, \mat{u^{\pix{k}}}, \mat{u_\alpha^{\pix{k}}}}_{k=1}^K} \\
\MoveEqLeft = \prod_{k=1}^K\prod_{n=1}^N \Prob*{\mat{\alpha_n^{\pix{k}}} \given \mat{u_\alpha^{\pix{k}}}, \mat{x_n}}\Variat*{\mat{u_\alpha^{\pix{k}}}}
\prod_{k=1}^K \prod_{n=1}^N \Prob*{\mat{f_n^{\pix{k}}} \given \mat{u^{\pix{k}}}, \mat{x_n}}\Variat*{\mat{u^{\pix{k}}}}.
\end{align}
\smallskip
\item Variational joint
\begin{align}
\Variat*{\mat{Y}, \mat{A}} &=
\int
\Prob*{\mat{Y} \given \mat{F}, \mat{A}}
\Prob*{\mat{A} \given \mat{\alpha}}
\Variat*{\mat{F}, \mat{\alpha}}
\diff \mat{F} \diff \mat{\alpha},
\end{align}
\smallskip
\item Variational bound
\begin{align}
\Ell_{\text{DAGP}}
&= \Moment*{\E_{\Variat*{\mat{F}, \mat{\alpha}, \mat{U}}}}{\log\frac{\Prob*{\mat{Y}, \mat{A}, \mat{F}, \mat{\alpha}, \mat{U} \given \mat{X}}}{\Variat*{\mat{F}, \mat{\alpha}, \mat{U}}}} \\
&= \sum_{n=1}^N \Moment*{\E_{\Variat*{\mat{f_n}}}}{\log \Prob*{\mat{y_n} \given \mat{f_n}, \mat{a_n}}}
+ \sum_{n=1}^N \Moment*{\E_{\Variat*{\mat{\alpha_n}}}}{\log \Prob*{\mat{a_n} \given \mat{\alpha_n}}} \\
&\quad - \sum_{k=1}^K \KL*{\Variat*{\mat{u^{\pix{k}}}}}{\Prob*{\mat{u^{\pix{k}}} \given \mat{Z^{\pix{k}}}}}
- \sum_{k=1}^K \KL*{\Variat*{\mat{u_\alpha^{\pix{k}}}}}{\Prob*{\mat{u_\alpha^{\pix{k}}} \given \mat{Z_\alpha^{\pix{k}}}}}
\end{align}
\smallskip
\item Predictive posterior
\begin{align}
\Variat*{\mat{f_\ast} \given \mat{x_\ast}}
&= \int \sum_{k=1}^K \Variat*{a_\ast^{\pix{k}} \given \mat{x_\ast}} \Variat*{\mat{f_\ast^{\pix{k}}} \given \mat{x_\ast}} \diff \mat{a_\ast^{\pix{k}}} \\
&\approx \sum_{k=1}^K \hat{a}_\ast^{\pix{k}} \mat{\hat{f}_\ast^{\pix{k}}}
\end{align}
\end{itemize}
\end{block}
\end{column}

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