larger-graph-intro.md

\ifndef{largerGraphIntro} \define{largerGraphIntro} \editme

\newslide{Parametric but Non-parametric} \slides{

• Augment with a vector of inducing variables, $\inducingVector$.
• Form a variational lower bound on true likelihood.
• Bound factorizes given inducing variables.
• Inducing variables appear in bound similar to parameters in a parametric model.
• But number of inducing variables can be changed at run time. }

\newslide{Inducing Variable Approximations}

• Date back to \small{@Williams:nystrom00; @Smola:sparsegp00; @Csato:sparse02; @Seeger:fast03; @Snelson:pseudo05}. See \small{@Quinonero:unifying05; @Thang:unifying17} for reviews.
• We follow variational perspective of \small{@Titsias:variational09}.
• This is an augmented variable method, followed by a collapsed variational approximation \small{@King:klcorrection06; @Hensman:fast12}.

\newslide{Augmented Variable Model: Not Wrong but Useful?}

\columns{ \only<1-2>{Augment standard model with a set of $\numInducing$ new inducing variables, $\inducingVector$.} \only<3>{\textbf{Important:} Ensure inducing variables are \emph{also} Kolmogorov consistent (we have $\numInducing^\ast$ other inducing variables we are not \emph{yet} using.)} \only<4>{Assume that relationship is through $\mappingFunctionVector$ (represents `fundamentals'---push Kolmogorov consistency up to here).} \only<5>{Convenient to assume factorization (\emph{doesn't} invalidate model---think delta function as worst case).} \only<6-7>{Focus on integral over $\mappingFunctionVector$.} \only<1>{[ p(\dataVector) = \int p(\dataVector, \inducingVector) \text{d}\inducingVector ]} \only<2>{[ p(\dataVector) = \int p(\dataVector| \inducingVector) p(\inducingVector)\text{d}\inducingVector ]} \only<3>{[ p(\inducingVector) = \int p(\inducingVector, \inducingVector^\ast) \text{d}\inducingVector^\ast ]} \only<4>{[ p(\dataVector) = \int p(\dataVector| \mappingFunctionVector) p(\mappingFunctionVector|\inducingVector) p(\inducingVector) \text{d}\mappingFunctionVector \text{d}\inducingVector ]} \only<5>{[ p(\dataVector) = \int \prod_{i=1}^\numData p(\dataScalar_i| \mappingFunction_i) p(\mappingFunctionVector|\inducingVector) p(\inducingVector) \text{d}\mappingFunctionVector \text{d}\inducingVector ]} \only<6>{[ p(\dataVector) = \int \int \prod_{i=1}^\numData p(\dataScalar_i| \mappingFunction_i) p(\mappingFunctionVector|\inducingVector) \text{d}\mappingFunctionVector p(\inducingVector) \text{d}\inducingVector ]} \only<7>{[ p(\dataVector|\inducingVector) = \int \prod_{i=1}^\numData p(\dataScalar_i| \mappingFunction_i) p(\mappingFunctionVector|\inducingVector) \text{d}\mappingFunctionVector
]} }{ \begin{center} \begin{tikzpicture}

    % Define nodes
    \draw<1-4> node[obs] (y) {$\dataVector$};
    \draw<4> node[latent, above=of y] (f) {$\mappingFunctionVector$};
    \draw<5-> node[obs] (y) {$\dataScalar_i$};
    \draw<5-> node[latent, above=of y] (f) {$\mappingFunction_i$};
    \draw<2> node[latent, above=of y] (u) {$\inducingVector$};
    \draw<3> node[latent, above left=of y] (u) {$\inducingVector$};
    \draw<3> node[latent, above right=of y] (ustar) {$\inducingVector^\ast$};
    \draw<4-6> node[latent, above=of f] (u) {$\inducingVector$};
    \draw<4-7> node[latent, above right=of f, draw=gray] (ustar) {\color{gray}$\inducingVector^\ast$};
    \draw<7-> node[const, above=of f] (u) {$\inducingVector$};
    
    % Connect the nodes
    \draw<2-3> [->] (u) to (y);%
    \draw<3> [->] (ustar) to (y);%
    \draw<3> [-] (ustar) to (u);%
    \draw<4-> [-, draw=gray] (ustar) to (u);%
    \draw<4-> [->, draw=gray,color=gray] (ustar) to (f);%
    \draw<4-> [->] (f) to (y);%
    \draw<4-> [->] (u) to (f);%

    \only<5->{\plate[inner sep=10pt] {fy} {(f)(y)} {$i=1\dots\numData$} ;}
    
  \end{tikzpicture}
\end{center}

}{60%}{40%}

\endif