\ifndef{bayesGplvmIntro} \define{bayesGplvmIntro} \editme
\newslide{Selecting Data Dimensionality}
- GP-LVM Provides probabilistic non-linear dimensionality reduction.
- How to select the dimensionality?
- Need to estimate marginal likelihood.
- In standard GP-LVM it increases with increasing
$\latentDim$ .
\newslide{Integrate Mapping Function and Latent Variables}
\columns{Bayesian GP-LVM
- Start with a standard GP-LVM.
- Apply standard latent variable approach:
-
Define Gaussian prior over \emph{latent space},
$\latentMatrix$ . -
Integrate out \emph{latent variables}.
-
Unfortunately integration is intractable. }{ \only<1->{\begin{tikzpicture}
% Define nodes \node[obs] (Y) {$\dataMatrix$}; \node[latent, above=of Y] (X) {$\latentMatrix$}; \node[const, right=1cm of Y] (sigma) {$\dataStd^2$}; % Connect the nodes \edge {X,sigma} {Y} ; % % Plates % \plate {YX} {(Y)} {$i=1\dots\numData$} ; % \plate {} {(W)(Y)(YX.north west)(YX.south west)} {$j=1\dots\dataDim$} ;\end{tikzpicture} } \aligncenter{ {\scriptsize \only<1->{[ p\left(\dataMatrix|\latentMatrix\right)=\prod_{j=1}^{\dataDim}\gaussianDist{\dataVector_{:,j}}{\zerosVector}{\kernelMatrix} ] }\only<3->{[ p\left(\latentMatrix\right)=\prod_{j=1}^{\latentDim}\gaussianDist{\latentVector_{:,j}}{\zerosVector}{\alpha_i^{-2}\eye} ] }\only<4->{[ p\left(\dataMatrix|\boldsymbol{\alpha}\right)= ?? ] } }} }{40%}{60%} \endif
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