A Short Presentation on Probabilistic Programming and Variational Inference








Probabilistic Programming for Bayesian Computation: Part I

A presentation on variational inference and probabilistic programming.





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An effective kernelization of logistic regression

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I will present a sparse kernelization of logistic regression where the prototypes are not necessarily from the training data. Traditional sparse kernel logistic regression Consider an $latex M$ class logistic regression model given by $latex P(y|x)\propto\mbox{exp}(\beta_{y0} + \sum_{j}^{d}\beta_{yj}x_j)$ for $latex y =0,1,\ldots,M$ where $latex j$ indexes the $latex d$ features. Fitting the model to a data set $latex D = \{x_i, y_i\}_{i=1,\ldots,N}$ involves estimating the betas to maximize the likelihood of $latex D$. The above logistic regression model is quite simple (because the classifier is a linear function of the features of the example), and in some circumstances we might want a classifier that can produce a more complex decision boundary. One way to achieve this is by kernelization . We write $latex P(y|x) \propto \mbox{exp}(\beta_{y0} + \sum_{i=1}^N \beta_{yi} k(x,x_i))$ for $latex y=0,1,\ldots,M$. where $latex k(.,.)$ is a kernel function. In order to be able to use this c
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Incremental complexity support vector machine

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One of the problems with using complex kernels with support vector machines is that they tend to produce classification boundaries that are odd, like the ones below. (I generated them using a java SVM applet from here , whose reliability I cannot swear to, but have no reason to doubt.) Both SVM boundaries are with Gaussian RBF kernels: the first with $latex \sigma = 1$ and the second with $latex \sigma = 10$ on two different data sets. Note the segments of the boundary to the east of the blue examples in the bottom figure, and those to the south and to the north-east of the blue examples in the top figure. They seem to violate intuition. The reason for these anomalous boundaries is of course the large complexity of the function class induced by the RBF kernel with large $latex \sigma$, which gives the classifier a propensity to make subtle distinctions even in regions of  somewhat low example density. A possible solution: using complex kernels only where they are needed We propose to b
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Random Fourier Features for Kernel Density Estimation

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The NIPS paper Random Fourier Features for Large-scale Kernel Machines , by Rahimi and Recht presents a method for randomized feature mapping where dot products in the transformed feature space approximate (a certain class of) positive definite (p.d.) kernels in the original space. We know that for any p.d. kernel there exists a deterministic map that has the aforementioned property but it may be infinite dimensional. The paper presents results indicating that with the randomized map we  can get away with only a "small" number of features (at least for a classification setting). Before applying the method to density estimation let us review the relevant section of the paper briefly. Bochner's Theorem and Random Fourier Features Assume that we have data in $R^d$ and a continuous p.d. kernel $K(x,y)$ defined for every pair of points $x,y \in R^d$. Assume further that the kernel is shift-invariant, i.e., $K(x,y) = K(x-y) \triangleq K(\delta)$ and that the kernel is
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