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Kernel Methods – Gaussian Processes

Presented by Shankar Bhargav

Arizona State University DMML

Gaussian Processes

• Extending role of kernels to probabilistic discriminative models leads to framework of Gaussian processes

• Linear regression model

– Evaluate posterior distribution over W

• Gaussian Processes: Define probability distribution over functions directly

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Linear regression

x - input vectorw – M Dimensional weight vector

Prior distribution of w given by the Gaussian form

Prior distribution over w induces a probability distribution over function y(x)

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Linear regressionY is a linear combination of Gaussian distributed variables

given by elements of W,

where is the design matrix with elements

We need only mean and covariance to find the joint distribution of Y

where K is the Gram matrix with elements

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Gaussian Processes

• Defn. : Probability distributions over functions y(x) such that the set of values of y(x) evaluated at an arbitrary set of points jointly have a gaussian distribution

– Mean is assumed zero– Covariance of y(x) evaluated at any two values of x is

given by the kernel function

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Gaussian Processes for regression

To apply Gaussian process models for regression we need to take account of noise on observed target values

Consider noise processes with gaussian distribution

with

To find marginal distribution over ‘t’ we need to integrate over ‘Y’

where covariance matrix C

has elements

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Gaussian Processes for regression

Joint distribution over is given by

Conditional distribution of is a Gaussian distribution with mean and covariance given by

where and is N*N covariance matrix

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Learning the hyperparameters

• Rather than fixing the covariance function we can use a parametric family of functions and then infer the parameter values from the data

• Evaluation of likelihood function where denotes the hyperparameters of Gaussian process model

• Simplest approach is to make a point estimate of by maximizing the log likelihood function

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Gaussian Process for classification

• We can adapt gaussian processes to classification problems by transforming the output using an appropriate nonlinear activation function– Define Gaussian process over a function a(x),

and transform using Logistic sigmoid function ,we obtain a non-Gaussian

stochastic process over functions

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The left plot shows a sample from the Gaussian process prior over functions a(x). The right plot shows the result of transforming this sample using a logistic sigmoid function.

Probability distribution function over target variable is given by Bernoulli distribution on one dimensional input space

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Gaussian Process for classification• To determine the predictive distribution

we introduce a Gaussian process prior over vector , the Gaussian prior takes the form

The predictive distribution is given by

where

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Gaussian Process for classification• The integral is analytically intractable so may be

approximated using sampling methods.

• Alternatively techniques based on analytical approximation can be used– Variational Inference– Expectation propagation– Laplace approximation

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Illustration of Gaussian process for classification

Optimal decision boundary – Green

Decision boundary from Gaussian Process classifier - Black

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Connection to Neural Networks• For a broad class of prior distributions over w,

the distribution of functions generated by a neural network will tend to a Gaussian process as M -> Infinity

• In this Gaussian process limit the ouput variables of the neural network become independent.

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Thank you