Kernel Methods – Gaussian Processes

1. Kernel Methods – Gaussian Processes Presented by Shankar Bhargav Arizona State University DMML

2. 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 Arizona State University DMML

3. Linear regression x - input vector w – 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) Arizona State University DMML

4. Linear regression Y 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 Arizona State University DMML

5. 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 Arizona State University DMML

6. 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 Arizona State University DMML

7. 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 Arizona State University DMML

8. 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 Arizona State University DMML

9. 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 Arizona State University DMML

10. 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 Arizona State University DMML

11. 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 Arizona State University DMML

12. 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 Arizona State University DMML

13. Illustration of Gaussian process for classification Optimal decision boundary – Green Decision boundary from Gaussian Process classifier - Black Arizona State University DMML

14. 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. Arizona State University DMML

15. Thank you Arizona State University DMML