Gaussian Processes

1. Gaussian Processes Li An anli@temple.edu

2. The Plan • Introduction to Gaussian Processes • Revisit Linear regression • Linear regression updated by Gaussian Processes • Gaussian Processes for Regression • Conclusion

3. Why GPs? • Here are some data points! What function did they come from? • I have no idea. • Oh. Okay. Uh, you think this point is likely in the function too? • I have no idea.

4. Why GPs? • You can’t get anywhere without making some assumptions • GPs are a nice way of expressing this ‘prior on functions’ idea. • Can do a bunch of cool stuff • Regression • Classification • Optimization

5. Gaussian • Unimodal • Concentrated • Easy to compute with • Sometimes • Tons of crazy properties

6. Linear Regression Revisited • Linear regression model: Combination of M fixed basis functions given by , so that • Prior distribution • Given training data points , what is the joint distribution of ? • is the vector with elements , this vector is given by where is the design matrix with elements

7. Linear Regression Revisited • , y is a linear combination of Gaussian distributed variables given by the elements of w, hence itself is Gaussian. • Find its mean and covariance

8. Definition of GP • A Gaussian process is defined as a probability distribution over functions y(x), such that the set of values of y(x) evaluated at an arbitrary set of points x1,.. Xn jointly have a Gaussian distribution. • Probability distribution indexed by an arbitrary set • Any finite subset of indices defines a multivariate Gaussian distribution • Input space X, for each x the distribution is a Gaussian, what determines the GP is • The mean function µ(x) = E(y(x)) • The covariance function (kernel) k(x,x')=E(y(x)y(x')) • In most applications, we take µ(x)=0. Hence the prior is represented by the kernel.

9. Linear regression updated by GP • Specific case of a Gaussian Process • It is defined by the linear regression model with a weight prior the kernel function is given by

10. Kernel function • We can also define the kernel function directly. • The figure show samples of functions drawn from Gaussian processes for two different choices of kernel functions

11. GP for Regression Take account of the noise on the observed target values, which are given by

12. GP for regression • From the definition of GP, the marginal distribution p(y) is given by • The marginal distribution of t is given by • Where the covariance matrix C has elements

13. GP for Regression • The sampling of data points t

14. GP for Regression • We’ve used GP to build a model of the joint distribution over sets of data points • Goal: • To find , we begin by writing down the joint distribution

15. GP for Regression • The conditional distribution is a Gaussian distribution with mean and covariance given by • These are the key results that define Gaussian process regression. • The predictive distribution is a Gaussian whose mean and variance both depend on

16. A Example of GP Regression

17. GP for Regression • The only restriction on the kernel is that the covariance matrix given by must be positive definite. • GP will involve a matrix of size n*n, for which require computations.

18. Conclusion • Distribution over functions • Jointly have a Gaussian distribution • Index set can be pretty much whatever • Reals • Real vectors • Graphs • Strings • … • Most interesting structure is in k(x,x’), the ‘kernel.’ • Uses for regression to predict the target for a new input

19. Questions • Thank you!