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- 1. Gaussian Processes: Applications in Machine Learning Abhishek Agarwal (05329022) Under the Guidance of Prof. Sunita Sarawagi KReSIT, IIT Bombay Seminar Presentation March 29, 2006 Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 2. Outline Introduction to Gaussian Processes(GP) Prior & Posterior Distributions GP Models: Regression GP Models: Binary Classiﬁcation Covariance Functions Conclusion. Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 3. Introduction Supervised Learning Gaussian Processes Deﬁnes distribution over functions. Collection of random variables, any ﬁnite number of which have joint Gaussian distributions.[1] [2] f ∼ GP(m, k) Hyperparameters and Covariance function. Predictions Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 4. Prior Distribution Represents our belief about the function distribution, which we pass through parameters Example: GP(m, k) 1 m(x) = x 2 , k(x, x ) = exp(− 1 (x − x )2 ). 2 4 To draw sample from the distribution: Pick some data points. Find distribution parameters at each point. µi = m(xi ) & Σij = k(xi , xj ) i, j = 1, . . . , n Pick the function values from each individual distribution. Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 5. Prior Distribution(contd.) 9 8 7 6 function values 5 4 3 2 1 −5 −4 −3 −2 −1 0 1 2 3 4 5 data points Figure: Prior distribution over function using Gaussian Process Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 6. Posterior Distribution Distribution changes in presence of Training data D(x, y ). Functions which satisy D are given higher probability. 8 7 6 5 function values 4 3 2 1 0 −1 −5 −4 −3 −2 −1 0 1 2 3 4 5 data points Figure: Posterior distribution over functions using Gaussian Processes Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 7. Posterior Distribution (contd.) Prediction for unlabeled data x∗ GP outputs the function distribution at x∗ Let f be the distribution at data points in D and f∗ at x∗ f and f∗ will have a joint Gaussian distribution, represented as: f µ Σ Σ∗ ∼ f∗ µ∗ Σ∗ T Σ∗∗ Conditional distribution of f∗ given f can be expressed as: f∗ |f ∼ N ( µ∗ + Σ∗ T Σ−1 (f − µ), Σ∗∗ − Σ∗ T Σ−1 Σ∗ ) (1) Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 8. Posterior Distribution (contd.) Parameters of the posterior in Eq. 1 are: f∗ |D ∼ GP(mD , kD ) , where mD (x) = m(x) + Σ(X , x)T Σ−1 (f − m) kD (x, x ) = k(x, x ) − Σ(X , x)T Σ−1 Σ(X , x ) 8 7 6 5 function values 4 3 2 1 0 −5 −4 −3 −2 −1 0 1 2 3 4 5 data points Figure: Prediction from GP Applications in Machine Learning Abhishek Agarwal (05329022) Gaussian Processes:
- 9. GP Models: Regression GP can be directly applied to Bayesian Linear Regression model like: f (x) = φ(x)T w with prior w ∼ N (0, Σ) Parameters for this distribution will be: E[f (x)] = φ(x)T E[w ] = 0, E[f (x)f (x )] = φ(x)T E[ww T ]φ(x ) = φ(x)T Σp φ(x ) So, f (x) and f (x ) are jointly Gaussian with zero mean and covariance φ(x)T Σp φ(x ). Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 10. GP Models: Regression (contd.) In Regression, posterior distribution over the weights, is given as (9): likelhood ∗ prior posterior = marginal likelihood Both prior p(f|X ) and likelihood p(y |f, X ) are Gaussian: prior: f|X ∼ N (0, K ) (5) likelihood: y|f ∼ N (f, σ n 2 I) Marginal Likelihood p(y |X ) is deﬁned as (6): p(y |X ) = p(y |f, X )p(f|X )df (2) Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 11. GP Models: Classiﬁcation Modeling Binary Classiﬁer Squash the output of a regression model using a response function, like sigmoid. Ex: Linear logistic regression model: 1 p(C1 |x) = λ(x T w ), λ(z) = 1 + exp(−z) Likelihood is expressed as (7): p(yi |xi , w ) = σ(yi fi ), fi ∼ f (xi ) = x i T w and therefore its non-Gaussain. Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 12. GP Models: Classiﬁcation (contd.) Distribution over latent function, after seeing the test data, is given as: p(f∗ |X , y , x∗ ) = p(f∗ |X , x∗ , f)p(f|X , y )df, (3) where p(f|X , y ) = p(y |f)p(f|X )/p(y |X ) is the posterior over the latent variable. Computation of the above integral is analytically intractable Both, likelihood and posterior are non-Gaussian. Need to use some analytic Approximation of integrals. Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 13. GP Models: Laplace Approximations Gaussian Approximation of p(f|X , y ): Using second order Taylor expansion, we obtain: q(f|X , y ) = N (f|ˆ A−1 ) f, where where ˆ = argmaxf p(f|X , y ) and f A=− log p(f|X , y )|f=ˆ f To ﬁnd ˆ we use Newton’s method, because of non-linearity of f, log p(f|X , y ) (9) Prediction is given as: π∗ = p(y∗ = +1|X , y , x∗ ) = σ(f∗ )p(f∗ |X , y , x∗ )df∗ , (4) Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 14. Covariance Function Encodes our belief about the prior distribution over function Some properties: Staionary Isotropic Dot-Product Covariance Ex: Squared Exponential(SE) covarince function: 1 cov (f (xp ), f (xq )) = exp(− |xp − xq |2 ) 2 Learned with other hyper-parameters. Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 15. Summary and Future Work Current Research: Fast sparse approximation algorithm for matrix inversion. Approximation algorithm for non-Gaussian likelihoods. GP approach has outperformed traditional methods in many applications. Gaussin Process based Positioning System (GPPS) [6] Multi user Detection (MUD) in CDMA [7] GP models are more powerful and ﬂexible than simple linear parametric models and less complex in comparison to other models like multi-layer perceptrons. [1] Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 16. Rasmussen and Williams. Gaussian Process for Machine Learning, The MIT Press, 2006. Matthias Seeger. Gaussian Process for Machine Learning, 2004. International Journal of Neural Systems, 14(2):69-106, 2004. Christopher Williams, Bayesian Classiﬁcation with Gaussian Processes, In IEEE Trans. Pattern analysis and Machine Intelligence, 1998 Rasmussen and Williams, Gaussian Process for Regression. In Proceedings of NIPS’ 1996. Rasmussen, Evaluation of Gaussian Processes and Other Methods for Non-linear Regression. PhD thesis, Dept. of Computer Science, University of Toronto, 1996. Available from http://www.cs.utoronto.ca/ carl/ Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 17. Anton Schwaighofer, et. al. GPPS: A Gaussian Process Positioning System for Cellular Networks, In proceedings of NIPS’ 2003. Murillo-Fuentes, et. al. Gaussian Processes for Multiuser Detection in CDMA receivers, Advances in Neural Information Processing System’ 2005 David Mackay, Introduction to Gaussian Processes C. Williams. Gaussian processes. In M. A. Arbib, editor, Handbook of Brain Theory and Neural Networks, pages 466-470. The MIT Press, second edition, 2002. Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 18. Thank You !! Questions ?? Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 19. Extra Prior: 1 1 n log p(f|X ) = − f T K −1 f − log |K | − log 2π (5) 2 2 2 Mariginal likelihood 1 1 n log p(y|X ) = − yT (K +σ n 2 I)−1 y− log |K +σ n 2 I|− log 2π 2 2 2 (6) Likelihood p(y = +1|x, w ) = σ(x T w ), (7) For symmetric like hood σ(−z) = 1 − σ(z). p(yi |xi , w ) = σ(x i T w ), (8) Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning
- 20. Extra (contd.) ﬁrst derivative of posterior ˆ = K( f log p(f|X , y )) Prediction p(y|X, w) ∗ p(w) p(w |y , X ) = p(y |X ) Abhishek Agarwal (05329022) Gaussian Processes: Applications in Machine Learning

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