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Pattern Recognition and Machine Learning

Institute of Empirical Research in Economics (IEW). Laboratory for Social & Neural Systems Research (SNS). Pattern Recognition and Machine Learning. Course schedule. Date Topic Chapter 13-10-2010 Density Estimation, Bayesian Inference 2

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Pattern Recognition and Machine Learning

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  1. Institute of Empirical Research in Economics (IEW) Laboratory for Social & Neural Systems Research (SNS) Pattern Recognition and Machine Learning ComputationalNeuroeconomicsandNeuroscience

  2. Course schedule Date Topic Chapter 13-10-2010 Density Estimation, Bayesian Inference 2 Adrian Etter, Marco Piccirelli, Giuseppe Ugazio 20-10-2010 Linear Models for Regression 3 Susanne Leiberg, Grit Hein 27-10-2010 Linear Models for Classification 4 Friederike Meyer, ChaohuiGuo 03-11-2010 Kernel Methods I: Gaussian Processes 6 Kate Lomakina 10-11-2010 Kernel Methods II: SVM and RVM 7 Christoph Mathys, MortezaMoazami 17-11-2010 Probabilistic Graphical Models 8 Justin Chumbley Computational Neuroeconomics and Neuroscience

  3. Course schedule Date Topic Chapter 24-11-2010 Mixture Models and EM 9 BastiaanOud, Tony Williams 01-12-2010 Approximate Inference I: Deterministic Approximations 10 Falk Lieder 08-12-2010 Approximate Inference II: Stochastic Approximations 11 Kay Brodersen 15-12-2010 Inference on Continuous Latent Variables: PCA, Probabilistic PCA, ICA 12 Lars Kasper 22-12-2010 Sequential Data: Hidden Markov Models, Linear Dynamical Systems 13 Chris Burke, Yosuke Morishima Computational Neuroeconomics and Neuroscience

  4. Institute of Empirical Research in Economics (IEW) Sandra Iglesias Laboratory for Social & Neural Systems Research (SNS) Chapter 1: Probability, Decision, and Information Theory ComputationalNeuroeconomicsandNeuroscience

  5. Outline Introduction Probability Theory Probability Rules Bayes’Theorem Gaussian Distribution Decision Theory Information Theory Computational Neuroeconomics and Neuroscience

  6. Pattern recognition • computer algorithms •  automatic discovery of regularities in data • use of these regularities to take actions such • as classifying the data into different categories • classify data (patterns) based either on • a priori knowledge or • statistical information extracted from the patterns Computational Neuroeconomics and Neuroscience

  7. Machine learning • 'How can we program systems to automatically learn and to improve with experience?' • the machine is programmed to learn from • an incomplete set of examples (training set) • the core objective of a learner is to • generalize from its experience Computational Neuroeconomics and Neuroscience

  8. Polynomial Curve Fitting Computational Neuroeconomics and Neuroscience

  9. Sum-of-Squares Error Function  Computational Neuroeconomics and Neuroscience

  10. Plots of polynomials Computational Neuroeconomics and Neuroscience

  11. Over-fitting Root-Mean-Square (RMS) Error: Computational Neuroeconomics and Neuroscience

  12. Regularization Penalize large coefficient values M = 9 M = 9 Computational Neuroeconomics and Neuroscience

  13. Regularization: vs. M = 9 Computational Neuroeconomics and Neuroscience

  14. Outline Introduction Probability Theory Decision Theory Information Theory Computational Neuroeconomics and Neuroscience

  15. Probability Theory Noise on measurements Uncertainty Probability theory: consistent framework for the quantification and manipulation of uncertainty Finite size of data sets Computational Neuroeconomics and Neuroscience

  16. Probability Theory Marginal Probability Conditional Probability Joint Probability Computational Neuroeconomics and Neuroscience

  17. Probability Theory i = 1, …,M j = 1, …,L nij: numberoftrials in which X = xiand Y = yj ci: numberoftrials in which X = xiirrespectiveofthevalueof Y rj: numberoftrials in which X = xiirrespectiveofthevalueof Y Computational Neuroeconomics and Neuroscience

  18. Probability Theory Marginal Probability Conditional Probability Joint Probability Computational Neuroeconomics and Neuroscience

  19. Probability Theory Marginal Probability Conditional Probability Joint Probability Computational Neuroeconomics and Neuroscience

  20. Probability Theory Marginal Probability Conditional Probability Joint Probability Computational Neuroeconomics and Neuroscience

  21. Probability Theory Sum Rule Computational Neuroeconomics and Neuroscience

  22. Probability Theory Product Rule Computational Neuroeconomics and Neuroscience

  23. The Rules of Probability Sum Rule Product Rule Computational Neuroeconomics and Neuroscience

  24. Bayes’ Theorem p(X,Y) = p(Y,X) T. Bayes (1702-1761) P.-S. Laplace (1749-1827) Computational Neuroeconomics and Neuroscience

  25. Bayes’ Theorem Polynomial curve fitting problem T. Bayes (1702-1761) posterior  likelihood × prior P.-S. Laplace (1749-1827) Computational Neuroeconomics and Neuroscience

  26. Probability Densities Computational Neuroeconomics and Neuroscience

  27. Expectations Expectation of f(x) is the average value of some function f(x) under a probability distribution p(x) Expectation for a discrete distribution: Expectation for a continuous distribution: Computational Neuroeconomics and Neuroscience

  28. The Gaussian Distribution Computational Neuroeconomics and Neuroscience

  29. Gaussian Parameter Estimation Likelihood function Computational Neuroeconomics and Neuroscience

  30. Maximum (Log) Likelihood Computational Neuroeconomics and Neuroscience

  31. Curve Fitting Re-visited Computational Neuroeconomics and Neuroscience

  32. Maximum Likelihood Determine by minimizing sum-of-squares error, . Computational Neuroeconomics and Neuroscience

  33. Outline Introduction Probability Theory Decision Theory Information Theory Computational Neuroeconomics and Neuroscience

  34. Decision Theory Used with probability theory to make optimal decisions Input vector x, target vector t Regression: t is continuous Classification: t will consist of class labels Summary of uncertainty associated is given by Inference problem: is to obtain from data Decision problem: make specific prediction for value of t and take specific actions based on t Inference step Decision step Determine either or . For given x, determine optimal t. Computational Neuroeconomics and Neuroscience

  35. Medical Diagnosis Problem X-ray image of patient Whether patient has cancer or not Input vector x: set of pixel intensities Output variable t: whether cancer or not C1 = cancer; C2 = no cancer General inference problem is to determine which gives most complete description of situation In the end we need to decide whether to give treatment or not  Decision theory helps do this Computational Neuroeconomics and Neuroscience

  36. Bayes’ Decision How do probabilities play a role in making a decision? Given input x and classes Ck using Bayes’ theorem Quantities in Bayes theorem can be obtained from p(x,Ck) either by marginalizing or conditioning with respect to the appropriate variable Computational Neuroeconomics and Neuroscience

  37. Minimum Expected Loss Example: classify medical images as ‘cancer’ or ‘normal’ • Unequal importance of mistakes • Loss or Cost Function given by Loss Matrix • Utility is negative of Loss • Minimize Average Loss Decision Truth • Regions are chosen to minimize Computational Neuroeconomics and Neuroscience

  38. Why Separate Inference and Decision? Classification problem  broken into two separate stages: – Inference stage: training data is used to learn a model for – Decision stage: posterior probabilities used to make optimal class assignments Three distinct approaches to solving decision problems 1. Generative models: 2. Discriminative models 3. Discriminant functions Computational Neuroeconomics and Neuroscience

  39. Generative models 1. solve inference problem of determining class-conditional densities for each class separately and the prior probabilities 2. use Bayes’ theorem to determine posterior probabilities 3. use decision theory to determine class membership Computational Neuroeconomics and Neuroscience

  40. Discriminative models 1. solve inference problem to determine posterior class probabilities 2. Use decision theory to determine class membership Computational Neuroeconomics and Neuroscience

  41. Discriminant functions 1. Find a function f(x) that maps each input x directly to a class label e.g. two-class problem: f (·) is binary valued f =0 represents C1, f =1 represents C2  Probabilities play no role Computational Neuroeconomics and Neuroscience

  42. Decision Theory for Regression Inference step Determine Decision step For given x, make optimal prediction, y(x), for t Loss function: Computational Neuroeconomics and Neuroscience

  43. Outline Introduction Probability Theory Decision Theory Information Theory Computational Neuroeconomics and Neuroscience

  44. Information theory Quantification of information Degree of surprise: highly improbable  a lot of information highly probable  less information certain  no information Based on probability theory Most important quantity: entropy Computational Neuroeconomics and Neuroscience

  45. Entropy Entropy is the average amount of information expected, weighted with the probability of the random variable  quantifies the uncertainty involved when we encounter this random variable H[x] 0 p(x) Computational Neuroeconomics and Neuroscience

  46. The Kullback-Leibler Divergence • Non-symmetric measure of the difference • between two probability distributions • Also called relative entropy Computational Neuroeconomics and Neuroscience

  47. Mutual Information Two sets of variables: x and y If independent: If not independent: Computational Neuroeconomics and Neuroscience

  48. Mutual Information Mutual information  mutual dependence  shared information  related to the conditional entropy Computational Neuroeconomics and Neuroscience

  49. Course schedule Date Topic Chapter 22-09-2010 Probability, Decision, and Information Theory 1 13-10-2010 Density Estimation, Bayesian Inference 2 20-10-2010 Linear Models for Regression 3 27-10-2010 Linear Models for Classification 4 03-11-2010 Kernel Methods I: Gaussian Processes 6 10-11-2010 Kernel Methods II: SVM and RVM 7 17-11-2010 Probabilistic Graphical Models 8 24-11-2010 Mixture Models and EM 9 01-12-2010 Approximate Inference I: Deterministic Approximations 10 08-12-2010 Approximate Inference II: Stochastic Approximations 11 15-12-2010 Inference on Continuous Latent Variables: PCA, Probabilistic PCA, ICA 12 22-12-2010 Sequential Data: Hidden Markov Models, Linear Dynamical Systems 13 Computational Neuroeconomics and Neuroscience

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