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Pattern Recognition and Machine LearningChapter 1: Introduction
Example Handwritten Digit Recognition
Polynomial Curve Fitting
Sum-of-Squares Error Function
0th Order Polynomial
1st Order Polynomial
3rd Order Polynomial
9th Order Polynomial
Over-fitting Root-Mean-Square (RMS) Error:
Data Set Size: 9th Order Polynomial
Data Set Size: 9th Order Polynomial
Regularization Penalize large coefficient values
Probability Theory Apples and Oranges
Probability Theory Marginal Probability Conditional ProbabilityJoint Probability
Probability Theory Sum Rule Product Rule
The Rules of ProbabilitySum Rule Product Rule
Bayes’ Theorem posterior likelihood × prior
Expectations Conditional Expectation (discrete)Approximate Expectation (discrete and continuous)
Variances and Covariances
The Gaussian Distribution
Gaussian Mean and Variance
The Multivariate Gaussian
Gaussian Parameter EstimationLikelihood function
Maximum (Log) Likelihood
Properties of and
Curve Fitting Re-visited
Maximum Likelihood Determine by minimizing sum-of-squares error,
MAP: A Step towards BayesDetermine by minimizing regularized sum-of-squares error,
Bayesian Curve Fitting
Bayesian Predictive Distribution
Model Selection Cross-Validation
Curse of Dimensionality
Curse of DimensionalityPolynomial curve fitting, M = 3 Gaussian Densities in higher dimensions
Decision Theory Inference step Determine either or . Decision step For given x, determine optimal t.
Minimum Misclassification Rate
Minimum Expected Loss Example: classify medical images as ‘cancer’ or ‘normal’ Decision Truth
Minimum Expected Loss Regions are chosen to minimize
Why Separate Inference and Decision?Minimizing risk (loss matrix may change over time) Reject option Unbalanced class priors Combining models
Decision Theory for RegressionInference step Determine . Decision step For given x, make optimal prediction, y(x), for t. Loss function:
The Squared Loss Function
Generative vs DiscriminativeGenerative approach: Model Use Bayes’ theorem Discriminative approach: Model directly
Entropy Important quantity in coding theory statistical physicsmachine learning
Entropy Coding theory: x discrete with 8 possible states; how many bits to transmit the state of x? All states equally likely
Entropy In how many ways can N identical objects be allocated M bins? Entropy maximized when
Differential Entropy Put bins of width ¢ along the real line Differential entropy maximized (for fixed ) when in which case
The Kullback-Leibler Divergence
Pattern Recognition and Machine Learning
: INTRODUCTION TO Machine Learning Parametric Methods.
Notes Sample vs distribution “m” vs “µ” and “s” vs “σ” Bias/Variance Bias: Measures how much the learnt model is wrong disregarding noise Variance: Measures.
INTRODUCTION TO MACHINE LEARNING Bayesian Estimation.
Polynomial Curve Fitting BITS C464/BITS F464 Navneet Goyal Department of Computer Science, BITS-Pilani, Pilani Campus, India.
ECE 8443 – Pattern Recognition LECTURE 05: MAXIMUM LIKELIHOOD ESTIMATION Objectives: Discrete Features Maximum Likelihood Resources: D.H.S: Chapter 3 (Part.
Supervised Learning Recap
Chapter 2: Bayesian Decision Theory (Part 2) Minimum-Error-Rate Classification Classifiers, Discriminant Functions and Decision Surfaces The Normal Density.
Pattern Classification, Chapter 2 (Part 2) 0 Pattern Classification All materials in these slides were taken from Pattern Classification (2nd ed) by R.
Pattern Classification. Chapter 2 (Part 1): Bayesian Decision Theory (Sections ) Introduction Bayesian Decision Theory–Continuous Features.
Pattern Classification Chapter 2 (Part 2)0 Pattern Classification All materials in these slides were taken from Pattern Classification (2nd ed) by R. O.
Chapter 4: Linear Models for Classification
What is Statistical Modeling
Laboratory for Social & Neural Systems Research (SNS) PATTERN RECOGNITION AND MACHINE LEARNING Institute of Empirical Research in Economics (IEW)
Classification and risk prediction
G. Cowan Lectures on Statistical Data Analysis 1 Statistical Data Analysis: Lecture 10 1Probability, Bayes’ theorem, random variables, pdfs 2Functions.
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