5/17/2012 1 A N COGNI ION PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 1: INTRODUCTION Example Handwritten Digit Recognition
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5/17/2012
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A N COGNI IONPATTERN RECOGNITION AND MACHINE LEARNINGCHAPTER 1: INTRODUCTION
Example
Handwritten Digit Recognition
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Polynomial Curve Fitting
Sum‐of‐Squares Error Function
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0th Order Polynomial
1st Order Polynomial
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3rd Order Polynomial
9th Order Polynomial
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Over‐fitting
Root‐Mean‐Square (RMS) Error:
Polynomial Coefficients
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Data Set Size:
9th Order Polynomial
Data Set Size:
9th Order Polynomial
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Regularization
Penalize large coefficient values
Regularization:
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Regularization:
Regularization: vs.
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Polynomial Coefficients
Probability Theory
Apples and Oranges
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Probability Theory
Marginal ProbabilityMarginal Probability
Conditional ProbabilityJoint Probability yy
Probability Theory
Sum RuleSum Rule
Product Rule
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The Rules of Probability
Sum Rule
Product Rule
Bayes’ Theorem
posterior likelihood × priorposterior likelihood × prior
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Probability Densities
Transformed Densities
JFMS5
Slide 24
JFMS5 This figure was taken from Solution 1.4 in the web-edition of the solutions manual for PRML, available at http://research.microsoft.com/~cmbishop/PRML. A more thorough explanation of what the figure shows is provided in the text of the solution.Markus Svensén, 11/14/2007
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Expectations
Conditional Expectation(discrete)
i iApproximate Expectation(discrete and continuous)
Variances and Covariances
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The Gaussian Distribution
Gaussian Mean and Variance
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The Multivariate Gaussian
Gaussian Parameter Estimation
Likelihood functionLikelihood function
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Maximum (Log) Likelihood
Properties of and
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Curve Fitting Re‐visited
Maximum Likelihood
Determine b minimi in s m of sq ares errorDetermine by minimizing sum‐of‐squares error, .
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Predictive Distribution
MAP: A Step towards Bayes
Determine by minimizing regularized sum‐of‐squares error, .
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Bayesian Curve Fitting
Bayesian Predictive Distribution
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Model Selection
Cross‐Validation
Curse of Dimensionality
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Curse of Dimensionality
Polynomial curve fitting, M = 3
Gaussian Densities in higher dimensions
Decision Theory
Inference step
Determine either orDetermine either or .
Decision step
For given x, determine optimal t.
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Minimum Misclassification Rate
Minimum Expected Loss
Example: classify medical images as ‘cancer’ or ‘normal’
Decision
Truth
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Minimum Expected Loss
Regions are chosen to minimize
Reject Option
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Why Separate Inference and Decision?
• Minimizing risk (loss matrix may change over time)
• Reject option• Reject option
• Unbalanced class priors
• Combining models
Decision Theory for Regression
Inference step
DetermineDetermine .
Decision step
For given x, make optimal prediction, y(x), for t.
Loss function:
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The Squared Loss Function
Generative vs Discriminative
Generative approach:
ModelModel
Use Bayes’ theorem
Discriminative approach:
Model directly
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Entropy
Important quantity in• coding theory• statistical physics
hi l i• machine learning
Entropy
Coding theory: xdiscrete with 8 possible states; how many bits to transmit the state of x?bits to transmit the state of x?
All states equally likely
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Entropy
Entropy
In how many ways can N identical objects be allocated Mbins?bins?
Entropy maximized when
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Entropy
Differential Entropy
Put bins of width ∆ along the real line
Differential entropy maximized (for fixed ) when
in which case
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Conditional Entropy
The Kullback‐Leibler Divergence
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Mutual Information
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