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**Machine Learning CMPT 726 Simon Fraser University**

CHAPTER 1: INTRODUCTION

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**Outline Comments on general approach. Probability Theory.**

Joint, conditional and marginal probabilities. Random Variables. Functions of R.V.s Bernoulli Distribution (Coin Tosses). Maximum Likelihood Estimation. Bayesian Learning With Conjugate Prior. The Gaussian Distribution. More Probability Theory. Entropy. KL Divergence.

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Our Approach The course generally follows statistics, very interdisciplinary. Emphasis on predictive models: guess the value(s) of target variable(s). “Pattern Recognition” Generally a Bayesian approach as in the text. Compared to standard Bayesian statistics: more complex models (neural nets, Bayes nets) more discrete variables more emphasis on algorithms and efficiency

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**Things Not Covered Within statistics:**

Hypothesis testing Frequentist theory, learning theory. Other types of data (not random samples) Relational data Scientific data (automated scientific discovery) Action + learning = reinforcement learning. Could be optional – what do you think?

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Probability Theory Apples and Oranges

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**Probability Theory Marginal Probability Conditional Probability**

Joint Probability Key point to remember: from joint probability can get everything else

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**Probability Theory Sum Rule Product Rule Proof of product rule:**

Sum rule: In logical terms, X = xi, is equivalent to OR(X=xi, Y=yj) where the OR is over all yj.

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**The Rules of Probability**

Sum Rule Product Rule

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Bayes’ Theorem posterior likelihood × prior Exercise: prove this

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**Bayes’ Theorem: Model Version**

Let M be model, E be evidence. P(M|E) proportional to P(M) x P(E|M) Intuition prior = how plausible is the event (model, theory) a priori before seeing any evidence. likelihood = how well does the model explain the data? Exercise: prove this

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**Probability Densities**

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**Transformed Densities**

Important concept: function of random variable. X = g(y), non-linear invertible transformation.

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**Expectations Conditional Expectation (discrete)**

Approximate Expectation (discrete and continuous)

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**Variances and Covariances**

Exercise: prove the variance formula

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**The Gaussian Distribution**

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**Gaussian Mean and Variance**

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**The Multivariate Gaussian**

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**Reading exponential prob formulas**

In infinite space, cannot just form sum Σx p(x) grows to infinity. Instead, use exponential, e.g. p(n) = (1/2)n Suppose there is a relevant feature f(x) and I want to express that “the greater f(x) is, the less probable x is”. Use p(x) = exp(-f(x)).

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**Example: exponential form sample size**

Fair Coin: The longer the sample size, the less likely it is. p(n) = 2-n. ln[p(n)] Try to do matlab plot Sample size n

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**Exponential Form: Gaussian mean**

The further x is from the mean, the less likely it is. ln[p(x)] 2(x-μ)

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**Smaller variance decreases probability**

The smaller the variance σ2, the less likely x is (away from the mean). ln[p(x)] -σ2

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**Minimal energy = max probability**

The greater the energy (of the joint state), the less probable the state is. ln[p(x)] E(x)

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**Gaussian Parameter Estimation**

Likelihood function Independent identically distributed data points

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**Maximum (Log) Likelihood**

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**Properties of and Sample Mean is unbiased estimator**

Sample variance is not

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**Curve Fitting Re-visited**

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Maximum Likelihood Determine by minimizing sum-of-squares error,

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**Predictive Distribution**

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**Frequentism vs. Bayesianism**

Frequentists: probabilities are measured as the frequencies of repeatable events. E.g., coin flips, snow falls in January. Bayesian: in addition, allow probabilities to be attached to parameter values (e.g., P(μ=0). Frequentist model selection: give performance guarantees (e.g., 95% of the time the method is right). Bayesian model selection: choose prior distribution over parameters, maximize resulting cost function (posterior).

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**MAP: A Step towards Bayes**

Key point: Bayesian hyperparameters leads to regularization Determine by minimizing regularized sum-of-squares error,

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**Bayesian Curve Fitting**

Probably skip

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**Bayesian Predictive Distribution**

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Model Selection Cross-Validation

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**Curse of Dimensionality**

Rule of Thumb: 10 datapoints per parameter.

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**Curse of Dimensionality**

Polynomial curve fitting, M = 3 Gaussian Densities in higher dimensions

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Decision Theory Inference step Determine either or . Decision step For given x, determine optimal t.

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**Minimum Misclassification Rate**

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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

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**Why Separate Inference and Decision?**

Minimizing risk (loss matrix may change over time) Unbalanced class priors Combining models

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**Decision Theory for Regression**

Inference step Determine . Decision step For given x, make optimal prediction, y(x), for t. Loss function: Probably skip

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**The Squared Loss Function**

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**Generative vs Discriminative**

Generative approach: Model Use Bayes’ theorem Discriminative approach: Model directly

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**Entropy Important quantity in coding theory statistical physics**

machine learning

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Entropy

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Entropy Coding theory: x discrete with 8 possible states; how many bits to transmit the state of x? All states equally likely

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Entropy General compression principle, widely used, intuitive.

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**The Maximum Entropy Principle**

Commonly used principle for model selection: maximize entropy. Example: In how many ways can N identical objects be allocated M bins? Entropy maximized when

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**Differential Entropy and the Gaussian**

Put bins of width ¢ along the real line Differential entropy maximized (for fixed ) when in which case

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Conditional Entropy

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**The Kullback-Leibler Divergence**

Used in many ML applications as predictive quality measure

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Mutual Information

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