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PATTERN RECOGNITION AND MACHINE LEARNING CHAPTER 3: LINEAR MODELS FOR REGRESSION

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Linear Basis Function Models (1) Example: Polynomial Curve Fitting

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Sum-of-Squares Error Function

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0 th Order Polynomial

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1 st Order Polynomial

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3 rd Order Polynomial

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9 th Order Polynomial

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Linear Basis Function Models (2) Generally where Á j (x) are known as basis functions. Typically, Á 0 (x) = 1, so that w 0 acts as a bias. In the simplest case, we use linear basis functions : Á d (x) = x d.

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Linear Basis Function Models (3) Polynomial basis functions: These are global; a small change in x affect all basis functions.

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Linear Basis Function Models (4) Gaussian basis functions: These are local; a small change in x only affect nearby basis functions. ¹ j and s control location and scale (width).

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Linear Basis Function Models (5) Sigmoidal basis functions: where Also these are local; a small change in x only affect nearby basis functions. ¹ j and s control location and scale (slope).

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Maximum Likelihood and Least Squares (1) Assume observations from a deterministic function with added Gaussian noise: which is the same as saying, Given observed inputs,, and targets,, we obtain the likelihood function where

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Maximum Likelihood and Least Squares (2) Taking the logarithm, we get where is the sum-of-squares error.

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Computing the gradient and setting it to zero yields Solving for w, we get where Maximum Likelihood and Least Squares (3) The Moore-Penrose pseudo-inverse,.

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Maximum Likelihood and Least Squares (4) Maximizing with respect to the bias, w 0, alone, we see that We can also maximize with respect to ¯, giving

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Geometry of Least Squares Consider S is spanned by. w ML minimizes the distance between t and its orthogonal projection on S, i.e. y. N -dimensional M -dimensional

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Sequential Learning Data items considered one at a time (a.k.a. online learning); use stochastic (sequential) gradient descent: This is known as the least-mean-squares (LMS) algorithm. Issue: how to choose ´ ?

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1 st Order Polynomial

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3 rd Order Polynomial

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9 th Order Polynomial

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Over-fitting Root-Mean-Square (RMS) Error:

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Polynomial Coefficients

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Regularized Least Squares (1) Consider the error function: With the sum-of-squares error function and a quadratic regularizer, we get which is minimized by Data term + Regularization term ¸ is called the regularization coefficient.

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Regularized Least Squares (1) Is it true that or

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Regularized Least Squares (2) With a more general regularizer, we have LassoQuadratic

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Regularized Least Squares (3) Lasso tends to generate sparser solutions than a quadratic regularizer.

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Understanding Lasso Regularizer Lasso regularizer plays the role of thresholding

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Multiple Outputs (1) Analogously to the single output case we have: Given observed inputs,, and targets,, we obtain the log likelihood function

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Multiple Outputs (2) Maximizing with respect to W, we obtain If we consider a single target variable, t k, we see that where, which is identical with the single output case.

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The Bias-Variance Decomposition (1) Recall the expected squared loss, where The second term of E [L] corresponds to the noise inherent in the random variable t. What about the first term?

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The Bias-Variance Decomposition (2) Suppose we were given multiple data sets, each of size N. Any particular data set, D, will give a particular function y(x; D ). We then have

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The Bias-Variance Decomposition (3) Taking the expectation over D yields

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The Bias-Variance Decomposition (4) Thus we can write where

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Bias-Variance Tradeoff

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The Bias-Variance Decomposition (5) Example: 25 data sets from the sinusoidal, varying the degree of regularization, ¸.

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The Bias-Variance Decomposition (6) Example: 25 data sets from the sinusoidal, varying the degree of regularization, ¸.

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The Bias-Variance Decomposition (7) Example: 25 data sets from the sinusoidal, varying the degree of regularization, ¸.

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The Bias-Variance Trade-off From these plots, we note that an over-regularized model (large ¸ ) will have a high bias, while an under- regularized model (small ¸ ) will have a high variance.

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Bayesian Linear Regression (1) Define a conjugate prior over w Combining this with the likelihood function and using results for marginal and conditional Gaussian distributions, gives the posterior where

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Bayesian Linear Regression (2) A common choice for the prior is for which Next we consider an example …

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Bayesian Linear Regression (3) 0 data points observed Prior Data Space

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Bayesian Linear Regression (4) 1 data point observed Likelihood Posterior Data Space

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Bayesian Linear Regression (5) 2 data points observed Likelihood Posterior Data Space

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Bayesian Linear Regression (6) 20 data points observed Likelihood Posterior Data Space

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Predictive Distribution (1) Predict t for new values of x by integrating over w : where

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Predictive Distribution (1) How to compute the predictive distribution ?

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Predictive Distribution (2) Example: Sinusoidal data, 9 Gaussian basis functions, 1 data point

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Predictive Distribution (3) Example: Sinusoidal data, 9 Gaussian basis functions, 2 data points

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Predictive Distribution (4) Example: Sinusoidal data, 9 Gaussian basis functions, 4 data points

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Predictive Distribution (5) Example: Sinusoidal data, 9 Gaussian basis functions, 25 data points

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Salesman’s Problem Given a consumer a described by a vector x a product b to sell with base cost c estimated price distribution of b in the mind of a is Pr(t|x, w) = N(x T w, 2 ) What is price should we offer to a ?

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Salesman’s Problem

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Bayesian Model Comparison (1) How do we choose the ‘right’ model?

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Bayesian Model Comparison (1) How do we choose the ‘right’ model? Assume we want to compare models M i, i=1, …,L, using data D ; this requires computing PosteriorPrior Model evidence or marginal likelihood

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Bayesian Model Comparison (1) How do we choose the ‘right’ model? PosteriorPrior Model evidence or marginal likelihood

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Bayesian Model Comparison (3) For a model with parameters w, we get the model evidence by marginalizing over w

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Bayesian Model Comparison (4) For a given model with a single parameter, w, con- sider the approximation where the posterior is assumed to be sharply peaked.

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Bayesian Model Comparison (4) For a given model with a single parameter, w, con- sider the approximation where the posterior is assumed to be sharply peaked.

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Bayesian Model Comparison (5) Taking logarithms, we obtain With M parameters, all assumed to have the same ratio, we get Negative Negative and linear in M.

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Bayesian Model Comparison (5) Data matching Model Complexity

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Bayesian Model Comparison (6) Matching data and model complexity

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What You Should Know Least square and its maximum likelihood interpretation Sequential learning for regression Regularization and its effect Bayesian regression Predictive distribution Bias variance tradeoff Bayesian model selection

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