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INTRODUCTION TO MACHINE LEARNING Bayesian Estimation

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Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 2 Estimating parameters of a model from the data Regression Classification Have some prior knowledge on possible parameter range Before looking at the data Distribution of the parameter

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Generative Model Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 3

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Bayes Rule Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 4

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Multinomial variable Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 5 Sample of multinomial data taking one of K state Sample Likelihood Good way to specify prior distribution on state probabilities q

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Dirichlet Distribution Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 6 Probability of each combination of state probabilities Parameters: approximate proportions of data in state q i

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Posteriori Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 7 Likelihood Posteriori

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Conjugate Prior Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 8 Posteriori and prior have the same form Sequential learning Instance by instance Calculate posteriori for the current item Make it prior for the next item

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Continuous Variable Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 9 Instances are Gaussian Distributed with unknown parameters Conjugate prior

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Continuous Variable Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 10 Posteriori Mean is weighted combination of sample mean and prior mean More samples, estimate is closer to m Little prior uncertainty=>closer to prior mean

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Precision/Variance Prior Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 11 More convenient to work with precision Conjugate prior is a Gamma Distribution

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Precision Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 12 Posteriori is a weighted sum of prior and sample statistics

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Parameter Estimation Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 13 Used prior to refine distribution parameter estimates User prior to refine parameter of some function of the input Regression Classification discriminant

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Regression Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 14

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Regression Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 15 Maximum Likelihood Prediction Gaussian Prior

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Prior on weights Based on E Alpaydın 2004 Introduction to Machine Learning © The MIT Press (V1.1) 16

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Examples 17

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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.

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.

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