Readings listed in class website Bayesian point estimation Gaussians Linear Regression Bias Variance Tradeoff Machine Learning 10701 15781 Carlos Guestrin Carnegie Mellon University September 14th 2009 Carlos Guestrin 2005 2009 1 What about prior Billionaire says Wait I know that the thumbtack is close to 50 50 What can you do for me now You say I can learn it the Bayesian way Rather than estimating a single we obtain a distribution over possible values of Carlos Guestrin 2005 2009 2 1 Bayesian Learning Use Bayes rule Or equivalently Carlos Guestrin 2005 2009 3 Bayesian Learning for Thumbtack Likelihood function is simply Binomial What about prior Represent expert knowledge Simple posterior form Conjugate priors Closed form representation of posterior For Binomial conjugate prior is Beta distribution Carlos Guestrin 2005 2009 4 2 Beta prior distribution P Mean Mode Likelihood function Posterior Carlos Guestrin 2005 2009 5 Posterior distribution Prior Data H heads and T tails Posterior distribution Carlos Guestrin 2005 2009 6 3 Using Bayesian posterior Posterior distribution Bayesian inference No longer single parameter Integral is often hard to compute Carlos Guestrin 2005 2009 7 MAP Maximum a posteriori approximation As more data is observed Beta is more certain MAP use most likely parameter Carlos Guestrin 2005 2009 8 4 MAP for Beta distribution MAP use most likely parameter Beta prior equivalent to extra thumbtack flips As N prior is forgotten But for small sample size prior is important Carlos Guestrin 2005 2009 9 What you need to know Go to the recitation on intro to probabilities And other recitations too Point estimation MLE Bayesian learning MAP Carlos Guestrin 2005 2009 10 5 What about continuous variables Billionaire says If I am measuring a continuous variable what can you do for me You say Let me tell you about Gaussians Carlos Guestrin 2005 2009 11 Some properties of Gaussians affine transformation multiplying by scalar and adding a constant N 2 Y aX b Y N a b a2 2 X Sum of Gaussians N X 2X Y N Y 2Y Z X Y Z N X Y 2X 2Y X Carlos Guestrin 2005 2009 12 6 Learning a Gaussian Collect a bunch of data Hopefully i i d samples e g exam scores Learn parameters Mean Variance Carlos Guestrin 2005 2009 13 MLE for Gaussian Prob of i i d samples D x1 xN Log likelihood of data Carlos Guestrin 2005 2009 14 7 Your second learning algorithm MLE for mean of a Gaussian What s MLE for mean Carlos Guestrin 2005 2009 15 MLE for variance Again set derivative to zero Carlos Guestrin 2005 2009 16 8 Learning Gaussian parameters MLE BTW MLE for the variance of a Gaussian is biased Expected result of estimation is not true parameter Unbiased variance estimator Carlos Guestrin 2005 2009 17 Bayesian learning of Gaussian parameters Conjugate priors Mean Gaussian prior Variance Wishart Distribution Prior for mean Carlos Guestrin 2005 2009 18 9 MAP for mean of Gaussian Carlos Guestrin 2005 2009 19 Prediction of continuous variables Billionaire says Wait that s not what I meant You says Chill out dude He says I want to predict a continuous variable for continuous inputs I want to predict salaries from GPA You say I can regress that Carlos Guestrin 2005 2009 20 10 The regression problem Instances xj tj Learn Mapping from x to t x Hypothesis space Given basis functions Find coeffs w w1 wk Why is this called linear regression model is linear in the parameters Precisely minimize the residual squared error 21 Carlos Guestrin 2005 2009 The regression problem in matrix notation N sensors K basis func N sensors K basis functions Carlos Guestrin 2005 2009 measurements weights 22 11 Regression solution simple matrix operations where k k matrix for k basis functions Carlos Guestrin 2005 2009 k 1 vector 23 Announcements 1 Readings associated with each class See course website for specific sections extra links and further details Visit the website frequently Recitations Thursdays 5 00 6 20pm in Gates Hillman 6115 Special recitation on Matlab Today Carlos Guestrin 2005 2009 5 00 6 20pm GHC 6115 24 12 Announcement 2 First homework out later today Download from course website Start early Due Sept 30th Also HW0 To expedite grading Due this Thursday Just to make sure you can access the submission directory there are 4 questions please hand in 4 stapled separate parts one for each question Privacy policy for returning homeworks and exams We write grades in second page of homework or exam We want to handout graded homeworks in class but to do that CMU requires you to sign a waiver acknowledging that someone may turn the page and find your grade If you are not comfortable with this possibility let us know and your homework will be available for pick up from Michelle Martin at GHC 8001 Carlos Guestrin 2005 2009 25 But why Billionaire again says Why sum squared error You say Gaussians Dr Gateson Gaussians Model prediction is linear function plus Gaussian noise t i wi hi x Learn w using MLE Carlos Guestrin 2005 2009 26 13 Maximizing log likelihood Maximize Least squares Linear Regression is MLE for Gaussians Carlos Guestrin 2005 2009 27 Applications Corner 1 Predict stock value over time from past values other relevant vars e g weather demands etc Carlos Guestrin 2005 2009 28 14 Applications Corner 2 50 OFFICE 12 9 54 OFFICE 51 QUIET PHONE 11 8 52 49 53 16 15 10 CONFERENCE 13 14 17 7 18 STORAGE 48 LAB ELEC Measure temperatures at some locations Predict temperatures throughout the environment COPY 5 47 19 6 4 46 20 21 3 45 2 SERVER 44 KITCHEN 39 37 23 33 35 29 40 42 41 22 1 43 27 31 38 36 34 25 32 30 28 24 26 Guestrin et al 04 29 Carlos Guestrin 2005 2009 Applications Corner 3 Predict when a sensor will fail based several variables age chemical exposure number of hours used Carlos Guestrin 2005 2009 30 15 Bias Variance tradeoff Intuition Model too simple does not fit the data well A biased solution Model too complex small changes to the data solution changes a lot A high variance solution Carlos Guestrin 2005 2009 31 Squared Bias of learner Given dataset D with m samples learn function h x If you sample a different datasets you will learn different h x Expected hypothesis ED h x Bias difference between what you expect to learn and truth Measures how well you expect to represent true solution Decreases with more complex model Carlos Guestrin 2005 2009 32 16 Variance of learner Given a dataset D with m samples you learn function h x If you sample a different datasets you will learn different h x Variance difference between
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