11Bayesian point estimationGaussiansLinear RegressionBias-Variance TradeoffMachine Learning – 10701/15781Carlos GuestrinCarnegie Mellon UniversitySeptember 14th, 2009Readings listed in class website©Carlos Guestrin 2005-20092What 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 θ, we obtain a distribution over possible values of θ©Carlos Guestrin 2005-200923Bayesian Learning Use Bayes rule: Or equivalently:©Carlos Guestrin 2005-20094Bayesian 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 2005-200935Beta prior distribution – P(θ) Likelihood function: Posterior:Mean:Mode: ©Carlos Guestrin 2005-20096Posterior distribution Prior: Data: αHheads and αTtails Posterior distribution: ©Carlos Guestrin 2005-200947Using Bayesian posterior Posterior distribution: Bayesian inference: No longer single parameter: Integral is often hard to compute©Carlos Guestrin 2005-20098MAP: Maximum a posteriori approximation As more data is observed, Beta is more certain MAP: use most likely parameter:©Carlos Guestrin 2005-200959MAP 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 2005-200910What you need to know Go to the recitation on intro to probabilities And, other recitations too Point estimation: MLE Bayesian learning MAP©Carlos Guestrin 2005-2009611©Carlos Guestrin 2005-2009What 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…12©Carlos Guestrin 2005-2009Some properties of Gaussians affine transformation (multiplying by scalar and adding a constant) X ~ N(µ,σ2) Y = aX + b → Y ~ N(aµ+b,a2σ2) Sum of Gaussians X ~ N(µX,σ2X) Y ~ N(µY,σ2Y) Z = X+Y → Z ~ N(µX+µY, σ2X+σ2Y)713©Carlos Guestrin 2005-2009Learning a Gaussian Collect a bunch of data Hopefully, i.i.d. samples e.g., exam scores Learn parameters Mean Variance14©Carlos Guestrin 2005-2009MLE for Gaussian Prob. of i.i.d. samples D={x1,…,xN}: Log-likelihood of data:815©Carlos Guestrin 2005-2009Your second learning algorithm:MLE for mean of a Gaussian What’s MLE for mean?16©Carlos Guestrin 2005-2009MLE for variance Again, set derivative to zero:917©Carlos Guestrin 2005-2009Learning Gaussian parameters MLE: BTW. MLE for the variance of a Gaussian is biased Expected result of estimation is not true parameter! Unbiased variance estimator:18©Carlos Guestrin 2005-2009Bayesian learning of Gaussian parameters Conjugate priors Mean: Gaussian prior Variance: Wishart Distribution Prior for mean:1019©Carlos Guestrin 2005-2009MAP for mean of Gaussian20©Carlos Guestrin 2005-2009Prediction 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…1121©Carlos Guestrin 2005-2009The 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:22©Carlos Guestrin 2005-2009The regression problem in matrix notationN sensorsK basis functionsN sensorsmeasurementsweightsK basis func1223©Carlos Guestrin 2005-2009Regression solution = simple matrix operationswherek×k matrix for k basis functions k×1 vector24©Carlos Guestrin 2005-2009Announcements 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! 5:00-6:20pm GHC 61151325©Carlos Guestrin 2005-2009Announcement 2 First homework out later today! Download from course website! Start early!!! :) Due Sept. 30th Also, HW0! Due this Thursday! Just to make sure you can access the submission directory To expedite grading: 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 800126©Carlos Guestrin 2005-2009 Billionaire (again) says: Why sum squared error??? You say: Gaussians, Dr. Gateson, Gaussians… Model: prediction is linear function plus Gaussian noise t = ∑iwihi(x) + ε Learn w using MLEBut, why?1427©Carlos Guestrin 2005-2009Maximizing log-likelihoodMaximize:Least-squares Linear Regression is MLE for Gaussians!!!28©Carlos Guestrin 2005-2009Applications Corner 1 Predict stock value over time from past values other relevant vars e.g., weather, demands, etc.1529©Carlos Guestrin 2005-2009Applications Corner 2 Measure temperatures at some locations Predict temperatures throughout the environmentSERVERLABKITCHENCOPYELECPHONEQUIETSTORAGECONFERENCEOFFICEOFFICE50515253544648494743454442 41373938 36333610111213141516171920212224252628303231272923189587434123540[Guestrin et al. ’04] 30©Carlos Guestrin 2005-2009Applications Corner 3 Predict when a sensor will fail based several variables age, chemical exposure, number of hours used,…1631©Carlos Guestrin 2005-2009Bias-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
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