gaussians-regression-annotated

# gaussians-regression-annotated - Readings listed in class...

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1 1 Bayesian point estimation Gaussians Linear Regression Bias-Variance Tradeoff Machine Learning – 10701/15781 Carlos Guestrin Carnegie Mellon University September 14 th , 2009 Readings listed in class website ©Carlos Guestrin 2005-2009 2 What about prior s Billionaire says: Wait, I know that the thumbtack is “close” to 50-50. What can you do for me now? s You say: I can learn it the Bayesian way… s Rather than estimating a single θ , we obtain a distribution over possible values of θ ©Carlos Guestrin 2005-2009

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2 3 Bayesian Learning s Use Bayes rule: s Or equivalently: ©Carlos Guestrin 2005-2009 4 Bayesian Learning for Thumbtack s Likelihood function is simply Binomial: s What about prior? b Represent expert knowledge b Simple posterior form s Conjugate priors: b Closed-form representation of posterior b For Binomial, conjugate prior is Beta distribution ©Carlos Guestrin 2005-2009
3 5 Beta prior distribution – P( θ ) s Likelihood function: s Posterior: Mean: Mode: ©Carlos Guestrin 2005-2009 6 Posterior distribution s Prior: s Data: α H heads and α T tails s Posterior distribution: ©Carlos Guestrin 2005-2009

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4 7 Using Bayesian posterior s Posterior distribution: s Bayesian inference: b No longer single parameter: b Integral is often hard to compute ©Carlos Guestrin 2005-2009 8 MAP: Maximum a posteriori approximation s As more data is observed, Beta is more certain s MAP: use most likely parameter: ©Carlos Guestrin 2005-2009
5 9 MAP for Beta distribution s MAP: use most likely parameter: s Beta prior equivalent to extra thumbtack flips

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gaussians-regression-annotated - Readings listed in class...

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