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Bark08 Ghahramani Samlbb 01

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Bayesian Learning
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  Should all Machine Learning be Bayesian?Should all Bayesian models be non-parametric? Zoubin Ghahramani Department of EngineeringUniversity of Cambridge, UK zoubin@eng.cam.ac.ukhttp://learning.eng.cam.ac.uk/zoubin/ BARK 2008  Some Canonical Machine Learning Problems ã  Linear Classification ã  Nonlinear Regression ã  Clustering with Gaussian Mixtures (Density Estimation)  Example: Linear Classification Data:  D = { ( x ( n ) ,y ( n ) ) }  for  n  = 1 ,...,N  data points x ( n ) ∈    D y ( n ) ∈ { +1 , − 1 } xoxxxxxxooooo oxxxxo Parameters:  θ  ∈    D +1 P  ( y ( n ) = +1 | θ , x ( n ) ) =  1 if  D  d =1 θ d x ( n ) d  +  θ 0 ≥ 00 otherwise Goal:  To infer  θ  from the data and to predict future labels  P  ( y |D , x )  Basic Rules of Probability P  ( x )  probability of   xP  ( x | θ )  conditional probability of   x  given  θP  ( x,θ )  joint probability of   x  and  θP  ( x,θ ) =  P  ( x ) P  ( θ | x ) =  P  ( θ ) P  ( x | θ ) Bayes Rule: P  ( θ | x ) =  P  ( x | θ ) P  ( θ ) P  ( x ) Marginalization P  ( x ) =    P  ( x,θ ) dθ
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