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# ex3_slides - CS221 Discussion Section 1 Least squares...

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CS221 Discussion Section October 12, 2007 October 12, 2007 1

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Least squares regression Training set : { . . . , ( x ( i ) , y ( i ) ) , . . . } m i =1 Parameters : θ R n Hypothesis : h θ ( x ) = n j =1 θ j x j Parameter estimation : Minimize J ( θ ) = m summationdisplay i =1 ( h θ ( x ( i ) ) - y ( i ) ) 2 October 12, 2007 2
Estimating the bias of a coin On each coin toss : P (head) = p . P (tail) = 1 - p . Experiment : m independent tosses with h heads. What is a good estimate for the bias of the coin ?? October 12, 2007 3

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Maximum likelihood estimation Maximum likelihood principle Estimate parameters to make data as likely as possible. θ MLE = arg max θ P (data; θ ) For the coin example P ( h heads in m tosses) = parenleftbigg m h parenrightbigg · p h (1 - p ) m - h Applying the maximum likelihood principle : p MLE = arg max p parenleftbigg m h parenrightbigg · p h (1 - p ) m - h = h m October 12, 2007 4
Least squares regression Parameter estimation : Minimize J ( θ ) = m i =1 ( h θ ( x ( i ) ) - y ( i ) ) 2 Consider the following model : P ( y | x ; θ ) = 1 2 πσ exp parenleftbigg - ( y - θ T x ) 2 2 σ 2 parenrightbigg bracehtipupleft

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