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Unformatted text preview: 1 Likelihood of Data Consider n I.I.D. random variables X 1 , X 2 , ..., X n X i a sample from density function f (X i | ) o Note: now explicitly specify parameter of distribution We want to determine how likely the observed data (x 1 , x 2 , ..., x n ) is based on density f (X i | ) Define the Likelihood function , L ( ): o This is just a product since X i are I.I.D. Intuitively: what is probability of observed data using density function f (X i | ), for some choice of n i i X f L 1 ) | ( ) ( Demo Maximum Likelihood Estimator The Maximum Likelihood Estimator (MLE) of , is the value of that maximizes L ( ) More formally: More convenient to use log-likelihood function , LL ( ): Note that log function is monotone for positive values o Formally: x y log(x) log(y) for all x, y &gt; 0 So, that maximizes LL ( ) also maximizes L ( ) o Formally: o Similarly, for any positive constant c (not dependent on ): ) ( max arg L MLE n i i n i i X f X f L LL 1 1 ) | ( log ) | ( log ) ( log ) ( ) ( max arg ) ( max arg L LL ) ( max arg ) ( max arg )) ( ( max arg L LL LL c Computing the MLE General approach for finding MLE of Determine formula for LL ( ) Differentiate LL ( ) w.r.t. (each) : To maximize, set Solve resulting (simultaneous) equation to get MLE o Make sure that derived is actually a maximum (and not a minimum or saddle point). E.g., check LL ( MLE ) &lt; LL ( MLE ) This step often ignored in expository derivations So, well ignore it here too (and wont require it in this class) For many standard distributions, someone has already done this work for you. (Yay!) ) ( LL ) ( LL M L E Maximizing Likelihood with Bernoulli Consider I.I.D. random variables X 1 , X 2 , ..., X n...
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- Spring '09