Introduction to Likelihood Statistics

Introduction to Likelihood Statistics 1. The Likelihood function. 2. Use of the Likelihood function to model data. 3. Comparison to standard frequentist and Bayesean Statistics . Edward L. Robinson* Department of Astronomy and McDonald Observatory University of Texas at Austin *Look for: Data Analysis for Scientists and Engineers Princeton University Press, Sept 2016. The Likelihood Function Let a probability distribution function for have m+1parameters ajf( ,a0,a1, ,am)=f( ,~a),The joint probability distribution for n samples of isf( 1, 2, , n,a0,a1, ,am)=f(~ ,~a). Now make measurements. For each variable ithere is a measuredvalue xi. To obtain the Likelihood function L(~x,~a), replace each variable iwiththe numerical value of the corresponding data point xi:L(~x,~a) f(~x,~a)=f(x1,x2, ,xn,~a).

From elementary probability theory we know that the mean value of the Gaussian is a. Let us estimate the mean value. The likelihood equation for ˆa is 0 = @` @a ˆa = @ @a " 1 2 Xn i=1 (x i a)2 2 i # ˆa. After some algebra we arrive at ˆa = P n Pi=1 w ix i n i=1 w i, where w i= 1/2 This is the same as the weighted average in freqentist ...

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  Elementary, Probability, Elementary probability

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