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).In the Likelihood function the~x are known and fixed, while the~aarethe Simple Example Suppose the probabilitydistribution for the data isf( ,a)=a2 e a.

The Maximum Likelihood Principle The maximum likelihood principle is one way to extract information from the likelihood function. It says, in e↵ect, “Use the modal values of the parameters.” The Maximum Likelihood Principle Given data points ~x drawn from a joint probability dis-tribution whose functional form is known to be f(~⇠,~a),

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