Berkeley COMPSCI 294 - The Likelihood Principle (6 pages)

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The Likelihood Principle



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The Likelihood Principle

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6
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University of California, Berkeley
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Compsci 294 - Special Topics
Special Topics Documents

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ISyE8843A Brani Vidakovic 1 Handout 2 The Likelihood Principle Likelihood principle concerns foundations of statistical inference and it is often invoked in arguments about correct statistical reasoning Let f x be a conditional distribution for X given the unknown parameter For the observed data X x the function f x considered as a function of is called the likelihood function The name likelihood implies that given x the value of is more likely to be the true parameter than 0 if f x f x 0 Likelihood Principle In the inference about after x is observed all relevant experimental information is contained in the likelihood function for the observed x Furthermore two likelihood functions contain the same information about if they are proportional to each other Remark The maximum likelihood estimation does satisfy the likelihood principle Figure 1 Leonard Jimmie Savage Born November 20 1917 Detroit Michigan Died November 1 1971 New Haven Connecticut The following example quoted by Lindley and Phillips 1976 is an argument of Leonard Savage discussed at Purdue Symposium 1962 It shows that the inference can critically depend on the likelihood principle Example 1 Testing fairness Suppose we are interested in testing the unknown probability of heads for possibly biased coin Suppose H0 1 2 v s H1 1 2 1 An experiment is conducted and 9 heads and 3 tails are observed This information is not sufficient to fully specify the model f x A rashomonian analysis follows Scenario 1 Number of flips n 12 is predetermined Then number of heads X is binomial B n with probability mass function n x 12 9 n x P X x f x 1 1 3 220 9 1 3 x 9 For a frequentist the p value of the test is 12 X 12 1 12 66 220 P X 9 H0 1 2 x 1 1 2 12 x 0 073 x 212 x 9 and if you recall the classical testing the H0 is not rejected at level 0 05 Scenario 2 Number of tails successes 3 is predetermined i e the flipping is continued until 3 tails are observed Then X number of heads failures until 3 tails appear is Negative



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