UI STAT 4520 - The Likelihood Principle

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122S:138The Likelihood PrincipleLecture 22Nov. 29, 2006Kate Cowles374 SH, [email protected] likelihood principle• Suppose that two different experiments mayinform about an unknown parameter θ• Suppose the outcomes of the experiementsare respectively y∗and z∗• Suppose the likelihoods for θ resulting fromthe two experiements are proportional; thatisp(y∗; θ) = c p(z∗; θ)where c is a constant• Then the information about θ contained inboth experiments is equivalent3Another way to state the likelihood prin-ciple• For a given sample of data, any two proba-bility models p(y|θ) that have the same like-lihood function yield the same inference forθ.• With regard to the information contained inthe data about the unknown parameter(s),only the actual observed data y is relevant.– No other possible outcomes∗ Contrast this with the frequents p-valuethe probability assuming H0is true, ofgetting a test statistic as extreme as, ormore extreme than, the value that wasactually obtained– Not the researchers’ intentions4Example• We are given a coin. We are interested inestimating θ, the probability of obtaining ahead on a single flip.• We want to test the hypotheses:H0: θ =12Ha: θ >12• Experiment consists of flipping coin 12 timesindependently.• Result is 9 heads and 3 tails.5Example, continued• There are (at least) two possible ways theexperiment might have been conducted:– Design 1: do 12 flips. The random variableY is the number of heads obtained in n =12 flips.– Design 2: Flip the coin until 9 heads areobtained. Random variable Y is the num-ber of tails that are obtained before theninth head.• Frequentist inference for θ would be differentdepending on which design is used.• Bayesian inference would be the same underboth designs because the likelihoods are pro-portional.• The negative binomial distribution– Y = the number of failures observed ina sequence of independent Bernoulli trials6before the kthsuccess– Y ∼ NB(k, p)– p(y|p) =k + y − 1ypk(1 − p)y– E(Y ) =k(1−p)p7Implications of the likelihood principle• the stopping rule principle• the likelihood principle and reference priors8“Stopping rules” are often used in de-signing frequentist statistical studies• instead of a fixed sample size• to make it possible to stop a study early ifthe results are in• particularly common in clinical trials– reducing the size and duration of a clinicaltrial reduces the number of patients whoare exposed to the treatment that will befound to be inferior and speeds up the dis-semination of the results to the medicalcommunity• Frequentist statisticians must choose the stop-ping rule before the experiment is conductedand adhere to it exactly– deviations can produce serious errors if afrequentist analysis is used9• Large frequentist literature on how to controlthe overall probability of Type I error whileallowing for more than one analysis of thedata10Stopping Rule Principle• In a sequential experiment, the evidence pro-vided by the experiment about the value ofthe unknown parameter(s) θ should not de-pend on the stopping rule• follows directly from the likelihood principle11Jeffreys’ priors and the likelihood prin-ciple• recall Jeffreys’ prior– “reference” prior– noninformative– invariant to transformations of parameters– p(θ) ∝ [I(θ)]12where I(θ) is the expected Fisher informa-tion for θ• Jeffreys’ prior when likelihood is Binomial(n,θ)p(θ) ∝ θ−12(1 − θ)−12= Beta(12,12)• Jeffreys’ prior when likelihood is negative binomial(k,θ)p(θ) ∝ θ−1(1 − θ)−12= Beta(0,12)12• So use of Jeffreys’ prior in some cases canviolate the likelihood


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UI STAT 4520 - The Likelihood Principle

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