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Dealing with Nuisances : Principled and Ad Hoc Methods. Xiao-Li Meng Department of Statistics, Harvard University Joint work with Jingchen Liu (and CHASC). Dealing with Nuisance Parameters. Bringing in a little “Bee”: Posterior Predictive Assessment Giving up a bit of power:

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## Dealing with Nuisances : Principled and Ad Hoc Methods

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**Dealing with Nuisances: Principled and Ad Hoc Methods**Xiao-Li Meng Department of Statistics, Harvard University Joint work with Jingchen Liu (and CHASC) Harvard University**Dealing with Nuisance Parameters**• Bringing in a little “Bee”: Posterior Predictive Assessment • Giving up a bit of power: Using an alternative alternative (or a “working” alternative) • Being further away from the big “Bee”: Profiling via moments Harvard University**A Simple Spectral Model**• A source spectrum with two components: a continuum modeled by a power law E- , and an emission line modeled as a Gaussian profile with a total flux F. • The expected observed flux Fj from the source within an energy bin Ej for a “perfect” instrument is given by where dEj is the energy width of bin j, and j is the Gaussian proportion in bin j. • If the exact energy is observed, then the distribution follows • Reference: Protassov et al (2002) Harvard University**Hypothesis Testing – Notation**• Likelihood L(q|x) = f(x|q), = 0[1, 0\1=; • Null Hypothesis H0: 20 • Alternative Hypothesis HA: 21 • Critical region: C ) Reject null hypothesis if x 2C. • Type I error: P(X 2C | 20) – False negative rate Type II error: P(X 2Cc| 2A) – False positive rate • Power function: p() = P(X 2C | ) • Hypothesis testing of size : p() ·, 8 20 Harvard University**Hypothesis Testing – Likelihood Ratio Test**• Uniformly most powerful (UPM) test: the most powerful test among all the tests with size • Likelihood ratio test (LRT): C(c) = {x : LR(x) > c} • In a simple null hypothesis case, if the UMP test exists, it is likelihood ratio test. Harvard University**Seeking Pivotal Quantity**• Hypothesis testing of size : max20 P(X 2C | ) = , hard to maximize. • Ideally, we seek a pivotal quantity: T(X) -- its distribution is completely known under the null 0 • Then type I error P(T(X)>t| ) = , 820, • Easy to control type I error, but typically it is very hard to find a useful/powerful pivotal quantity. Harvard University**Posterior Predictive Assessment**• p-value = P(T(X) > T(x)| 0), • In the presence of nuisance parameter , under the null, the p-value will be a function of , p() = P(T(X) > T(x) | ). • Posterior predictive p-value: ppp=E(p() | x) = s p() f( | x) d , where f( | x) is the posterior density of . That is, the p-value is calculated under the posterior predictive distribution: f(Xrep|x) = s f(Xrep| 0, ) f( | x) d • Casting doubt on the null hypothesis/model if a ppp is extreme. • Can use realized discrepancy D(X, ): p() = P(D(X , ) > D(x, ) | ). • Can assess the entire posterior distribution of p(). • References: Rubin (1984), Meng (1994), Gelman, Meng and Stern (1996) Harvard University**MODEL 0. There is no emission line.**• MODEL 1. There in an emission line with fixed location in the spectrum, but unknown intensity. • MODEL 2. There is an emission line with unknown location and intensity. • Reference: van Dyk & Kang (2004) Harvard University**The posterior predictive check. The two histograms compare**the observed likelihood ratio test statistics (vertical lines) with 1000 simulations from the posterior predictive distribution. The left plot is the comparison between Model 0 and Model 1, and the right plot is the comparison between Model 0 and Model 2. Both model checks indicate strong evidence for including the emission line. Harvard University**Mixture Model - Testing p = 0**• Hypothesis testing of mixture model • Particularly, f(x | ) / x-, g(x | , ) = (x| , ) (To avoid singularity at the 0, when > 1, we need to truncate the density away from 0. Without losing generality, we assume x > 1.) • LR is not a pivotal quantity under this model. But if we use a different model for the g component, then we can construct a LR test that is a pivotal quantity. • Let y = log (x) and = 1 / ( - 1), then we can model Harvard University**Difference between the Two Choices**Density: normal(1, 0.2) Vs log-normal(0,0.2) Density: normal(1, 0.02) Vs log-normal(0,0.02) Harvard University**Power Comparison: LR under log-normal mixture vs LR under**normal mixture when the true model is (almost) normal mixture =1, = 1, = 0.02 are treated as known p = 0.0001, 0.005, 0.01, 0.015, 0.02, 0.03 Only one free parameter, p. =1, = 1, = 0.3 are treated as known p = 0.0001, 0.005, 0.01, 0.015, 0.02, 0.03 Only one free parameter, p. Harvard University**Likelihood Ratio Test and Pivotal Quantity**• H0: p = 0, HA: p > 0 • The LRT is pivotal quantity, i.e., the distribution of likelihood ratio is free of . • The maximization can be done via the EM algorithm by viewing the subgroup membership as missing data. Harvard University**Multiple Modes log Likelihood**Likelihood of given that = 1, = 0.02, p = 0.01, the sample size is 500 Harvard University**A “Profiled” Likelihood Ratio Test**• “Profile likelihood” via moment • Lp( p, , | y) can be maximized via numerical optimization method (the correct likelihood was harder to maximize without using EM). • Let’s define critical region C( c ) = {y | LRp(y) > c} Harvard University**A Sketch of Proof**Harvard University**Demonstrating a pivot: QQ-plot of LRs when = 1 vs =**10 Profile Likelihood EM Harvard University**Distribution of 2 log (LR )’s under the null hypothesis**Profile Likelihood EM: Starting from E( | y)= 0.5 Harvard University**Power Comparison: “Profile” LRT vs “EM” LRT**Harvard University**References**• Gelman, A., Meng, X.L., and Stern, H. (1996). Posterior predictive assessment of model fitness via realized discrepancies (with discussions). Statistica Sinica, 6, 733-807 • Meng, X. L. (1994). Posterior predictive p-values. Ann. Stat. 22:1142 - 1160. • Protassov, R., van Dyk, D.A., Connors, A., Kashyap, V.L., and Siemiginowska, A. (2002) Statistics: Handle with Care, Detecting Multiple Model Components with the Likelihood Ratio Test. The Astrophysical Journal, 571:545–559 • Rubin, DB (1984). Bayesianly justifiable and relevant frequency calculations for the applied statistician. Annals of Statistics, 12(4), 1151–1172 • van Dyk, D.A., and Kang, H. (2004). Highly Structured Models for Spectral Analysis in High-Energy Astrophysics. Statistical Science, 9, no. 2, 275–293 Harvard University**Topic “B” reinstated:**• How to measure “ego”? • How to classify professions by such “ego” measures? • Finding the most powerful test for testing Ego_Particle Physicists > Ego_Astrophysicists> Ego_Statisticians Harvard University

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