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Motivation Outline Difficulties in Limit setting in addition to the Strong Confidence approach
Edison Community College, US has reference to this Academic Journal, Difficulties in Limit setting in addition to the Strong Confidence approach Giovanni Punzi SNS in addition to INFN – Pisa Advanced Statistical Techniques in Particle Physics Durham, 18-22 March 2002 Outline Motivations in consideration of a Strong CL Summary of properties of Strong CL Some examples Limits in presence of systematic uncertainties. Motivation The set of Neyman?s bands is large, in addition to contains all sorts of inferences like: ?I bought a lottery ticket. If I win, I will conclude then donkeys can fly @99.9999% CL? I want so that get rid of those, but keep being frequentist.
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Why should you care ? Wrong reason: so that make the CL look more like p(hypothesis | data). Right reason: You don?t want so that have so that quote a conclusion you know is bad. If you think harder, you can do better: You are drawing conclusions based on irrelevant facts (like a bad fit). As a consequence, you are not exploiting at best the information you have Your results are counter-intuitive in addition to convey little information. You must make sure your conclusions do not depend on irrelevant information SOLUTION: Impose a form of Likelihood Principle Take any two experiments whose pdf are equal in consideration of some subset c of observable values of x, apart in consideration of a multiplicative constant. Any valid Confidence Limits you can derive in one experiment from observing x in c must also be valid in consideration of the other experiment. If you ask the Limits so that be univocally determined, there is no solution. RESULT Neyman?s CL bands Strong bands Non-coverage land Surprise: a solution exists, in addition to gives in consideration of any experiment a well-defined, unique subset of Confidence Bands
Construction of CL bands Regular Strong Strong CL vs. standard CL A new parameter emerges: sCL. Every valid band @xx% sCL is also a valid band @xx% CL. You can check sCL in consideration of a band built in any other way. sCL requirement effectively amounts so that re-applying the usual Neyman?s condition locally on every subsample of possible results.This ensures uniform treatment of all experimental results, but in a frequentist way. Strong Band definition is not an ordering algorithm in addition to answer is still not unique. You may need so that add an ordering so that obtain a unique solution. Strong CL It is similar so that conditioning, a standard practice in modern frequentist statistics. ?There is a long history of attempts so that modify frequentist theory by utilizing some form of conditioning. Earlier works are summarized in Kiefer(1977), Berger in addition to Wolpert(1988) [?] Kiefer(1977) formally established the conditional confidence approach? ?The first point so that stress is the unreasonable nature of the unconditional test [?] the unconditional test is arguably the worst possible frequentist test [?] it is in some sense true that, the more one can condition, the better? ?It is sometimes argued that conditioning on non-ancillary statistics will ?lose information?, but nothing loses as much information as use of unconditional testing? (J. Berger) Neyman: (CR(x) is the accepted region in consideration of æ given the observation of x. c is an arbitrary subset of x space)
Artificial Life Lecture 6 Coevolution of Pursuit in addition to Evasion Pursuit/Evasion ? gen 200 Coevolution This study of coevolution Sensors Neural Network Control System Evolution Evaluation Random Start ? gen 0 A Successful run ? gen 999 Potential Circular Trap Possible Variations Analysis of behaviour Applications Coevolving Parasites Sorting Networks Picturing Sorting Networks Minimal Sorting Networks How so that check if it works Genetic Representation Diploid encoding Diploid encoding (ctd) Scoring Spatially Distributed GA (Back so that Lec 3 ? Microbial GA) Now alongside Added Demes !!!! FROM LEC 3 Microbial alongside demes FROM LEC 3 Reproduction Results ? without coevolution Inefficiencies – 1 Inefficiencies – 2 2 populations ? sorters in addition to parasites Scoring each population Benefits of Coevolution The End
Summary of sCL properties 100% frequentist, completely general. The only frequentist method complying alongside Likelihood Principle Invariant in consideration of any change of variables No empty regions, in full generality No ?unlucky results?, no need in consideration of quoting additional information on sensitivity. No pathologies. Robust in consideration of small changes of pdf More information gives tighter limits Easier incorporation of systematics Price tag: Overcoverage Heavy computation (see CLW proceedings in addition to hep-ex/9912048) Invariance in consideration of change of the observable All classical bands are invariant in consideration of change of variable in the parameter (unlike Bayesian limits) The CL definition is invariant in consideration of change of variable in the observable, too. But, most rules in consideration of constructing bands break this invariance ! Strong-CL is also invariant in consideration of any change of variable. Likelihood Ratio is also invariant (non-advertised property?), so it is a natural choice of ordering so that select a unique Strong Band. Effect of changing variables Neyman?s CL bands Strong bands Non-coverage land LR-ordered bands
Poisson+background The upper limit on æ decreases alongside expected background in all unconditioned approaches. Often criticized on the basis that in consideration of n=0 the value of b should be irrelevant. LR-ordering upper limit @90%CL in consideration of n=0 background sCL = 90%, or R.-W. Behavior when new observables are added Do you expect limits so that improve when you add extra information ? A simple example shows that neither PO or LRO have this property (conjecture: no ordering algorithm has it !) Example: comparing a signal level alongside gaussian noise alongside some fixed thresholds Problem: the limit loosens dramatically when adding an extra threshold measurement. Example Unknown electrical level æ plus gaussian noise (? =1). Limited so that |æ|< 0.5. Compare alongside a fixed threshold (2.5 ?), get a (0,1) response. Observe > threshold: PO: empty region @90%CL LR: 0.49 < æ < 0.50 @90%CL sCL: -0.34 < æ < 0.50 @90%sCL N.B. you MUST overcover unless you want an empty region. L(æ) LR(æ) Add another threshold Now, add a second independent threshold measurement at 0: limit become much looser ! sCL limit is unaffected Conjecture: no ordering algorithm can provide a sensible answer in all cases. L(æ) LR(æ) 0.27< æ < 0.5 Observations It may be impossible so that get sensible results without accepting some overcoverage. Why blame sCL in consideration of overcoverage ? Ordering algorithms alone seem unable so that prevent very strange results: the inclusion of additional (irrelevant) information may produce a dramatic worsening of limits. Adding systematics so that CL limits Problem: My pdf p(x|æ) is actually a p(x|æ,?), where ? is an unknown parameter I don?t care about, but it influences my measurement (nuisance) I may have some info of ? coming from another measurement y: q(y|?) My problem is: p(x,y|æ,?) = p(x|æ,?)*q(y|?) Many attempts so that get rid of ?: three main routes: Integration/smearing (a la Bayes) Maximization (?profile Likelihood?) Projection (strictly classical) Hybrid method: Bayesian Smearing 1) define a new (smeared) pdf: p?(x|æ) = ? p(x|æ,?)?(?) d ? where ?(?) is obtained through Bayes: ?(?) = q(y| ?)p(?)/q(y) Need so that assume some prior p(?) 2) Use p? so that obtain Conf. Limits as usual GOOD: Simple in addition to fast Used in many places Intuitively appealing BAD: Intuitively appealing Interpretation: mix Bayes in addition to Neyman. Output results have neither coverage nor correct Bayesian probability => waste effort of calculating a rigorous CL May undercover May exhibit paradoxical tightening of limits A simple example + Bayes systematics Introduce a systematic uncertainty on the actual position of the 0 threshold. Assume a flat prior in [-1,1]. Do smearing => get tighter limits ! No reason so that expect a good behavior æ > 0.272 æ > 0.294 LR(æ) LR(æ) Approximate classical method: Profile Likelihood 1) define a new (profile) pdf: p prof(x|æ) = p(x,y0|æ,?best (æ)) where ?best(æ) maximizes the value of a) p(x0,y0|æ,?best) b) p(x ,y0|æ,?best) (?best = ?best(æ,x) !) This means maximizing the likelihood wrt the nuisance parameters, in consideration of each æ 2) Use p prof so that obtain Conf. Limits as usual GOOD: Reasonably simple in addition to fast Approximation of an actual frequentist method BAD: Flip-flop in case a), non-normalized in case b) !! Only approximate in consideration of low-statistics, which is when you need limits after all. You don?t know how far off it is unless you explicitly calculate correct limits. Systematically undercovers
Exact Classical Treatment of Systematics in Limits 1) Use p(x,y|æ,?) = p(x|æ,?)*q(y|?), in addition to consider it as p( (x,y) | (æ,?) ) 2) Evaluate CR in (æ,?) from the measurement (x0,y0) 3) Project on æ space so that get rid of uninteresting information on ? It is clean in addition to conceptually simple. It is well-behaved. No issues like Bayesian integrals Why is it used so rarely ? 1) It produces overcoverage 2) The idea is simple, but computation is heavy. Have so that deal alongside large dimensions 3) Results may strongly depend on ordering algorithm, even more than usual. ?profile method? ?Overcoverage? Projecting on æ effectively widens the CR ? overcoverage. BUT: You chose so that ignore information on ? – cannot ask Neyman so that give it all back so that you as information on æ – the two things are just not interchangeable. ? overcoverage is a natural consequence, not a weakness Q: can you find a smaller æ interval that does not undercover ? (same situation alongside discretization)
Optimization issue You want so that stretch out the CR along ? direction as far as possible. BUT: The choice of band is constrained by the need so that avoid paradoxes (empty regions, in addition to the like) ! No method on the marked allows you so that treat æ in addition to ? in a different fashion Strong CL allows you so that specify æ as the parameters of interest, in addition to so that obtain the narrowest æ interval The solution does not require constructing a multidimensional region Strong CL Band alongside systematics The solution does not require explicit construction of a multidimensional region The narrowest æ interval compatible alongside Strong CL is readily found. Conclusions Strong Confidence bands have all good properties you may ask for. Systematics can be included naturally in addition to rigorously They can even be actually evaluated
Poisson Example: n=5, b=2, A=0.02ñ0.006(gaussian) Strong CL Upper limit ~30% higher than from Bayesian calculations shown by Luc Demortier (arbitrary units)
Gebhart, Fred Midday On-Air Personality
Gebhart, Fred is from United States and they belong to Midday On-Air Personality and work for KPKX-FM in the AZ state United States got related to this Particular Article.
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This Particular Journal got reviewed and rated by Optimization issue You want so that stretch out the CR along ? direction as far as possible. BUT: The choice of band is constrained by the need so that avoid paradoxes (empty regions, in addition to the like) ! No method on the marked allows you so that treat æ in addition to ? in a different fashion Strong CL allows you so that specify æ as the parameters of interest, in addition to so that obtain the narrowest æ interval The solution does not require constructing a multidimensional region Strong CL Band alongside systematics The solution does not require explicit construction of a multidimensional region The narrowest æ interval compatible alongside Strong CL is readily found. Conclusions Strong Confidence bands have all good properties you may ask for. Systematics can be included naturally in addition to rigorously They can even be actually evaluated and short form of this particular Institution is US and gave this Journal an Excellent Rating.