Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede

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1 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Hypothess Testng, Lkelhoo Functons an Parameter Estmaton: We conser estmaton of (one or more parameters to be the expermental etermnaton (aka measurement of those parameters (whch are assume to have fxe, but apror unknown values, an whch s base on a lmte/fnte number of expermental observatons. We have alreay encountere the sample mean, x x, as an estmator of x ˆ. ow we wll be more general But before we get nto a full-scale stuy of estmaton, we begn by lookng at Hypothess Testng usng Lkelhoo Ratos. Suppose t s known that ether Hypothess A or Hypothess B s true. An suppose further f x, whle that f A s true, then the ranom varable x s apror known to have a P.D.F. A f B s true, then the ranom varable x s apror known to have a fferent P.D.F. fb ( x. Suppose that we carry out nepenent measurements of a ranom varable x: x,x,, x : If A s true, the probablty that the results are x,x,, x s: PA( x, x,, x f A( x x f A( x x f A( x x f A( x x On the other han, f B ha nstea been true, the probablty of the same strng of results woul have been: PB( x, x,, x fb( x x fb( x x fb( x x fb( x x The Lkelhoo Rato R s efne as: In other wors, the Lkelhoo Rato R s: (,,, (,,, P x x x R P x x x A B f x x A f x x B {The probablty that the partcular expermental result of measurements turne out the way that t, assumng A s true} {The probablty that the partcular expermental result of measurements turne out the way that t, assumng B s true}. In effect, the Lkelhoo Rato R s the bettng os of A aganst B,.e. we assgn probabltes to A an B proportonal to ther Lkelhoos : A A A B PB x x x fb x x L P ( x, x,, x f ( x x an L (,,, Thus, the Lkelhoo Rato s: (,,, (,,, L P x x x R L A A B PB x x x f x x A f x x B n.b. L A an L B are numbers whch wll change (slghtly, e.g. f the entre experment s repeate hence they are ranom varables but they are not ranom strbutons of any kn. P598AEM Lecture otes

2 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree ow n physcs, t s common to have an nfnte number (e.g. a contnuum of hypotheses! For example, n the weak ecays of ve muons ( μ e ve v μ whose spns s μ are fully (.e. % polarze (.e. algne along the ẑ axs {the flght recton of the μ n the lab frame}, the ecay postrons are emtte wth a (normalze polar angle strbuton n the μ center of mass frame: ( cosθ P cosθ ( α cosθ s μ ẑ μ θ e Here, P cosθ αcosθ cosθ s the (nfntesmal probablty that the ecay e s emtte at angle θ whose cosne s wthn the (nfntesmal range cosθ cosθ cosθ. If the muons are % spn-polarze, then the asymmetry parameter α s a number that epens on how Party (space nverson symmetry, {here, θ θ } s volate (along wth Charge Conjugaton,.e. μ μ n the weak ecays of the muon. The value of α s physcally constrane to le between an. The Stanar Moel electroweak (VA.6 precton s ˆ α.. The expermentally measure worl-average value s α.9..7 We efne x cosθ, whch here n ths stuaton s seen as a ranom varable rangng from x cosθ <, snce θ < π. The Probablty Densty Functon (PDF for spn-polarze P( x P( cosθ f x, α αx αcosθ, wth normalzaton conton: μ ecay s x cosθ ( α ( α α α f x x x x x x,. The Cumulatve Dstrbuton Functon (CDF for fully spn-polarze μ ecays s: ( <, α X (, α ( α( CDF x X f x x X X. Plots of the PDF an CDF for spn-polarze μ ecay are shown n the two fgures below, for four fferent physcally allowe values of the asymmetry parameter, α : P598AEM Lecture otes

3 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree An expermentalst wants to measure the α parameter,.e. so as to be able to choose among all possble hypotheses here, a contnuum: α., α.99999,..., α., α., α.,..., α , α.. muon ecays are measure an yel the (nepenent ranom varable set of angles θ. If we efne x cosθ, then the P.D.F. becomes f ( x, α ( α x an the Lkelhoo s: A ( α f ( x, α L L. As before, we wll thnk n terms of the Lkelhoo Rato ( α ( α Lkelhoo { probablty that α m s true of L ( α m }. ext, we plot ( α R L L an assgn a L vs. α : L ( α L ( α -OR- Δ α α α The Most Probable Value of α, α s calle the Maxmum Lkelhoo Soluton. α α The Root Mean Square (RMS Sprea the square root of the varance of α aroun α s a measure of the accuracy wth whch Δα α. α s etermne, call t If s large, then L ( α wll be a Gaussan (ue to/because of the Central Lmt Theorem. But f s small, then we may have a stuaton lke the one epcte on the LHS of the above fgure. In that case, Δ α has no real meanng an shoul not be quote. Instea, the plot shoul be shown. P598AEM Lecture otes 3

4 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree ow let us be very careful... If we nterpret L ( α as a measure of the probablty of α, then we must make certan that t s properly normalze. Thus, we calculate ( α α L an replace L ( α by L ( α. Then we know that: L ( α α ( L L ( α α α α α α Δα α E[( α α ] or: Δ α α ( L L ( α α α α α α Ths approach to etermnng parameters (such as α an ther uncertantes α s calle the Maxmum Lkelhoo Metho (M.L.M. A Detale Example of the Use of the M.L.M. : Suppose we are tryng to rectly measure a physcal parameter α. Let each measurement be calle x, an let be the stanar evaton assocate wth each nvual measurement x. Let us further assume that the nepenent nvual measurements ( x are Gaussanstrbute, wth x ˆ α as ther expectaton value. ( x Then for any nvual measurement:, α f x α e π ( x Thus, for nepenent measurements: α L α e π n.b. Here, we have use the parameter α nstea of α, snce we are tryng to fn/etermne α (whch s apror unknown. It s by varyng α (as a free parameter that we fn / etermne / measure the partcular value, α that maxmzes the lkelhoo functon L ( α. α (whch we entfy wth ote further that we also use the shorthan L ( α nstea of the more correct ( α x x ;,, L. We wll carry out the maxmzaton of the Lkelhoo functon explctly,.e. we wll fn L ( α α an then look for a zero corresponng to a maxmum n L ( α. ln α L α. Defne: ote also that n practce, we maxmze L, the log lkelhoo nstea of ( xα ln e π ( x α ln π ( α ( α P598AEM Lecture otes 4

5 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree ote that f α L has a maxmum at some α α maxmum at the same value of α. Thus: Then: ( α, then ( α ln ( α ( x α ( α ( α constant ( α x α α ( α α α ( α α an solve: α An: Fnally, set: x α x α α L wll also have a x α Snce ths s <, the extremum s a maxmum α α Thus we see that the Maxmum Lkelhoo value of the parameter α s just a Weghte Mean. ote that f all of the nvual measurements x ha the same varances x, the above expresson woul reuce to the smple/arthmetc/unweghte sample mean: α x So we recover a result that we shoul have antcpate, an see how to use M.L.M. n the smplest case. But has ths elegant proceure gven us anythng new? In orer to unerstan ths, we now step back an look n general at estmaton. For smplcty, let us conser experments where we perform a sngle measurement x k, n each, an where we are tryng to etermne (.e. estmate a common sngle parameter. Later, when we look at practcal schemes, we wll scuss the case(s of several measurements per experment, an also the smultaneous etermnaton of several parameters. After we have mae nepenent measurements of a ranom varable x: x,,x, we construct a functon S(x,, x whose numercal value s the estmate of the apror unknown parameter of nterest (e.g. xˆ, x,... Thus the estmator S cannot epen on. The numercal value of the functon S ( the estmator s tself a ranom varable. For example, f we wante to estmate the expectaton value of x, E[ x] xˆ for a set of ( xxˆ Gaussan-strbute measurements (whose P.D.F. s e, all wth the same π stanar evaton we coul use e.g. the sample mean: Sx (,, x x x P598AEM Lecture otes 5

6 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Ths estmator s unbase, snce ES [ ] xˆ. Ths s nce! In general, the estmator S that we choose may n fact be base. That s, ES [ ] may n fact not be, the true value of the parameter. We efne the Bas B ( as the fference between E[ S ] an : B( E[ S] A goo estmator S wll be unbase,.e. have B E[ S]. Another mportant property of the estmator S s ts own varance, S. It s obvously hghly esrable to nvent estmators wth S as small as possble. Can S be arbtrarly small? (ans: o! It s not obvous (an n general not true! that one can fn an estmator S wth both mnmum bas an mnmum varance. For now, we wll eal only wth unbase estmators, but later on, we wll look at the base kn. For now, we focus on/concern ourselves wth possble bas ssues assocate wth varance: Let f ( ; x be the P.D.F. assocate wth the measurement of x. s the parameter we wsh to etermne by carryng out a seres of nepenent expermental measurements x,,x. The jont P.D.F. of ths seres s f ( x, x,, x ; f ( x ; f ( x ; f ( x ; prove that the nvual measurements of x are nepenent. Let S(x,, x be an unbase estmator of,.e. ES [ ]. E S S x x x f x f x f x xx x Then: [ ] (,,, ( ; ( ; ( ; We now go through a seres of manpulatons n orer to arrve at a result concernng the mnmum varance assocate wth the estmator S. Along the way, we wll also efne a quantty known as nformaton. n.b. In all that follows, we assume all ntegrals are efne, ntegraton an fferentaton commute, etc... So: E[ S] S( x, x,, x f ( x; f ( x; f ( x; xx x Dfferentate both ses of the above relaton wth respect to the parameter : Then let us S( x, x,, x { f ( x ; f ( x ; f ( x ; } xx x Defne: L ( x, x,, x ; f ( x; f ( x ; f ( x ; f ( x ; P598AEM Lecture otes 6

7 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Defne: ( x, x,, x ; ( x, x,, x ; ln f ( x; Defne: Defne: (,,, ; x x x x, x,, x ; ( x, x,, x; ln f ( x; (,,, ; (,,, ; x x x x x x ( x, x,, x ; x, x,, x ; ln f ( x ; f ( x ; Defne: f ( x; an: f ( x; ( ; ( ; f x f x ( Then: ext: (,,, ; x x x ( x, x,, x ; ( x, x,, x; ln f ( x; f ( x; f ( x; ( ( ; f x f ( x; ln f x; φ x; { f ( x; f ( x; f ( x ; } ( x ; ( x; ( f x ; f x ; f x ; f x f x f x ( ; ( ; ( ; ( x, x,, x ; f ( x ; ( ; ( ; ( ; f x; f x; f x ; f f x f x f x f ( x, x,, x ; f x ; ; f x { } S x x x f x f x f x xx x to get: Plug ths result nto: (,,, ( ; ( ; ( ; S x x x x x x f x f x f x xx x E S.e. ES [ ],,,,,, ; ; ; ; [ ] ow: E [ ], whch follows from takng of both ses of the P.D.F. normalzaton conton: f x ; f x ; f x ; xx x P598AEM Lecture otes 7

8 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Proof: f x ; f x ; f x ; xx x ( f x ; f x ; f x ; f x ; f x ; f x ; xx x (,,, ; ( ; ( ; ( ; f x; f x; f x ; ( x, x,, x ; x x x f x f x f x xx x E[ ] QE.. D. E[ ] Then: E[ ] E[ S] (assumng ES [ ] s fnte Snce: ES [ ] Then: ES [ ] ES [ ] E[ ] But: ES [ ] ES [ ] E[ ] cov ( S, cov S, S an are postvely correlate! cov( x, y Recall that the correlaton coeffcent ρ ( xy, has magntue ρ ( xy Thus, here: ρ ( S ( S cov,, varance of the estmator S: S S L s efne as: However, the varance of ln ( Thus: E E E E [( [ ] ] [ ] [ ] S E[ ] E ( x y E[ ] ( xy cov,, We efne the so-calle nformaton I ( of the sample (wth respect to the parameter as: I E[ ] E lnl Then: I. S Thus we see that large nformaton I ( small varance S of the estmator S an vce versa. ( S x y P598AEM Lecture otes 8

9 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Ha the estmator S been base, then wth B E[ S] at the general form of ths nequalty: we woul have nstea arrve B B S I E ( whch s known as the Rao-Cramér-Frechet (RCF Inequalty, aka the Informaton Inequalty. It gves a rgorous lower boun on the varance S assocate wth a base estmator S of the parameter. We re not qute one here an nterestng relatonshp exsts between or equvalently, exsts between E an L ( E [ ] an E [ ] ln E, namely that: E[ ] E[ ], or equvalently, that: E ( E ln ( L. We showe above that the expectaton value of ( x, x,, x ; What s the expectaton value of ( x, x,, x ; was zero,.e. E [ ]?.e. what s:. ( E[ ] E E E ln? L Repeatng what we above n the etermnaton of E [ ], ths tme we take of the P.D.F. normalzaton conton: f x; f x ; f x ; xx x of both ses f x ; f x ; f x ; xx x ( x, x,, x ; ( f x ; f x ; f x ; f ( x ; f ( x ; f ( x ; xx x f x; f x; f x ; ( x, x,, x; f ( x; f ( x; f ( x; x x x P598AEM Lecture otes 9

10 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree { ( x, x,, x; f ( x; f ( x; f ( x; } xx x (,,, ; ( ; ( ; ( ; x x x f x f x f x xx x ( x, x,, x; { f ( x; f ( x; f ( x; } xx x (,,, ; ( ; ( ; ( ; x x x f x f x f x xx x ( x, x,, x ; ( x, x,, x ; f ( x; f ( ; ( ; ( f x; f x ; f x ; ( x, x,, x ; f x; f x ; f x ; f x ; f x ; f x ; x x x x f x xx x E[ ] x, x,, x ; f x ; f x ; f x ; xx x E[ ] E[ ] E[ ] or: E[ ] E[ ] QED... Thus, we see that: L ( E[ ] E E E ln [ ] lnl ( E E E The relaton between these two expectaton values, E[ ] E[ ] s qute an amazng, an very general result snce t was erve wthout reference to a specfc form of P.D.F. t s therefore val/hols for any P.D.F! The relaton E[ ] E[ ] says that the negatve of the expectaton value of the (negatve! curvature the n ervatve of the log of the lkelhoo functon (whch s an -mensonal ntegral convolvng ( x, x,, x ; wth the prouct factor of P.D.F. s, ntegrate over all of the x s, over ther entre allowe physcal ranges s equal to the expectaton value of the square of the slope the st ervatve of the log of the lkelhoo functon (whch s another -mensonal ntegral convolvng ( x, x,, x ; wth the prouct factor of P.D.F. s, ntegrate over all of the x s, over ther entre allowe physcal ranges. P598AEM Lecture otes

11 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Restatng ths somewhat less rgorously, but n a more physcal sense: E[ ] E[ ] tells us physcally that the negatve of the ntegrate-over average value of the (negatve! curvature of the log of the lkelhoo functon s equal to the ntegrate-over average value of the square of the slope of the log of the lkelhoo functon! Agan, E[ ] E[ ] s a very general result t s val/hols for any P.D.F! One can also physcally unerstan now why E [ ]. The above Informaton Inequaltes can therefore atonally be expresse n terms of E [ ] the Informaton Inequalty for unbase estmators, S s: S E[ ] E[ ] I E ( E ( The Rao-Cramér-Frechet (RCF Inequalty (aka Informaton Inequalty for base estmators, S wth B E[ S] s: B B B S E[ ] E[ ] B B B I ( E ( E ( These Informaton Inequaltes gve rgorous lower bouns on the varance S assocate wth the estmator S of the parameter. The physcal meanng of the relaton E ( E ( [ ] [ ] also enables us to unerstan/realze some amazng physcal/mathematcal propertes of the (log of the lkelhoo functon (, because physcally, ( ( s the local slope (.e. st ervatve of the ( vs. curve at the pont, an ( ( s the local (negatve! vs. curve at the pont. curvature (.e. n ervatve of the In the lmt of very large (.e. the -parameter lkelhoo functon L ( s Gaussan/ normal n that parameter: ( ( Ce (, L Then: ln ln L C. P598AEM Lecture otes

12 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree At the maxmum of the lkelhoo: thus we can wrte: ( then: ( ln ( L ln C, thus: C L (, ( L L e or: ( Then for: ± ± or: ( ± ± then: L( ( e ± L an: ( ( ± or: ( ln ( ± L or: ( ( ± Δ Then, we have the conventonal wsom that 68.3% of the tme the true value ˆ wll be wthn ± of, etc. In the very large lmt, the ln ( L vs. curve s an (nverte parabola of the general form: L ln constant y x A xx B Ths relaton s shown graphcally n the fgure below the RHS (LHS plot has a we (narrow parabola, respectvely: ( ( ± ( -OR- ( When Δ ( ( ± s evaluate at the local maxmum Δ of the ln ( L functon: In the vcnty of the local maxmum, the curvature s negatve. When ( ( s evaluate at the ( ± ether se of the local maxmum : ponts of the ln ( ± ( ± ±. ± ± ± L functon on P598AEM Lecture otes

13 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Thus: ln ( L ±. ± ± ± In the lmt of very large (.e., f we fn/etermne the ± ponts assocate wth reucng the value of ( functon by a factor of / from the value of the ( functon at ts local maxmum,, ths correspons to the ±, 68.3% central/oublese confence level.e. an are respectvely the (low an (hgh ponts on ether se of the local maxmum,. In the lmt of very large (.e., we have four equvalent methos of etermnng :. Compute the Root Mean Square Devaton the square root of the varance Lkelhoo Dstrbuton, L (. of the. Determne the ( ± ln ± ponts on ether se of the local maxmum L ( functon from: ln ln ( ln ( ± Δ L L L. of the of the 3. Compute the (negatve! curvature at the local maxmum functon: ( 4. Compute the square of the local slope(s of the ( functon at the ( ± low, hgh ponts: ln ( ± P598AEM Lecture otes 3

14 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree Metho. above can also be use for obtanng more than just the ± (68.3% oublese/central Confence Level lmts on the parameter! From the table on central/ouble- se Confence Level Lmts for Gaussan Dstrbutons n P598AEM Lect. otes 8 (p. 6, we re-wrte t for use here, n terms of the Δ ( ( ± value assocate wth n. As can be seen from the table below, the relaton s very smple: n Δ ( ( ± n. n #.. (% CL Δ ( s What oes the ln ( of the PDF assocate wth the ranom varable x. In general, for fnte statstcs, the ( L vs. curve look lke when s fnte? Ths epens on the nature vs. can become nosy.e. t may not be a perfectly smooth curve havng ncrease statstcal fluctuatons n t as ecreases from. Depenng on the nature of the PDF assocate wth the ranom varable x, t may also become ncreasngly asymmetrcal as ecreases from, an may appear somethng lke the curves shown n the fgure below: ( ( ± ( -OR- ( ( Δ ( ( ± ( Δ Clearly, n ths asymmetrcal stuaton, the nvual results from usng the 4 above methos of etermnng can begn to verge from each other as ecreases from. Of the four, metho # ( Δ ln ( n L s the most robust /most wely accepte. P598AEM Lecture otes 4

15 Fall Analyss of Expermental Measurements B. Esensten/rev. S. Erree We see e.g. for the ± ouble-se/central 68.3% Confence Interval, wth Δ (, that ( (, hence the (asymmetrcal ± lowse/hgh-se uncertantes assocate wth the M.L.M. s most probable value result are quote. as:, e.g. 5.4 Volts.4. Clearly, n ths asymmetrcal stuaton, for metho # 4, the (local slopes of the ln ( L vs. curve at low vs. hgh are not equal, but that s smply because they are ant-correlate wth the numercal values of ther respectve sgmas, snce they are nversely relate to each other: ln ( L ( an: ( Or smply: ( an: ( For metho # 3, etermnng the curvature (.e. n ervatve of the ln ( L curve at yels only a sngle number for, whch s actually a weghte average of an, Ths coul be unfole/unweghte, e.g. usng the local slope nformaton of the ( vs. curve at low & hgh, but f one goes to all that effort, why not just use metho # 4 nstea? Metho # suffers from the same problems as metho # 3. Hence why metho # ( Δ ( n s the most popular/most wely use metho t s very easy to carry out, t works wth any lkelhoo functon, an s also easly unerstoo by others... Please see/rea Muon Decay Asymmetry MLM Ft Example, poste on the Physcs 598AEM Software webpage, for a etale scusson of the use of the MLM to obtan an estmate of the asymmetry parameter, α an corresponng ± α statstcal uncertantes, from a large sample of fully-polarze μ e ve v μ ecays. P598AEM Lecture otes 5