Physics 3700 Probability, Statistics, & Data Analysis

Transcription

1 R.Kass/Sp15 P3700 Lecture 1 1 Physics 3700 Probability, Statistics, & Data Aalysis Itroductio: I) The uderstadig of may physical pheomea relies o statistical ad probabilistic cocepts: Statistical Mechaics (physics of systems composed of may parts: gases, liquids, solids) 1 mole of aythig cotais 6x10 23 particles (Avogadro's umber) Eve though the force betwee particles (Newto s laws) is kow it is impossible to keep track of all 6x10 23 particles eve with the fastest computer imagiable We must resort to learig about the group properties of all the particles: use the partitio fuctio: calculate average eergy, etropy, pressure... of a system Quatum Mechaics (physics at the atomic or smaller scale, < m) wavefuctio = probability amplitude talk about the probability of a electro beig located at (x,y,z) at a certai time. II) Our uderstadig/iterpretatio of experimetal data relies o statistical ad probabilistic cocepts: how do we extract the best value of a quatity from a set of measuremets? how do we decide if our experimet is cosistet/icosistet with a give theory? how do we decide if our experimet is iterally cosistet? how do we decide if our experimet is cosistet with other experimets? I this course we will cocetrate o II), the above experimetal issues Note: The theory of probability is a area of pure mathematics while statistics is a area of applied mathematics that uses the axioms ad defiitios of probability theory.

2 A Example from Particle Physics May of the process ivolved with detectio of particles are statistical i ature: Number of io pairs created whe proto goes through 1 cm of gas Eergy lost by a electro goig through 1 mm of lead The uderstadig ad iterpretatio of all experimetal data deped o statistical ad probabilistic cocepts: The result of the experimet was icoclusive so we had to use statistics how do we extract the best value of a quatity from a set of measuremets? how do we decide if our experimet is cosistet/icosistet with a give theory? how do we decide if our experimet is iterally cosistet? how do we decide if our experimet is cosistet with other experimets? how do we decide if we have a sigal (i.e. evidece for a ew particle)? Cosider the recet discovery of the Higgs particle by ATLAS ad CMS i What is a Higgs particle? Resposible for you havig MASS 2013 Nobel Prize to Eglert & Higgs for predictig this particle What is the evidece that such a particle has bee discovered? How did statistics play a role i its discovery? Importat to covice skeptics that that there was a very low probability for two experimets to make a mistake at the same time. Fraçois Eglert Ca estimate this probability usig stuff we lear i this class Also, how do we use the data i a optimal way to calculate the mass of the Higgs? Peter Higgs 2

3 Higgs decay ito 2 high eergy gamma rays data take over a two year period R.Kass/Sp15 P3700 Lecture 1 3

4 Higgs decay ito 4 leptos lepto=electro or muo data take over a two year period R.Kass/Sp15 P3700 Lecture 1 4

5 Defiitios of probability: Frequetist defiitio: Suppose we have N trials ad a specified evet occurs r times. example: the trial could be rollig a dice ad the evet could be rollig a 6. defie probability (P) of a evet (E) occurrig as: P(E) = r/n whe N # examples: six sided dice: P(6) = 1/6 for a hoest dice: P(1) = P(2) = P(3) = P(4) =P(5) = P(6) =1/6 coi toss: P(heads) = P(tails) =0.5 P(heads) should approach 0.5 the more times you toss the coi. For a sigle coi toss we ca ever get P(heads) = 0.5 Mathematical defiitio (Kolmogorov) : By defiitio probability (P) is a o-egative real umber bouded by 0 P 1 if P = 0 the the evet ever occurs if P = 1 the the evet always occurs Let A ad B be subsets of S the P(A) 0, P(B) 0 Evets are idepedet if: P(A B) = P(A)P(B) Coi tosses are idepedet evets, the result of the ext toss does ot deped o previous toss. Evets are mutually exclusive (disjoit) if: P(A B) = 0 or P(A B) = P(A) + P(B) I tossig a coi we either get a head or a tail. How do we defie Probability? Sum (or itegral) of all probabilities if they are mutually exclusive must = 1. Bayesia defiitio: Istead of repeatability relies o degree of belief, obeys Kolmogorov s axioms. Avoids the difficult of N which ever happes i practice. R.Kass/Sp15 P3700 Lecture 1 itersectio uio Α= {1,2,3}# Β={1,3,5}# Α Β={1,3} A Β= {1,2,3,5} 5

6 R.Kass/Sp15 P3700 Lecture 1 6 Probability ca be a discrete or a cotiuous variable. Discrete probability: P ca have certai values oly. examples: tossig a six-sided dice: P(x i ) = P i here x i = 1, 2, 3, 4, 5, 6 ad P i = 1/6 for all x i. tossig a coi: oly 2 choices, heads or tails. NOTATION for both of the above discrete examples (ad i geeral) x whe we sum over all mutually exclusive possibilities: i is called a radom variable P( x i ) =1 i Cotiuous probability: P ca be ay umber betwee 0 ad 1. defie a probability desity fuctio, pdf, f(x): f ( x)dx = dp( x α x + dx) with α a cotiuous variable Probability for x to be i the rage a x b is: P(a x b) = b a ( ) f x dx Just like the discrete case the sum of all probabilities must equal 1. + f x dx =1 ( ) Probability= area uder the curve We say that f(x) is ormalized to oe. Probability for x to be exactly some umber is zero sice: x=a f ( x) dx = 0 x=a Note: i the above example the pdf depeds o oly 1 variable, x. I geeral, the pdf ca deped o may variables, i.e. f=f(x,y,z, ). I these cases the probability is calculated usig from multi-dimesioal itegratio.

7 R.Kass/Sp15 P3700 Lecture 1 7 Examples of some commo P(x) s ad f(x) s: Discrete = P(x) Cotiuous = f(x) biomial uiform, i.e. costat Poisso Gaussia expoetial chi square How do we describe a probability distributio? mea, mode, media, ad variace for a ormalized cotiuous distributio, these quatities are defied by: + f x dx =1 Mea Mode Media Variace average most probable 50% poit width of distributio + a + f x µ = xf (x)dx = = f (x)dx σ 2 = f (x) x µ ( ) x for a discrete distributio, the mea ad variace are defied by: x i i=1 µ = 1 ( ) x = a ( ) 2 dx σ 2 = 1 (x i µ) 2 i=1

8 Some Cotiuous Probability Distributios Remember: Probability is the area uder these curves For may pdfs its itegral ca ot be doe i closed form, use a table to calculate probability. mode media mea For a Gaussia pdf the mea, mode, ad media are all at the same x. σ symmetric distributio (gaussia) Asymmetric distributio showig the mea, media ad mode For may pdfs the mea, mode, ad media are i differet places. v=1 Cauchy (Breit-Wiger) v= gaussia Chi-square distributio u Studet t distributio R.Kass/Sp15 P3700 Lecture 1 8

9 R.Kass/Sp15 P3700 Lecture 1 9 Calculatio of mea ad variace: example: a discrete data set cosistig of three umbers: {1, 2, 3} average (µ) is just: x µ = i = = 2 i=1 3 Complicatio: suppose some measuremets are more precise tha others. Let each measuremet x i have a weight w i associated with it the: µ = x i w i / w i i=1 i=1 variace (σ 2 ) or average squared deviatio from the mea is just: σ 2 = 1 (x i µ) 2 i=1 # # # #σ is called the stadard deviatio rewrite the above expressio by expadig the summatios: σ 2 = 1 % x 2 i + µ 2 ( ' 2µ x i * & ) i=1 i=1 = 1 x 2 i + µ 2 2µ 2 i=1 weighted average The variace describes the width of the pdf This is sometimes writte as: i=1 <x 2 >-<x> 2 with <> average = 1 x 2 i µ 2 of what ever is i the brackets i=1 Note: The i the deomiator would be -1 if we determied the average (µ) from the data itself.

10 R.Kass/Sp15 P3700 Lecture 1 10 Usig the defiitio of µ from above we have for our example of {1,2,3}: σ 2 = 1 x 2 i µ 2 = = 0.67 i=1 The case where the measuremets have differet weights is more complicated: i= i ( xi µ ) / wi = wi xi / wi µ i= 1 i= 1 i= 1 2 σ = w Here µ is the weighted mea If we calculated µ from the data, σ 2 gets multiplied by a factor /( 1). Example: a cotiuous probability distributio, This pdf has two modes It has same mea ad media, but differ from the mode(s). 2 f ( x) = csi x for 0 x 2π,c = costat

11 R.Kass/Sp15 P3700 Lecture 1 11 For cotiuous probability distributios, the mea, mode, ad media are# calculated usig either itegrals or derivatives: f(x)=si 2 x is ot a true pdf sice it is ot ormalized f(x)=(1/π) si 2 x is a ormalized pdf (c=1/π). Note : µ = x si 2 xdx / 0 example: Gaussia distributio fuctio, a cotiuous probability distributio 2π 2π 0 si 2π 0 = π si 2 xdx = π mode = x si2 x = 0 π 2, 3π 2 2 xdx α media = si 2 2π xdx / si 2 xdx = α = π I this class you should feel free to use a table of itegrals ad/or derivatives.

12 R.Kass/Sp15 P3700 Lecture 1 12 Accuracy ad Precisio Accuracy: The accuracy of a experimet refers to how close the experimetal measuremet is to the true value of the quatity beig measured. Precisio: This refers to how well the experimetal result has bee determied, without regard to the true value of the quatity beig measured. Just because a experimet is precise it does ot mea it is accurate example: measuremets of the eutro lifetime over the years: The size of bar reflects the precisio of the experimet This figure shows various measuremets of the eutro lifetime over the years. Steady icrease i precisio of the eutro lifetime but are ay of these measuremets accurate?

13 Measuremet Errors (or ucertaities) Use results from probability ad statistics as a way of calculatig how good a measuremet is. most commo quality idicator: relative precisio = [ucertaity of measuremet]/measuremet example: we measure a table to be 10 iches with ucertaity of 1 ich. relative precisio = 1/10 = 0.1 or 10% (% relative precisio) Ucertaity i measuremet is usually square root of variace: # # #σ = stadard deviatio σ is usually calculated usig the techique of propagatio of errors. Statistical ad Systematic Errors Results from experimets are ofte preseted as: N ± XX ± YY N: value of quatity measured (or determied) by experimet. XX: statistical error, usually assumed to be from a Gaussia distributio. With the assumptio of Gaussia statistics we ca say (calculate) somethig about how well our experimet agrees with other experimets ad/or theories. Expect ~ 68% chace that the true value is betwee N - XX ad N + XX. YY: systematic error. Hard to estimate, distributio of errors usually ot kow. examples:mass of proto = ± GeV (oly statistical error give) mass of W boso = 80.8 ± 1.5 ± 2.4 GeV (both statistical ad systematic error give) R.Kass/Sp15 P3700 Lecture 1 13

14 R.Kass/Sp15 P3700 Lecture 1 14 What s the differece betwee statistical ad systematic errors? N ± XX ± YY Statistical errors are radom i the sese that if we repeat the measuremet eough times: XX 0 as the umber of measuremets icreases Systematic errors, YY, do ot 0 with repetitio of the measuremets. examples of sources of systematic errors: voltmeter ot calibrated properly a ruler ot the legth we thik is (meter stick might really be < meter) Because of systematic errors, a experimetal result ca be precise, but ot accurate How do we combie systematic ad statistical errors to get oe estimate of precisio? BIG PROBLEM two choices: σ tot = XX + YY add them liearly σ tot = (XX 2 + YY 2 ) 1/2 add them i quadrature Some other ways of quotig experimetal results lower limit: the mass of particle X is > 100 GeV upper limit: the mass of particle X is < 100 GeV asymmetric errors: mass of particle +4 X = GeV

15 Probability, Set Theory ad Stuff The relatioships ad results from set theory are essetial to the uderstadig of probability. Below are some defiitios ad examples that illustraτe the coectio betwee set theory, probability ad statistics. We defie a experimet as a process that geerates observatios ad a sample space (S) as the set of all possible outcomes from the experimet: simple evet: oly oe possible outcome compoud evet: more tha oe outcome As a example of simple ad compoud evets cosider particles (e.g. protos, eutros) made of u ( up ), d ( dow ), ad s ( strage ) quarks. The u quark has electric charge (Q) =2/3 e (e=charge of electro) while the d ad s quarks have charge =-1/3 e. Let the experimet be the ways we combie 3 quarks to make a Q=0, 1, or 2 state. Evet A: Q=0 {ssu, ddu, sdu} ote: a eutro is a ddu state Evet B: Q=1{suu, duu} ote: a proto is a duu state Evet C: Q=2 {uuu} For this example evets A ad B are compoud while evet C is simple. The followig defiitios from set theory are used all the time i the discussio of probability. Let A ad B be evets i a sample space S. Uio: The uio of A & B (A B) is the evet cosistig of all outcomes i A or B. Itersectio: The itersectio of A & B (A B) is the evet cosistig of all outcomes i A ad B. Complemet: The complemet of A (A ) is the set of outcomes i S ot cotaied i A. Mutually exclusive: If A & B have o outcomes i commo they are mutually exclusive. R.Kass/Sp15 P3700 Lecture 1 15

16 Probability, Set Theory ad Stuff Returig to our example of particles cotaiig 3 quarks ( baryos ): The evet cosistig of charged particles with Q=1,2 is the uio of B ad C: B C The evets A, B, C are mutually exclusive sice they do ot have ay particles i commo. A commo ad useful way to visualize uio, itersectio, ad mutually exclusive is to use a Ve diagram of sets A ad B defied i space S: B S A Ve diagram of A&B B S A A B: itersectio of A&B B S A A B: uio of A&B The axioms of probabilities (P): a) For ay evet A, P(A) 0. (o egative probabilities allowed) b) P(S)=1. c) If A 1, A 2,.A is a collectio of mutually exclusive evets the: (the collectio ca be ifiite (= )) S B A A & B mutually exclusive 1 A2 A ) = P( A i ) i= 1 P( A From the above axioms we ca prove the followig useful propositios: a) For ay evet A: P(A)=1-P(A ) b) If A & B are mutually exclusive the P(A B)=0 items b, c are obvious c) For ay two evets A & B: P(A B)=P(A)+P(B)-P(A B) from their Ve diagrams R.Kass/Sp15 P3700 Lecture 1 16

17 Probability, Set Theory ad Stuff Example: Everyoe likes pizza. Assume the probability of havig pizza for luch is 40%, the probability of havig pizza for dier is 70%, ad the probability of havig pizza for luch ad dier is 30%. Also, this perso always skips breakfast. We ca recast this example usig: P(A)= probability of havig pizza for luch =40% P(B)= probability of havig pizza for dier = 70% P(A B)=30% (pizza for luch ad dier) 1) What is the probability that pizza is eate at least oce a day? The key words are at least oce, this meas we wat the uio of A & B P(A B)=P(A)+P(B)-P(A B) = =0.8 2) What is the probability that pizza is ot eate o a give day? Not eatig pizza (Z ) is the complemet of eatig pizza (Z) so P(Z)+P(Z )=1 prop. c) P(Z )=1-P(Z) =1-0.8 = 0.2 prop. a) 3) What is the probability that pizza is oly eate oce a day? This ca be visualized by lookig at the Ve diagram ad realizig we eed to exclude the overlap (itersectio) regio. P(A B)-P(A B) = =0.5 The o-overlappig blue area is pizza for luch, o pizza for dier. The o-overlappig red area is pizza for dier, o pizza for luch. pizza for luch R.Kass/Sp15 P3700 Lecture 1 pizza for dier 17

18 Coditioal Probability Frequetly we must calculate a probability assumig somethig else has occurred. This is called coditioal probability. Here s a example of coditioal probability: Suppose a day of the week is chose at radom. The probability the day is Thursday is 1/7. P(Thursday)=1/7 Suppose we also kow the day is a weekday. Now the probability is coditioal, =1/5. P(Thursday weekday)=1/5 the otatio is: probability of it beig Thursday give that it is a weekday Formally, we defie the coditioal probability of A give B has occurred as: P(A B)=P(A B)/P(B) We ca use this defiitio to calculate the itersectio of A ad B: P(A B)=P(A B)P(B) For the case where the A i s are both mutually exclusive ad exhaustive we have: 1 ) P( A1 ) + P( B A2 ) P( A2 ) P( B A ) P( A ) = P( B Ai ) P( Ai ) i= 1 P( B) = P( B A For our example let B=the day is a Thursday, A 1 = weekday, A 2 =weeked, the: P(Thursday)=P(thursday weekday)p(weekday)+p(thursday weeked)p(weeked) P(Thursday)=(1/5)(5/7)+(0)(2/7)=1/7 R.Kass/Sp15 P3700 Lecture 1 18

19 Bayes s Theorem Bayes s Theorem relates coditioal probabilities. It is widely used i may areas of the physical ad social scieces. Let A 1, A 2,..A i be a collectio of mutually exclusive ad exhaustive evets with P(A i )>0 for all i. The for ay other evet B with P(B)>0 we have: P( Ai B) P( B Ai ) P( Ai ) P( Ai B) = = P( B) P( B A ) P( A ) j= 1 We call: P(A j ) the aprori probability of A j occurrig P(A j B) the posterior probability that A j will occur give that B has occurred P(B A j ) the likelihood Idepedece has a special meaig i probability: Evets A ad B are said to be idepedet if P(A B)=P(A) Usig the defiitio of coditioal probability A ad B are idepedet iff: P(A B)=P(A)P(B) j j R.Kass/Sp15 P3700 Lecture 1 19

20 Example of Bayes s Theorem While Bayes s theorem is very useful i physics, perhaps the best illustratio of its use is i medical statistics, especially drug testig. Assume a certai drug test: gives a positive result 97% of the time whe the drug is preset: P(positive test drug preset)=0.97 gives a positive result 0.4% of the time if the drug is ot preset ( false positive ) P(positive test drug ot preset)=0.004 Let s assume that the drug is preset i 0.5% of the populatio (1 out of 200 people). P(drug preset)=0.005 P(drug ot preset)=1-p(drug preset)=0.995 What is the probability that the drug is ot preset ad you have a positive test? P(drug is ot preset positive test)=???? Bayes s Theorem gives: P(drug ot preset test positive) = P(test positive drug ot preset) P(drug ot preset) P(test positive drug ot preset) P(drug ot preset) + P(test positive drug preset) P(drug preset) (0.004)( ) P( drug ot preset test positive) = = 0.45 (0.004)( ) + (0.97)(0.005) Thus there is a 45% chace that the test comes back positive eve if you are drug free The real life cosequece of this large probability is that drug tests are ofte admiistered twice R.Kass/Sp15 P3700 Lecture 1 20

21 Aother example from Particle Physics Cosider the (o-)discovery of the petaquark A petaquark is a boud state of 5 quarks These states are allowed but i ~40 years of searchig o evidece for them.. Util 2003 whe several expperimets reported positive evidece Cosider the petaquark from the SPrig-8 (LEPS) experimet Θ 5 =dudus Sigal: 19 evets Sigificace: 4.6σ (Assumig gaussia stats the prob. for a 4.6σ effect is ~4x10-6 ) What are the authors tryig to say here? If this bump is accidetal, the the accidet rate is 1 i 4 millio. Or If I repeated the experimet 4 millio times I would get a bump this big or bigger. What do the authors wat you to thik? Sice the accidet rate is so low it must ot be a accidet, therefore it is physics Θ 5 =dudus R.Kass/Sp15 P3700 Lecture 1 21

22 Petaquark Reality Sometimes it is ot a questio of statistical sigificace Agai, the petaquark state Θ + (1540) gives a great example: Cosider the CLAS experimet at JLAB: 2003/4 report a 7.8σ effect (~6x10-15 accordig to MATHEMATICA) 2005 report NO Sigal (better experimet) CLAS 2003/2004 CLAS 2005 g11 What size sigal should we expect? Lesso: This is ot a statistics issue, but oe of experimet desig ad implemetatio. R.Kass/Sp15 P3700 Lecture 1 22