概率ProbabilityA&BAWholeBAdefinitionofprobabilityspace,sConsider a set S with subsets A, B, ..For all A C S, P(A) ≥ OP(S) = 1KolmogorovIf AnB = 0, P(AUB) = P(A) +P(B)axioms(1933)P(AnB)Alsodefine conditionalP(A|B) :P(B)probability ofA given BP(AnB)= P(A)P(B)SubsetsA,BindependentifP(A)P(B): P(A)If A, B independent, P(A|B)P(B)
Whole space, SA B A&B 概率Probability
概率ProbabilityInterpretation of probabilityI. Relative frequencyA, B, ...are outcomes of a repeatable experimenttimes outcome is AP(A) =limn8ncf. quantum mechanics, particle scattering, radioactive decay..Il.SubjectiveprobabilityA,B,...arehypotheses(statementsthataretrueorfalse)P(A) = degree of belief that A is trueBoth interpretations consistent with Kolmogorov axiomsInparticlephysics frequencyinterpretationoftenmostusefulbut subjective probability can provide more natural treatment ofnon-repeatablephenomena:systematic uncertainties, probability thatHiggs boson exists
概率Probability
P(B)P(A)Bayes TheoremWhole spaceBP(A/B)P(BIA)P(AB)P(A) X P(BIA)PAB)P(B) X P(AIB)P(AOB)Fromthe definition of conditional probability we have,P(AnB)P(BnA)P(A|B) =P(B|A) =andP(B)P(A)but P(An B) = P(B n A), soBayes'theoremP(B|A)P(A)P(A|B) =P(B)Firstpublished(posthumously)bytheReverendThomasBayes(1702-1761)Anessay towards solving aprobleminthedoctrine of chances, Philos. Trans. R. Soc. 53(1763)370; reprinted inBiometrika,45(1958)293
Bayes Theorem
AnexampleusingBayestheoremSupposetheprobability (foranyone)tohaveadiseaseDis:P(D)=0.001-priorprobabilities,iebefore any test carried outP(noD)=0.999Consideratestforthedisease:resultis+or-P(+(D)0.98Mprobabilitiesto(in)correctlyidentifyapersonwiththediseaseP(-{D)0.02=0.03P(+|no D)=probabilitiesto(in)correctly-identify a healthy person=0.97P(-no D)Supposeyour resultis+,Howworried shouldyoube?Theprobabilitytohavethediseasegivena+resultisP(+/D)P(D)p(D/+)P(+D)P(D) + P(+|no D)P(no D)0.98x0.0010.98×0.001+0.03×0.999posteriorprobability0.032-i.e. you're probably OK!Your viewpoint:my degree of belief that I have the disease is 3.2%Your doctor's viewpoint:3.2%ofpeoplelike this have the disease
Probability density functionsOften a measurement of some observablequantityresults inacontinuous random variable. We can consider intervals in the possiblevaluesas subsetsof thetotal sample space, andderivetheprobabilitythatthemeasurementisincludedinthem P(xmeas in [x, X+dx[) =f(x) dx- Of course this has no meaning unless P(x in S)=1If x is single-dimensional, we realize that asetofprobabilityvaluesastheonedefinedabovecanbeeffectivelydrawnas0.5X0.45ahistogram,if wedefinethewidthof the1g→68.3%0.4bins to be dx.0.352G→95.5%0.3Bylettingdxgotozero,weobtainthe0.250.2continuous function f(x). This is called0.15probability densityfunction"of x (PDF).0.10.05
• If x is single-dimensional, we realize that a set of probability values as the one defined above can be effectively drawn as a histogram, if we define the width of the bins to be dx. • By letting dx go to zero, we obtain the continuous function f(x). This is called "probability density function" of x (PDF). • Often a measurement of some observable quantity results in a continuous random variable. We can consider intervals in the possible values as subsets of the total sample space, and derive the probability that the measurement is included in them – P(xmeas in [x, x+dx[) = f(x) dx – Of course this has no meaning unless P(x in S)=1 Probability density functions