July 9, 2007 Bayesian Inference and Maximum Entropy 20 07 1 Lessons about Likelihood Functions from Nuclear Physics Kenneth M. Hanson T-16, Nuclear Physics; Theoretical Division Los Alamos National Laboratory This presentation available at http://www.lanl.gov/home/k mh/ LA-UR-07- 2971 Bayesian Inference and Maximum Entropy Workshop, Saratoga Springs, NY, July 8-13, 2007
25
Embed
July 9, 2007Bayesian Inference and Maximum Entropy 20071 Lessons about Likelihood Functions from Nuclear Physics Kenneth M. Hanson T-16, Nuclear Physics;
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 1
Lessons about Likelihood Functions from Nuclear Physics
Kenneth M. Hanson
T-16, Nuclear Physics; Theoretical DivisionLos Alamos National Laboratory
This presentation available at http://www.lanl.gov/home/kmh/
LA-UR-07-2971
Bayesian Inference and Maximum Entropy Workshop, Saratoga Springs, NY, July 8-13, 2007
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 2
Overview• Uncertainties in physics experiments
• Particle Data Group (PDG)
• Lifetime data
• Outliers
• Uncertainty in the uncertainty
• Student t distributions vs. Normal distribution
• Analysis of lifetime data using t distributions
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 3
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 4
Physics experiments• Experimenters state their measurement of physical quantity y as
measurement ± standard error or d ± σd
• Experimenter’s degree of belief in measurement described by a normal distribution (Gaussian) and σd its standard deviation
• Experimental uncertainty often composed of two components:► statistical uncertainty
• from noise in signal or event counting (Poisson)
• Type A – determined by repeated meas., frequentist methods► systematic uncertainty
• from equipment calibration, experimental procedure, corrections
• Type B – determined by nonfrequentist methods
• based on experimenter’s judgment, hence subjective; difficult to assess ► these usually added in quadrature (rms sum)
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 5
Physics experiments – likelihood functions• In probabilistic terms, experimentalist’s statement y ± σd
is interpreted as a likelihood functionp( d | y σd I)
where I is background information about situation, including how experiment is performed
• Inference about the physical quantity y is obtained by Bayes lawp( y | d σy I) ~ p( y | d σd I) p( y | I)
where p( y | I) is the prior information about y
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 7
Exploratory data analysis• John Tukey (1977) suggested each set of
measurements of a quantity be scrutinized
► find quantile positions, Q1, Q2, Q3► calculate the inter-quantile range
IQR = Q3 – Q2 (for normal distr., IQR = 1.35 σ)
► determine fraction of data outside interval, SO Q1 – 1.5 IQR < y < Q3 + 1.5 IQRlabeling these as suspected outliers(for normal distr., 0.7%)
• IQR measures width of core• SO measures extent of tail
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 8
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 9
Particle Data Group (PDG)• Particle Data Group formed in 1957
► annually summarizes state-of-knowledge of properties of elementary particles
• For each particle property► list all relevant experimental data► committee decides which data to include in final analysis► state best current value (usually least-squares average)
and its standard error • often magnified by sqrt[χ2/(N – 1)] (avg of 2.0, 50% of time)
• PDG reports are excellent source of information about measurements of unambiguous physical quantities
► available online, free► provide insight into how physicists interpret data
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 11
Lambda lifetime measurement in the 60s• Hydrogen bubble chambers were used in
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 24
Measurements of lambda lifetime• Upper plot shows all
measurements of lambda lifetime
• Lower plot shows results based on all 27 data points:
► posterior for t-distr analysis (ν = 2.6, margin. over s)
► least-squares result (with χ2 scaling)
► PDG results (using 3 latest data points)
Lambda lifetime measurements
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 25
Tests • Draw 20 data points from various
t distrs. and analyze them using likelihoods: a) t distr with ν = 3 b) Normal distr
► scale uncertainties according to data variance
► results from 10,000 random trials
• Conclude► t distr results well behaved► normal distr results unstable
when data have significant outliers
t distr
Normal
dashed line = estimated σ
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 26
Summary• Technique presented for dealing gracefully with outliers
► is based on using for likelihood function the Student t distr. instead of the Normal distr.
► copes with outliers, while treating every datum identically
• Particle lifetime data distribution matched by t distr. with ν ≈ 2.6 to 3.0
► using likelihood functions based on t distr. produce stable results when outliers exist in data sets, whereas Normal distr. does not
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 27
Bibliography ► “A further look at robustness via Bayes;s theorem,” G.E.P. Box and G.C. Tiao,
Biometrica 49, pp. 419-432 (1962)► “On outlier rejection phenomena in Bayes inference,” A. O’Hagan, J. Roy. Statist. Soc. B
41, 358–367 (1979)► “Bayesian evaluation of discrepant experimental data,” F.H. Fröhner, Maximum Entropy
and Bayesian Methods, pp. 467–474 (Kluwer Academic, Dordrecht, 1989)► “Estimators for the Cauchy distribution,” K.M. Hanson and D.R. Wolf, Maximum Entropy
and Bayesian Methods, pp. 157-164 (Kluwer Academic, Dordrecht, 1993)► “Dealing with duff data,” D. Sivia, Maximum Entropy and Bayesian Methods, pp. 157-
164 (1996)► “Understanding data better with Bayesian and global statistical methods,” in W.H. Press,
Unsolved Problems in Astrophysics, pp. 49-60 (1997)► “Outlier-tolerant parameter estimation,” V. Dose and W. von der Linden, Maximum
Entropy and Bayesian Methods, pp. 157-164 (AIP, 2000)
This presentation available at http://www.lanl.gov/home/kmh/
July 9, 2007 Bayesian Inference and Maximum Entropy 2007 28