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Lessons about Likelihood Functions from Nuclear Physics. Kenneth M. Hanson T-16, Nuclear Physics; Theoretical Division Los Alamos National Laboratory. Bayesian Inference and Maximum Entropy Workshop, Saratoga Springs, NY, July 8-13, 2007 .

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Lessons about Likelihood Functions from Nuclear Physics

Kenneth M. Hanson

T-16, Nuclear Physics; Theoretical DivisionLos Alamos National Laboratory

Bayesian Inference and Maximum Entropy Workshop, Saratoga Springs, NY, July 8-13, 2007

This presentation available at http://www.lanl.gov/home/kmh/

LA-UR-07-2971

Bayesian Inference and Maximum Entropy 2007


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

Bayesian Inference and Maximum Entropy 2007


Bayesian Inference and Maximum Entropy 2007


Physics experiments

  • Experimenters state their measurement of physical quantity y asmeasurement ± 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)

Bayesian Inference and Maximum Entropy 2007


Physics experiments – likelihood functions

  • In probabilistic terms, experimentalist’s statement y± σdis 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 σyI) ~ p( y | d σdI) p( y | I)where p( y | I) is the prior information about y

Bayesian Inference and Maximum Entropy 2007


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

Bayesian Inference and Maximum Entropy 2007


Bayesian Inference and Maximum Entropy 2007


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

Bayesian Inference and Maximum Entropy 2007


Lambda lifetime measurement in the 60s

Hydrogen bubble- chamber photo

  • Hydrogen bubble chambers were used in 1950s and 60s to record elementary particles

  • Picture shows reaction sequence: K- + p →Ξ- + K+Ξ-→Λ0 + π-Λ0→ p + π-

  • Track lengths and particle momenta determined from curvature in magnetic field yield survival time of Ξ- and Λ

  • Hubbard et al. observed 828 such events to obtain lifetimes:τΞ = 1.69 ±0.06×10-10 sτΛ = 2.59 ±0.09×10-10 s

From J.R. Hubbard et al., Phys.Rev.135B (1964)

Bayesian Inference and Maximum Entropy 2007


Measurements of neutron lifetime

Neutron lifetime measurements

  • Because n lifetime is so long, it is difficult to measure without slowing neutrons or trapping them

  • Plot shows all measurements of neutron lifetime

  • Red line is PDG value, which includes 7 data sets,but excludes older ones and #2 because it is discrepant

  • χ2 for red line = 149/21 pts.

  • Evidence of outliers

  • Large systematic uncertainties

Bayesian Inference and Maximum Entropy 2007


Measurements of lifetimes of other particles

Bayesian Inference and Maximum Entropy 2007


Collection of lifetime measurements

  • Goal: determine distribution of measurements relative to their estimated uncertainties

  • Upper graph shows deviations of 99 lifetime meas. for 5 particles from PDG values, divided by their standard errors, i.e. Δx/σ

  • Lower graph shows histogram

  • Objective: characterize the distribution of Δx/σ for these expts

  • χ2 = 367/99 DOF

  • IQR = 1.83 (1.35 for normal)

  • Suspected outliers = 6.6 %

Bayesian Inference and Maximum Entropy 2007


Bayesian Inference and Maximum Entropy 2007


Uncertainty in the uncertainty

  • Suppose there is uncertainty in the stated standard error σ0 for measurement d

  • Dose and von der Linden (2000) gave plausible derivation:

    • assume likelihood has underlying Normal distr

    • assume uncertainty distr for ω, where σ is scaled by

    • marginalizing over ω, the likelihood is Student t distr., (2a = ν)

  • Many have contributed to outlier story: Box and Tiao, O’Hagan, Fröhner, Press, Sivia, Hanson and Wolf

Bayesian Inference and Maximum Entropy 2007


Student t distribution

ν = 1, 5, ∞

  • Student* t distribution

    • long tail for ν< 9 (SO > 1%)

    • outlier-tolerant likelihood function

    • ν= 1 is Cauchy distr (solid red)

    • ν= ∞ is Normal distr (solid blue)

* Student (1908) was pseudonym for W.S Gossett, who was not allowed to publish by his employer, Guiness brewery

Bayesian Inference and Maximum Entropy 2007


Physical analogy of probability

  • Drawing analogy between φ(Δx) = minus-log-posterior and a physical potential

    • is a force with which each datum pulls on fit model

  • Outlier-tolerant likelihoods

    • generally have long tails

    • restoring force eventually decreases for large residuals

  • G

    2G

    G

    G+E

    2G

    G+C

    G+C

    G+E

    Bayesian Inference and Maximum Entropy 2007


    Analysis of a collection of data

    • To calculate “average” value of a data set, use the Student t distribution for the likelihood of each datum:where s is scaling factor of standard error for whole data set

    • Select ν based on data using model selection

    • Scale factor s marginalized out of posterior

    Bayesian Inference and Maximum Entropy 2007


    Model selection

    t distr

    • Odds ratios of t distr (t) to Normal (N) is = 1.32x10-85 /2.2x10-90 = 6x104

      • for prior ratio on models = 1

      • evidence is integral over x (lifetime) and s; includes prior on s proportional to 1/s

    • Thus, t distr is strongly preferred by data to Normal distr

      • ν ≈ 2.6 (maximizes evidence)

    Normal distr

    Bayesian Inference and Maximum Entropy 2007


    Measurements of neutron lifetime

    Neutron lifetime measurements

    • Upper plot shows all measurements of neutron lifetime

    • Lower plot shows results based on all 21 data points:

      • posterior for t-distr analysis (ν = 2.6, margin. over s)

      • least-squares result (with and w/o χ2 scaling)

      • PDG results (using 7 selected data points, Serebrov rejected)

    Bayesian Inference and Maximum Entropy 2007


    Measurements of π0 lifetime

    π0 lifetime measurements

    • Upper plot shows all measurements of π0 lifetime

    • Lower plot shows results based on all 13 data points:

      • posterior for t-distr analysis (ν = 2.6, margin. over s)

      • least-squares result (with χ2 scaling)

      • PDG results (using 4 selected data points, excl. latest one)

    Bayesian Inference and Maximum Entropy 2007


    Measurements of lambda lifetime

    Lambda lifetime measurements

    • 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)

    Bayesian Inference and Maximum Entropy 2007


    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

    dashed line = estimated σ

    Normal

    t distr

    Bayesian Inference and Maximum Entropy 2007


    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

    Bayesian Inference and Maximum Entropy 2007


    Bibliography

    • “A further look at robustness via Bayes;s theorem,” G.E.P. Box and G.C. Tiao, Biometrica49, 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/

    Bayesian Inference and Maximum Entropy 2007


    Bayesian Inference and Maximum Entropy 2007


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