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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

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)

- statistical uncertainty

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

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

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 Outlier-tolerant likelihoods

- Drawing analogy between φ(Δx) = minus-log-posterior and a physical potential
- is a force with which each datum pulls on fit model

- 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

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