Propagation of Uncertainties in Measurements: Generalized/ Fiducial Inference

Transcription

1 Propagation of Uncertainties in Measurements: Generalized/ Fiducial Inference Jack Wang & Hari Iyer NIST, USA NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 1/31

2 Frameworks for Quantifying Uncertainty Frequentist Inference Likelihood Function Pivotal Quantities Confidence Intervals Bayesian Inference Priors on Parameters Posterior Distributions Probabiliy intervals Use of the Maximum Entropy Principle NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 2/31

3 A Simple Case Unknown value of measurand is µ. Reported value is x. Reported standard uncertainty is u. Reported degrees of freedom is k. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 3/31

4 Frequentist Approach µ is regarded as a fixed but unknown constant. Typical working model: X N(µ,σ 2 ) ku 2 /σ 2 = W χ 2 k. X µ U Student s t with k df. Obtain an interval of plausible values of µ. x ± t 1 α 2 :k u. Frequentist interpretation of a confidence interval. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 4/31

5 Bayesian Approach Describe our prior knowledge about plausible values of µ and σ by specifying prior distributions. Use Bayes theorem from probability theory to deduce the distribution of µ conditional on the observed data. In the absence of any detailed prior knowledge one could assume that µ has a distribution over some very wide interval. The wider this interval is the closer the Bayes solution is to the classical frequentist solution. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 5/31

6 Who Is This? NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 6/31

7 Sir Ronald Fisher NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 7/31

8 Fisher s Fiducial Method X µ σ k U 2 = Z, Z N(0, 1) σ 2 = W, W χ 2 k. µ = X σ Z = X U Z W/k = X U T k. Fiducial distribution: µ = x u T k. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 8/31

9 A More Realistic Case Value of a measurand is µ units. Measured value is x and reported uncertainty is u with k df. Measurement equation: δ is an unknown offset. k U 2 X = µ + δ + σ Z σ 2 = W χ 2 k. Information about δ is expressed in terms of an appropriate probability distribution. How to characterize our state of knowledge of µ? NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 9/31

10 Fisher s Fiducial Method X = µ + δ + σ Z, Z N(0, 1) k U 2 σ 2 = W, W χ 2 k. µ = X δ σ Z Z = X δ U W/k = X δ U T k. Fiducial distribution: µ = x δ u T k. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 10/31

11 A Numerical Illustration Measurand µ X = µ + δ + σ Z, Z N(0, 1) k U 2 σ 2 = W, W χ 2 k. Measurement result = mm; Standard Uncertainty = mm; df = 4. Fiducial distribution: µ = δ T 4. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 11/31

12 Scenario 1 δ N(0,sd = 0.017) Mean = percentile = percentile = NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 12/31

13 Scenario 2 δ skewed with mean = 0, sd = Mean = pctl = pctl = NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 13/31

14 About Fiducial Inference Introduced by R. A. Fisher in the 1930s. Fell into disrepute because of an apparent lack of a mathematical framework for its justification. Re-invented in the 1990 s by Weerahandi under the name Generalized Inference. Recent research results show Fiducial Inference is a valid method, justifiable within the frequentist setting. Fiducial Inference leads to POSTERIOR distributions on parameters without having to declare prior distributions. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 14/31

15 Bayesian Omelette Fiducial inference is an attempt to make the Bayesian Omelette without breaking the Bayesian eggs L. J. Savage. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 15/31

16 Fiducial Method in the Published Literature Weerahandi (1989, 1993). J. of American Statistical Association McNally, Iyer, Mathew. (2001). Statistics in Medicine J. Hannig, C. Wang, H. Iyer (2003). Uncertainty calculation for the ratio of dependent measurements, Metrologia, 40 (4), H. Iyer, C. Wang, T. Matthew (2004). Models and confidence intervals for true values in interlaboratory trials, J. of the American Statistical Association, 99, H. Iyer, C. Wang, D. F. Vecchia (2004). Consistency tests for key comparison data, Metrologia, 41 (4), C. Wang, H. Iyer (2005). Propagation of uncertainties in measurements using generalized inference, Metrologia, 42 (2), Hannig, Iyer, Patterson (2005). Fiducial Generalized Confidence Intervals, J. of American Statistical Association. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 16/31

17 GUM Example Example Annex H.1 of the GUM: Gauge Block Calibration NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 17/31

18 Gauge Blocks NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 18/31

19 Gauge Block Calibration NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 19/31

20 Gauge Block Calibration d = l[1 + (δ α + α s )θ] l s [1 + (θ δ θ )α s ] l = l s[1 + (θ δ θ )α s ] + d. 1 + (δ α + α s )θ NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 20/31

21 Gauge Block Calibration l is the length (at 20 C) of the test gauge block l s is the length (at 20 C) of the reference gauge block α is the coefficient of linear expansion of the test gauge block α s is the coefficient of linear expansion of the reference gauge block θ is the deviation in temperature from 20 C of the test gauge θ s is the deviation in temperature from 20 C of the reference gauge d = l l s δ α = α α s δ θ = θ θ s. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 21/31

22 Fiducial Distribution for d d i N(µ d,σ d ), i = 1,...,5. Mean of d i is 215 nm. Estimate of σ d is u d = 13 nm (24 df) based on an independent experiment. The value d differs from µ d due to errors associated with the comparator device. We have d µ d = d R + d S d R N(0,σ dr ) and d S N(0,σ ds ). u dr = 3.9 nm (5 df) u ds = 6.7 nm (8 df). d = ( µ d + d R + d S = T ) T T 8 5 (assume independence) NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 22/31

23 Fiducial Distribution for l s The estimated value of l s (i.e., the value given in the calibration certificate), denoted by ˆl s, is equal to nm. Uncertainty stated in the calibration certificate is u ls = 25 nm (with 18 degrees of freedom). Regard ˆl s N(l s,σ 2 l s ) and 18s2 l s σ 2 l s χ Fiducial distribution of l s is: ls = T 18. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 23/31

24 Subjective Distribution for θ θ = Devn of test bed temp from 20 C. µ θ = average devn from 20 C. We write θ = µ θ + represents the cyclic variation of the temperature of the test bed under a thermostat system. Estimate of µ θ = θ = 0.1 C with a standard uncertainty equal to 0.2 C (df = ). Regard θ N(µ θ, (0.2) 2 ). (units are C) is assumed to have a U-shaped density function g( ) = 2 π 1 4 2, 0.5 C < < 0.5 C. Mean = 0 and Standard deviation = 1/ 8 = NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 24/31

25 Subjective Distribution of Delta Observation: If U is a uniform [0, 1], then = cos(π U)/2 has the arcsine distribution. Fiducial distribution for θ is given by θ = Z cos(πu)/2 NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 25/31

26 Distributions for δ α,δ θ,α s Assume δ α is a uniform random variable over the interval ± C 1. Assume δ θ is uniform over the interval ±0.05 C. Assume α s is uniform over the interval ± C 1 l = l s[1 + (θ δ θ )α s ] + d. 1 + (δ α + α s )θ NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 26/31

27 Uncertainty Accounting Summary l = l s[1 + (µ θ + δ θ )α s ] + µ d + d R + d S 1 + (δ α + α s )θ. l = l s[1 + (θ δ θ )α s ] + d. 1 + (δ α + α s )θ NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 27/31

28 Fiducial Representation of l l = { ( T 18 )[1 ( Z + U 3 +cos(π U 1 )/2)( U 4 )] T 24 / 5 3.9T 5 6.7T 8 1 (U U 4 )( Z+ cos(π U 1 )/2) } Using simulation, the mean and the standard deviation of l are nm and 35 nm, respectively. A 99 % confidence interval for l can be obtained by using the interval between the and quantiles of the simulated distribution. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 28/31

29 Fiducial Density of the Measurand l percent of total A 99% CI is given by ( , ) nm. For comparison, the result from GUM is ( , ) nm length , nm Histogram of the Fiducial distribution of the true length l based on Monte Carlo samples NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 29/31

30 Final Remarks Fiducial Inference/Generalized Inference is a valid statistical method with good operating characteristics. Fiducial/Generalized intervals have satisfactory coverage Similarity with Bayesian approaches get posterior distribution" for measurand without assuming prior distributions for unknown parameters. The method allows the combination of type-b and type-a uncertainty information in a logical and straightforward manner. Principle of Maximum Entropy can be used when deemed appropriate. Generally requires a Monte-carlo approach for computing uncertainty intervals. Computations are straight-forward. NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 30/31

31 Thank You NMIJ-BIPM Workshop, AIST-Tsukuba, Japan, May 18-20, 2005 p. 31/31