Probabilistic interpretation of Peelle's pertinent puzzle and its resolution

Peelle’s Pertinent Puzzle (PPP) states a seemingly plausible set of measurements with their covariance matrix, which produce an implausible answer. To answer the PPP question, we describe a reasonable experimental situation that is consistent with the PPP solution. The confusion surrounding the PPP arises in part because of its imprecise statement, which permits to a variety of interpretations and resulting answers, some of which seem implausible. We emphasize the importance of basing the analysis on an unambiguous probabilistic model that reflects the experimental situation. We present several different models of how the measurements quoted in the PPP problem could be obtained, and interpret their solution in terms of a detailed probabilistic analysis. We suggest a probabilistic approach to handling uncertainties about which model to use. PEELLE’S PERTINENT PUZZLE Peelle’s original statement [1] of the PPP is as follows: “Suppose we are required to obtain the weighted average of two experimental results for the same quantity. The first result is 1.5, and the second result is 1.0. The full covariance matrix of these data is believed to be the sum of three components. The first component is fully correlated with standard error 20% of each representative value. The second and third components are independent of the first and each other, and correspond to 10% random uncertainties in each result. The weighted average from the least-squares method is 0.88 0.22, a value outside the range of the input values! Under what conditions is this the reasonable result that we sought to achieve by use of an advanced data reduction technique?” The PPP effect has been observed in numerous evaluations of nuclear cross sections. [2, 3, 4, 5, 6] Initially, the answer stated in the PPP seems implausible. The seeming paradox arises because the statement of the puzzle is ambiguous. For example, it is not stated whether the correlated uncertainty contributes to the measurements in an additive or multiplicative manner. We propose a plausible experimental situation that would correctly yield the answer of 0.88. Alternative interpretations of the PPP may be more appropriate in the context of nuclear cross-section evaluation. A preciselystated uncertainty model clarifies the probabilistic approach that needs to be taken. 1 Corresponding e-mail address: [email protected] LA-UR-04-6722 Standard solution We designate the two quantities that are measured by x1 and x2, and represent them by the vector x x1 x2 T. The measurements m1 = 1.5 and m2 = 1.0 are represented by m m1 m2 T. The measurements have relative independent standard errors of ρ 1 = ρ2 = 0.1, and a relative common error ρ c = 0.2. Assuming that normal distributions are appropriate, the probability density function (pdf) for x is given by [7, 8] p x m ∝ exp 2 x m TC 1 x m (1) where the covariance matrix is C m1 ρ2 1 ρ2 c m1m2ρ 2 c m1m2ρ 2 c m 2 2 ρ2 1 ρ2 c (2) The argument of the exponential in (1) is 2χ 2, generalized to include correlations. Figure 1a shows the joint distribution for x1 and x2, considered as separate variables. The strong correlation between x1 and x2 is evident from the tilt of the contours. Because x1 and x2 are the same quantity, we set x x1 x2. Figure 1b shows the resulting one-dimensional distribution p x , which is a normal distribution centered on x 0 882 and with a standard deviation of 0.228. An identical result is given by the familiar leastsquares solution, which maximizes the probability (1): x GTC 1G 1GTC 1m (3) and with the covariance V GTC 1G 1 where G is the sensitivity matrix, i.e, the matrix of derivatives of measured variables m with respect to the inferred variables x; G 1 1 . Proc. Int. Conf. Nuclear Data for Science and Technology R.C. Haight, M.B. Chadwick, T. Kawano, and P. Talou, eds. AIP Conf. Proc. 769, pp. 304-307 (AIP, Melville, 2005)