Estimators for the Cauchy Distribution

AIP Conference Proceedings

2007

Abstract. Least-squares data analysis is based on the assumption that the normal (Gaussian) distribution appropriately characterizes the likelihood, that is, the conditional probability of each measurement d, given a measured quantity y, p (d | y). On the other hand, there is ample evidence in nuclear physics of significant disagreements among measurements, which are inconsistent with the normal distribution, given their stated uncertainties. In this study the histories of 99 measurements of the lifetimes of five elementary particles are examined to determine what can be inferred about the distribution of their values relative to their stated uncertainties. Taken as a whole, the variations in the data are somewhat larger than their quoted uncertainties would indicate. These data strongly support using a Student t distribution for the likelihood function instead of a normal. The most probable value for the order of the t distribution is 2.6 ± 0.9. It is shown that analyses based on long-tailed t-distribution likelihoods gracefully cope with outlying data.

“…See Refs. [6][7][8][9][10][11][12][13] for other contributions to the outlier discussion. Figure 3a shows the Student 1 t ν distribution for three ν values, ν = 1, 3, and ∞.…”

Section: Uncertainty In the Uncertaintymentioning

confidence: 99%

Lessons about likelihood functions from nuclear physics

AIP Conference Proceedings

2007

“…By incorporating it probabilistically and considering it to be a nuisance variable, the signal is removed from the analysis by integrating over it. This idea grew out of recent Bayesian approaches to the treatment of outlying data in which it was recognized that the presence of a wide nonGaussian tail in the likelihood function effectively reduces the influence of outliers [11,12,13,14]. __We introduce the proposition Bi\ "datum di is purely background' and its complement Bi\ 'di contains some signal contribution'.…”

Section: The Likelihoodmentioning

confidence: 99%

“…The sum of the two types of likelihood in the mixture model for each datum results in a likelihood function with a central peak plus a long tail. The presence of such a long tail has the effect of reducing the influence of outlying data points when several data points are combined [11,12,13,14]. In the case of background estimation, the result is to reduce the influence of points that lie outside the uncertainty band of the measurement errors, which presumably contain significant signal contributions.…”

Section: J -Oomentioning

confidence: 99%

“…The denominator in (2) can be ignored at this point because it does not depend on c. What is actually done is to minimize i/j, defined in Eq. (14), with respect to the knot values c, which is a nonlinear optimization problem. To evaluate i/j, we use the likelihood given in Eq.…”

Section: Estimation Of Spline Valuesmentioning

confidence: 99%

“…(14). This optimization problem is somewhat harder than the one associated with finding c. The reason lies in the numerous constraints on the knot positions, namely that they must be ordered and they must be no closer to each other than a specified Arc.…”

Section: Estimation Of Knot Positionsmentioning

confidence: 99%

See 2 more Smart Citations

Bayesian background estimation

Fischer

Dose

AIP Conference Proceedings

et al. 2001

Abstract. The ubiquitous problem of estimating the background of a measured spectrum is solved with Bayesian probability theory. A mixture model is used to capture the defining characteristics of the problem, namely that the background is smoother than the signal. The smoothness property is quantified in terms of a cubic spline basis where a variable degree of smoothness is attained by allowing the number of knots and the knot positions to be adaptively chosen on the basis of the data. The fully Bayesian approach taken provides a natural way to handle knot adaptivity, allows uncertainties in the background to be estimated and data points to be classified in groups containing only background and groups with additional signal contribution. Our technique is demonstrated on a PIXE spectrum from a geological sample and an Auger spectrum from an 10 monolayer iron film on tungsten.

Uncertainty assessment for reconstructions based on deformable geometry

Int. J. Imaging Syst. Technol.

Cunningham

McKee

1997

Self Cite

Deformable geometric models can be used in the context of Bayesian analysis to solve ill‐posed tomographic reconstruction problems. The uncertainties associated with a Bayesian analysis may be assessed by generating a set of random samples from the posterior, which may be accomplished using a Markov Chain Monte Carlo (MCMC) technique. We demonstrate the combination of these techniques for a reconstruction of a two‐dimensional object from two orthogonal noisy projections. The reconstructed object is modeled in terms of a deformable geometrically defined boundary with a uniform interior density yielding a nonlinear reconstruction problem. We show how an MCMC sequence can be used to estimate uncertainties in the location of the edge of the reconstructed object. © 1997 John Wiley & Sons, Inc. Int J Imaging Syst Technol, 8, 506–512, 1997