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Register a new userA Paradox about Likelihood Ratios?
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Authors
Louis Lyons
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We consider whether the asymptotic distributions for the log-likelihood ratio test statistic are expected to be Gaussian or chi-squared. Two straightforward examples provide insight on the difference.
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We describe likelihood-based statistical tests for use in high energy physics for the discovery of new phenomena and for construction of confidence intervals on model parameters. We focus on the properties of the test procedures that allow one to account for systematic uncertainties. Explicit formulae for the asymptotic distributions of test statistics are derived using results of Wilks and Wald. We motivate and justify the use of a representative data set, called the Asimov data set, which provides a simple method to obtain the median experimental sensitivity of a search or measurement as well as fluctuations about this expectation.
We present a conclusive answer to Bertrands paradox, a long standing open issue in the basic physical interpretation of probability. The paradox deals with the existence of mutually inconsistent results when looking for the probability that a chord, drawn at random in a circle, is longer than the side of an inscribed equilateral triangle. We obtain a unique solution by substituting chord drawing with the throwing of a straw of finite length L on a circle of radius R, thus providing a satisfactory operative definition of the associated experiment. The obtained probability turns out to be a function of the ratio L/R, as intuitively expected.
We present a universal method to include residual un-modeled background shape uncertainties in likelihood based statistical tests for high energy physics and astroparticle physics. This approach provides a simple and natural protection against mismodeling, thus lowering the chances of a false discovery or of an over constrained confidence interval, and allows a natural transition to unbinned space. Unbinned likelihood allows optimal usage of information for the data and the models, and enhances the sensitivity. We show that the asymptotic behavior of the test statistic can be regained in cases where the model fails to describe the true background behavior, and present 1D and 2D case studies for model-driven and data-driven background models. The resulting penalty on sensitivities follows the actual discrepancy between the data and the models, and is asymptotically reduced to zero with increasing knowledge.
This paper presents a statistical method to subtract background in maximum likelihood fit, without relying on any separate sideband or simulation for background modeling. The method, called sFit, is an extension to the sPlot technique originally developed to reconstruct true distribution for each date component. The sWeights defined for the sPlot technique allow to construct a modified likelihood function using only the signal probability density function and events in the signal region. Contribution of background events in the signal region to the likelihood function cancels out on a statistical basis. Maximizing this likelihood function leads to unbiased estimates of the fit parameters in the signal probability density function.
In this paper, we consider a surrogate modeling approach using a data-driven nonparametric likelihood function constructed on a manifold on which the data lie (or to which they are close). The proposed method represents the likelihood function using a spectral expansion formulation known as the kernel embedding of the conditional distribution. To respect the geometry of the data, we employ this spectral expansion using a set of data-driven basis functions obtained from the diffusion maps algorithm. The theoretical error estimate suggests that the error bound of the approximate data-driven likelihood function is independent of the variance of the basis functions, which allows us to determine the amount of training data for accurate likelihood function estimations. Supporting numerical results to demonstrate the robustness of the data-driven likelihood functions for parameter estimation are given on instructive examples involving stochastic and deterministic differential equations. When the dimension of the data manifold is strictly less than the dimension of the ambient space, we found that the proposed approach (which does not require the knowledge of the data manifold) is superior compared to likelihood functions constructed using standard parametric basis functions defined on the ambient coordinates. In an example where the data manifold is not smooth and unknown, the proposed method is more robust compared to an existing polynomial chaos surrogate model which assumes a parametric likelihood, the non-intrusive spectral projection.
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