Interval estimation in the presence of nuisance parameters. 1. Bayesian approach

Coverage Of Confidence Intervals For Poisson Statistics In Presence Of Systematic Uncertainties

In this note we consider coverage of confidence intervals calculated with and without systematic uncertainties. These calculations follow the prescription originally proposed by Cousins & Highland but here extended to account for different shapes, size and types of systematic uncertainties. Also two different ordering schemes are considered: the Feldman & Cousins ordering and its variant where conditioning on the background expectation is applied as proposed by Roe & Woodroofe. Without uncertainties Feldman & Cousins method over-covers as expected because of the discreteness of the Poisson dis…

Likelihood ratio intervals with Bayesian treatment of uncertainties: coverage, power and combined experiments

In this note we present studies of coverage and power for confidence intervals for a Poisson process with known background calculated using the Likelihood ratio (aka Feldman & Cousins) ordering with Bayesian treatment of uncertainties in nuisance parameters. We consider both the variant where the Bayesian integration is done in both the numerator and the denominator and the modification where the integration is done only in the numerator whereas in the denominator the likelihood is taken at the maximum likelihood estimate of the parameters. Furthermore we discuss how measurements can be combin…

Two- and Multi-dimensional Curve Fitting using Bayesian Inference

Fitting models to data using Bayesian inference is quite common, but when each point in parameter space gives a curve, fitting the curve to a data set requires new nuisance parameters, which specify the metric embedding the one-dimensional curve into the higher-dimensional space occupied by the data. A generic formalism for curve fitting in the context of Bayesian inference is developed which shows how the aforementioned metric arises. The result is a natural generalization of previous works, and is compared to oft-used frequentist approaches and similar Bayesian techniques.

A Bayesian approach to evaluate confidence intervals in counting experiments with background

In this paper we propose a procedure to evaluate Bayesian confidence intervals in counting experiments where both signal and background fluctuations are described by the Poisson statistics. The results obtained when the method is applied to the calculation of upper limits will also be illustrated.

非参数有限样本可信区间与一维先验：介于贝叶斯与频率学区间之间的一种中间方法

Non-parametric finite-sample credible intervals with one-dimensional priors: a middle ground between Bayesian and frequentist intervals

本文提出一种非参数有限样本可信区间构造方法，结合贝叶斯与频率学方法，通过一维先验实现统计区间的中间方法，展示其在两个不同问题中的应用及优势。

The relation between frequentist confidence intervals and Bayesian credible intervals

We investigate the relation between frequentist and Bayesian approaches. Namely, we find the "frequentist" Bayes prior π_{f}(λ,x_{obs}) = -\frac{\int_{-\infty}^{x_{obs}}\frac{\partial f(x,λ)}{\partial λ}dx}{f(x_{obs},λ)} (here f(x,λ) is the probability density) for which the results of frequentist and Bayes approaches to the determination of confidence intervals coincide. In many cases (but not always) the "frequentist" prior which reproduces frequentist results coincides with the Jeffreys prior.

On the Estimation of Confidence Intervals for Binomial Population Proportions in Astronomy: The Simplicity and Superiority of the Bayesian Approach

I present a critical review of techniques for estimating confidence intervals on binomial population proportions inferred from success counts in small-to-intermediate samples. Population proportions arise frequently as quantities of interest in astronomical research; for instance, in studies aiming to constrain the bar fraction, AGN fraction, SMBH fraction, merger fraction, or red sequence fraction from counts of galaxies exhibiting distinct morphological features or stellar populations. However, two of the most widely-used techniques for estimating binomial confidence intervals--the 'normal a…

在NOvA实验中用于构建具有无约束和约束干扰参数的置信区间的蒙特卡罗方法

Monte Carlo method for constructing confidence intervals with unconstrained and constrained nuisance parameters in the NOvA experiment

本文记录了NOvA实验选用剖面构建法作为蒙特卡罗方法处理干扰参数以构建置信区间，并介绍了其选择依据、实现细节、验证测试及实用注意事项。

On Bayesian Treatment of Systematic Uncertainties in Confidence Interval Calculation

In high energy physics, a widely used method to treat systematic uncertainties in confidence interval calculations is based on combining a frequentist construction of confidence belts with a Bayesian treatment of systematic uncertainties. In this note we present a study of the coverage of this method for the standard Likelihood Ratio (aka Feldman & Cousins) construction for a Poisson process with known background and Gaussian or log-Normal distributed uncertainties in the background or signal efficiency. For uncertainties in the signal efficiency of upto 40 % we find over-coverage on the level…

通过放大不确定性来覆盖未知的相关性于贝叶斯先验中

Covering Unknown Correlations in Bayesian Priors by Inflating Uncertainties

本文提出通过放大不确定性来覆盖未知相关性，以确保贝叶斯先验中对感兴趣参数的保守后验不确定性。

Bayesian noise estimation for non-ideal CMB experiments

We describe a Bayesian framework for estimating the time-domain noise covariance of CMB observations, typically parametrized in terms of a 1/f frequency profile. This framework is based on the Gibbs sampling algorithm, which allows for exact marginalization over nuisance parameters through conditional probability distributions. In this paper we implement support for gaps in the data streams and marginalization over fixed time-domain templates, and also outline how to marginalize over confusion from CMB fluctuations, which may be important for high signal-to-noise experiments. As a by-product o…

Bayesian Analysis of Inertial Confinement Fusion Experiments at the National Ignition Facility

We develop a Bayesian inference method that allows the efficient determination of several interesting parameters from complicated high-energy-density experiments performed on the National Ignition Facility (NIF). The model is based on an exploration of phase space using the hydrodynamic code HYDRA. A linear model is used to describe the effect of nuisance parameters on the analysis, allowing an analytic likelihood to be derived that can be determined from a small number of HYDRA runs and then used in existing advanced statistical analysis methods. This approach is applied to a recent experimen…