Fiducial theory and optimal inference

Taraldsen, Gunnar; Lindqvist, Bo Henry

doi:10.1214/13-aos1083

Cited by 53 publications

(70 citation statements)

References 49 publications

(54 reference statements)

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“…Suppose that the number of points m and the scaling factor μ are chosen such that

P (ε \in N_{m, μ}) ⩾ 1 - α .

If the noise distribution is completely known, then one can for example simulate from the left‐hand side in inequality (8) to ensure that the constraint is satisfied. For a known noise distribution, the noise vector is clearly a pivotal quantity and this is exploited in the argument above and could possibly be extended to fiducial‐type inference (Cisewski and Hannig, ; Wang et al ., ; Taraldsen and Lindqvist, ). We shall return later to the question of the choice of m and μ if the variance of the noise is unknown (as it will be in practice).…”

Section: Confidence Intervals For Groups Of Variablesmentioning

confidence: 98%

Group Bound: Confidence Intervals for Groups of Variables in Sparse High Dimensional Regression Without Assumptions on the Design

Meinshausen

2014

Journal of the Royal Statistical Society Series B: Statistical Methodology

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It is in general challenging to provide confidence intervals for individual variables in highdimensional regression without making strict or unverifiable assumptions on the design matrix. We show here that a "group-bound" confidence interval can be derived without making any assumptions on the design matrix. The lower bound for the regression coefficient of individual variables can be derived via linear programming. The idea also generalises naturally to groups of variables, where we can derive a one-sided confidence interval for the joint effect of a group. While the confidence intervals of individual variables are by the nature of the problem often very wide, it is shown to be possible to detect the contribution of groups of highly correlated predictor variables even when no variable individually shows a significant effect. The assumptions necessary to detect the effect of groups of variables are shown to be weaker than the weakest known assumptions to detect the effect of individual variables.

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“…Suppose that the number of points m and the scaling factor μ are chosen such that

P (ε \in N_{m, μ}) ⩾ 1 - α .

Section: Confidence Intervals For Groups Of Variablesmentioning

confidence: 98%

Group Bound: Confidence Intervals for Groups of Variables in Sparse High Dimensional Regression Without Assumptions on the Design

Meinshausen

2014

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…To see this, suppose that there is a joint distribution for ( X , θ ) that is consistent with both the sampling model (‘ X | θ ’) and the fiducial distribution (‘ θ | X ’). Then θ , or some transformation thereof, must be a location parameter, and the fiducial distribution corresponds to the Bayesian posterior obtained from a flat prior on the location parameter (see Refs and for details).…”

Section: Difficulties In Reasoning Toward Prior‐free Inferencementioning

confidence: 99%

Frameworks for prior‐free posterior probabilistic inference

Liu

Martin

2014

WIREs Computational Stats

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The development of statistical methods for valid and efficient probabilistic inference without prior distributions has a long history. Fisher's fiducial inference is perhaps the most famous of these attempts. We argue that, despite its seemingly prior-free formulation, fiducial and its various extensions are not prior-free and, therefore, do not meet the requirements for prior-free probabilistic inference. In contrast, the inferential model (IM) framework is genuinely prior-free and is shown to be a promising new method for generating both valid and efficient probabilistic inference. With a brief introduction to the two fundamental principles, namely, the validity and efficiency principles, the three-step construction of the basic IM framework is discussed in the context of the validity principle. Efficient IM methods, based on conditioning and marginalization are illustrated with two benchmark examples, namely, the bivariate normal with unknown correlation coefficient and the Behrens--Fisher problem.Comment: 14 pages, 1 figure; to appear in WIREs Computational Statistic

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“…According to Fisher (1950, p.428): The importance of the paper lies, however, in setting forth a new mode of reasoning from observations to their hypothetical causes. Today, this new mode of reasoning is still in development and different lines of arguments have been published (Schweder and Hjort, 2002;Taraldsen and Lindqvist, 2013;Xie and Singh, 2013;Martin and Liu, 2014;Hannig et al, 2016).…”

Section: Introductionmentioning

confidence: 99%

Conditional fiducial models

Taraldsen

Lindqvist

2018

Journal of Statistical Planning and Inference

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The fiducial is not unique in general, but we prove that in a restricted class of models it is uniquely determined by the sampling distribution of the data. It depends in particular not on the choice of a data generating model. The arguments lead to a generalization of the classical formula found by Fisher (1930). The restricted class includes cases with discrete distributions, the case of the shape parameter in the Gamma distribution, and also the case of the correlation coefficient in a bivariate Gaussian model. One of the examples can also be used in a pedagogical context to demonstrate possible difficulties with likelihood-, Bayesian-, and bootstrap-inference. Examples that demonstrate non-uniqueness are also presented. It is explained that they can be seen as cases with restrictions on the parameter space. Motivated by this the concept of a conditional fiducial model is introduced. This class of models includes the common case of iid samples from a one-parameter model investigated by Hannig (2013), the structural group models investigated by Fraser (1968), and also certain models discussed by Fisher (1973) in his final writing on the subject.

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Fiducial theory and optimal inference

Cited by 53 publications

References 49 publications

Group Bound: Confidence Intervals for Groups of Variables in Sparse High Dimensional Regression Without Assumptions on the Design

Group Bound: Confidence Intervals for Groups of Variables in Sparse High Dimensional Regression Without Assumptions on the Design

Frameworks for prior‐free posterior probabilistic inference

Conditional fiducial models

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