Download What is fiducial inference

What is fiducial inference
- and why?
Gunnar Taraldsen
Statistics seminar at NTNU, October 17th 2016
Forslag til norske ord
•Fidus inferens = troverdig inferens
•Fidus fordeling = troverdighet
Abstract
This seminar contribution – with active discussion with the audience presents some of the historic background, the original example
presented by Fisher, and recent developments and trends as seen by
members of the BFF group (Bayes-Fiducial-Frequentist = Best-FriendsForever) and recent and upcoming JASA publications.
BFF: Min-ge Xie et al
http://stat.rutgers.edu/bff2016-program
BFF = Best-Friends-Forever = Bayes-Fiducial-Frequentist
What is fiducial inference?
• Fisher’s biggest blunder. Essentially dead. (Pederson,
1978)
• A big hit in the 21th century (Efron, 1998)
• A tool for fusion learning (Regina Liu, 2015)
• Data dependent priors ++(Hannig, 2016)
• Inferential models (Ryan Martin, 2015)
• Dempster (1967)-Shafer (1976) calculus and belief and
plausibility functions
• Confidence: Birnbaum (1961), Schweder-Hjort (2016).
Why fiducial inference?
• Fisher idea: A mode of inference that produces a posterior epistemic
distribution for cases without prior epistemic probability.
• Introduced by Fisher in 1930 and his arguments is then seemingly the
first formulation of what a confidence distribution is.
• He argued later that it should not be a confidence distribution (1973).
• An alternative to Bayesian and frequentist inference.
• Fraser introduced the term structural inference – and this gives a
very promissing idea for a general theory. This will be presented at
the end, but first history and some alternatives ….
Biologist and statistician Ronald Fisher
Fiducial inference is important in the history
of statistics since its development led to the
parallel development of concepts and tools
in theoretical statistics that are widely used.
It was invented in 1930 by Fisher in the
paper “Inverse probability”. The paper
considers the correlation coefficient in the
bivariate normal and the argument gives a
confidence distribution as the fiducial.
Neyman (1937) introduced the idea of
"confidence" in his paper on confidence
intervals: the frequentist property.
Arthur P. Dempster at the Workshop on Theory of
Belief Functions (Brest, 1 April 2010).
Zabell, S. L. (Aug 1992). "R. A. Fisher and Fiducial
Argument". Statistical Science. 7 (3): 369–387
Fisher admitted that "fiducial inference" had problems. Fisher
wrote to George A. Barnard that he was "not clear in the
head" about one problem on fiducial inference, and, also
writing to Barnard, Fisher complained that his theory seemed
to have only "an asymptotic approach to intelligibility". Later
Fisher confessed that "I don't understand yet what fiducial
probability does. We shall have to live with it a long time
before we know what it's doing for us. But it should not be
ignored just because we don't yet have a clear interpretation".
Hannig et al (JASA 2016, accepted): Generalized
Fiducial Inference: A Review and New Results
Hannig et al (2016): 1
The idea behind GFD is very similar to the idea
behind the likelihood function: what is the chance
of observing my data if any given parameter was
true. The added value of GFD is that it provides
likelihood function with an appropriate Jacobian
obtaining a proper probability distribution on the
parameter space.
Hannig et al (2016): 2
GFD does not presume that the parameter is
random. Instead it should be viewed as a distribution
estimator (rather than a point or interval estimator)
of the fixed true parameter. To validate this
distribution estimator in a specific example we then
typically demonstrate good small sample
performance by simulation and prove good large
sample properties by asymptotic theorems.
Hannig et al (2016): 3
From a Bayesian point of view, Bayes theorem
updates the distribution of U after the data are
observed. However, when no prior information is
present, changing the distribution of U only by
restricting it to the set “there is at least one θ
solving the equation y = G(U,θ)” seems to us as a
reasonable choice
Hannig et al (2016)
Hannig et al (2016)
Schweder-Hjort (2016) book
• Defines confidence inference and develops its basic
theory
• Includes many worked examples of/with confidence
inference, with emphasis on the confidence curve as a
good format of reporting
• Presents methods for meta-analysis and other forms
of combining information, which goes beyond present
day theory based on approximate normality
Taraldsen-Lindqvist (2013): Optimal rule
Fiducial inference and structure
• The structure of the fiducial equation is included.
• Information in the fiducial equation?
• Additional parameters a’la Fraser.
• Copulae, SEM, and other structural models can be studied to clearify
this point of view:
• The main difference between Bayesian and fiducial inference is
that the later takes the structure of the fiducial model into account
– and not just the resulting statistical model. A fiducial model is
more than a Bayesian model and generalizes Bayesian inference.
THE END