Improper Priors Are Not Improper

Taraldsen, Gunnar; Lindqvist, Bo Henry

doi:10.1198/tast.2010.09116

Cited by 31 publications

(46 citation statements)

References 8 publications

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“…Consider the case where the loss of an action a ∈ Ω A is of the form l = γ(θ, a) corresponding to a statistical model {P θ X | θ ∈ Ω Θ }. It is here assumed that the model parameter Θ is a σ-finite random quantity and this and all other random quantities are defined based on the underlying conditional probability space (Ω, E, P) as explained by Taraldsen and Lindqvist (2010). This means in particular that P θ X (B) = P(X ∈ B | Θ = θ), and X : Ω → Ω X , Θ : Ω → Ω Θ are measurable functions.…”

Section: Optimal Inferencementioning

confidence: 99%

Fiducial theory and optimal inference

Taraldsen¹,

Lindqvist²

2013

Ann. Statist.

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It is shown that the fiducial distribution in a group model, or more generally a quasigroup model, determines the optimal equivariant frequentist inference procedures. The proof does not rely on existence of invariant measures, and generalizes results corresponding to the choice of the right Haar measure as a Bayesian prior. Classical and more recent examples show that fiducial arguments can be used to give good candidates for exact or approximate confidence distributions. It is here suggested that the fiducial algorithm can be considered as an alternative to the Bayesian algorithm for the construction of good frequentist inference procedures more generally.Comment: Published in at http://dx.doi.org/10.1214/13-AOS1083 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

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Section: Optimal Inferencementioning

confidence: 99%

Fiducial theory and optimal inference

Taraldsen¹,

Lindqvist²

2013

Ann. Statist.

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“…In Eq. (14), p(x) is formally equal to 1 for the flat prior, and therefore is not defined as a probability distribution on x (see Taraldsen and Lindqvist, 2010). This illustrates the fact that the flat prior cannot be considered as the limit case of inference with π M and that limiting arguments are not valid.…”

Section: An Approach By a Limiting Argumentmentioning

confidence: 99%

On the Flatland paradox and limiting arguments

Druilhet

2017

Communications in Statistics - Theory and Methods

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We revisit the flatland paradox proposed by Stone (1976) which is an example of non-conglomerability. The main novelty in the analysis of the paradox is to consider marginal vs conditional models rather than proper vs improper priors. We show that in the first model a prior distribution should be considered as a probability measure whereas, in the second one, a prior distribution should be considered in the projective space of measure. This induce two different kinds of limiting arguments which are useful to understand the paradox. We also show that the choice of a flat prior is not adapted to the structure of the parameter space and we consider an improper prior based on reference priors with nuisance parameters for which the Bayesian analysis matches the intuitive reasoning.

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“…Our point of departure will be the paper by Taraldsen and Lindqvist (2010) which has a slightly different view than the above references. The idea is here simply to allow infinite probabilities in Kolmogorov's axioms.…”

Section: Introductionmentioning

confidence: 99%

“…Formally, this condition is the σ-finiteness of the random quantity that is conditioned on, here x. Details will be given in Section 2 which reviews the theoretical results of Taraldsen and Lindqvist (2010).…”

Section: Introductionmentioning

confidence: 99%

On the proper treatment of improper distributions

Lindqvist

Taraldsen

2018

Journal of Statistical Planning and Inference

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The axiomatic foundation of probability theory presented by Kolmogorov has been the basis of modern theory for probability and statistics. In certain applications it is, however, necessary or convenient to allow improper (unbounded) distributions, which is often done without a theoretical foundation. The paper reviews a recent theory which includes improper distributions, and which is related to Renyi's theory of conditional probability spaces. It is in particular demonstrated how the theory leads to simple explanations of apparent paradoxes known from the Bayesian literature. Several examples from statistical practice with improper distributions are discussed in light of the given theoretical results, which also include a recent theory of convergence of proper distributions to improper ones.Comment: Journal of Statistical Planning and Inference, 201

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Improper Priors Are Not Improper

Cited by 31 publications

References 8 publications

Fiducial theory and optimal inference

Fiducial theory and optimal inference

On the Flatland paradox and limiting arguments

On the proper treatment of improper distributions

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