arXiv subjects
Publications and source records attributed to Gunnar Taraldsen.
The philosophical foundations of statistics involve issues in theoretical statistics, such as goals and methods to meet these goals, and interpretation of the meaning of inference using statistics. They are related to the philosophy of science and to the philosophy of probability. We review the core and partly interrelated themes and place them in context.
Conditional Monte Carlo refers to sampling from the conditional distribution of a random vector X given the value T(X) = t for a function T(X). Classical conditional Monte Carlo methods were designed for estimating conditional expectations of functions of X by sampling from unconditional distributions obtained by certain weighting schemes. The basic ingredients were the use of importance sampling and change of variables. In the present paper we reformulate the problem by introducing an artificial parametric model, representing the conditional distribution of X given T(X)=t within this new model. The key is to provide the parameter of the artificial model by a distribution. The approach is illustrated by several examples, which are particularly chosen to illustrate conditional sampling in cases where such sampling is not straightforward. A simulation study and an application to goodness-of-fit testing of real data are also given.
The fiducial coincides with the posterior in a group model equipped with the right Haar prior. This result is here generalized. For this the underlying probability space of Kolmogorov is replaced by a $σ$-finite measure space and fiducial theory is presented within this frame. Examples are presented that demonstrate that this also gives good alternatives to existing Bayesian sampling methods. It is proved that the results provided here for fiducial models imply that the theory of invariant measures for groups cannot be generalized directly to loops: There exist a smooth one-dimensional loop where an invariant measure does not exist. Keywords: Conditional sampling, Improper prior, Haar prior, Sufficient statistic, Quasi-group
The purpose of this paper is to present a mathematical theory that can be used as a foundation for statistics that include improper priors. This theory includes improper laws in the initial axioms and has in particular Bayes theorem as a consequence. Another consequence is that some of the usual calculation rules are modified. This is important in relation to common statistical practice which usually include improper priors, but tends to use unaltered calculation rules. In some cases the results are valid, but in other cases inconsistencies may appear. The famous marginalization paradoxes exemplify this latter case. An alternative mathematical theory for the foundations of statistics can be formulated in terms of conditional probability spaces. In this case the appearance of improper laws is a consequence of the theory. It is proved here that the resulting mathematical structures for the two theories are equivalent. The conclusion is that the choice of the first or the second formulation for the initial axioms can be considered a matter of personal preference. Readers that initially have concerns regarding improper priors can possibly be more open toward a formulation of the initial axioms in terms of conditional probabilities. The interpretation of an improper law is given by the corresponding conditional probabilities. Keywords: Axioms of statistics, Conditional probability space, Improper prior, Projective space
Symmetry is key in classical and modern physics. A striking example is conservation of energy as a consequence of time-shift invariance from Noether's theorem. Symmetry is likewise a key element in statistics, which, as also physics, provide models for real world phenomena. Sufficiency, conditionality, and invariance are examples of basic principles. Galili and Meilijson (2016) and Mandel (2020) illustrate the first two principles very nicely by considering the scaled uniform model. We illustrate the third principle by providing further results which give optimal inference for the scaled uniform by symmetry considerations. The proofs are simplified by relying on fiducial arguments as initiated by Fisher (1930). Keywords: Data generating equation; Optimal equivariant estimate; Scale family; Conditionality principle; Minimal sufficient; Uniform distribution;
The theory of random sets is demonstrated to prove useful for the theory of random operators. A random operator is here defined by requiring the graph to be a random set. It is proved that the spectrum and the set of eigenvalues of random operators are random sets. These results seem to be a novelty even in the case of random bounded operators. The main technical tools are given by the measurable selection theorem, the measurable projection theorem, and a characterisation of the spectrum by approximate eigenvalues of the operator and the adjoint operator. A discussion of some of the existing definitions of the concept of a random operator is included at the end of the paper. Keywords: Random operators; Set-valued functions; General topics in linear spectral theory; Random operators and equations; Stochastic integrals; Disordered systems
In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional probability. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. This note introduces a general concept of conditional probability in Renyi spaces. Keywords: Measure theory; conditional probability space; conditional expectation
In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional probability. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. We introduce a general concept of conditional probability in Renyi spaces. In this theory improper priors are allowed, and the resulting posteriors can also be improper.
The Doob-Dynkin Lemma gives conditions on two functions $X$ and $Y$ that ensure existence of a function $ϕ$ so that $X = ϕ \circ Y$. This communication proves different versions of the Doob-Dynkin Lemma, and shows how it is related to optimal statistical learning algorithms. Keywords and phrases: Improper prior, Descriptive set theory, Conditional Monte Carlo, Fiducial, Machine learning, Complex data.
In 1933 Kolmogorov constructed a general theory that defines the modern concept of conditional expectation. In 1955 Renyi fomulated a new axiomatic theory for probability motivated by the need to include unbounded measures. We introduce a general concept of conditional expectation in Renyi spaces. In this theory improper priors are allowed, and the resulting posterior can also be improper. In 1965 Lindley published his classic text on Bayesian statistics using the theory of Renyi, but retracted this idea in 1973 due to the appearance of marginalization paradoxes presented by Dawid, Stone, and Zidek. The paradoxes are investigated, and the seemingly conflicting results are explained. The theory of Renyi can hence be used as an axiomatic basis for statistics that allows use of unbounded priors. Keywords: Haldane's prior; Poisson intensity; Marginalization paradox; Measure theory; conditional probability space; axioms for statistics; conditioning on a sigma field; improper prior
We prove universal compact-measurability of the intersection of a compact-measurable Souslin family of closed-valued multifunctions. This generalizes previous results on intersections of measurable multifunctions. We introduce the unique maximal part of a multifunction which is defined on the quotient given by an equivalence relation. Measurability of this part of a multifunction is proven in a special case. We show how these results apply to the spectral theory of measurable families of closed linear operators.
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.
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.
An image is here defined to be a set which is either open or closed and an image transformation is structure preserving in the following sense: It corresponds to an algebra homomorphism for each singly generated algebra. The results extend parts of results of J.F. Aarnes on quasi-measures, -states, -homomorphisms, and image-transformations from the setting compact Hausdorff spaces to locally compact Hausdorff spaces.
In 1991 J.F. Aarnes introduced the concept of quasi-measures in a compact topological space $Ω$ and established the connection between quasi-states on $C (Ω)$ and quasi-measures in $Ω$. This work solved the linearity problem of quasi-states on $C^*$-algebras formulated by R.V. Kadison in 1965. The answer is that a quasi-state need not be linear, so a quasi-state need not be a state. We introduce nonlinear measures in a space $Ω$ which is a generalization of a measurable space. In this more general setting we are still able to define integration and establish a representation theorem for the corresponding functionals. A probabilistic language is choosen since we feel that the subject should be of some interest to probabilists. In particular we point out that the theory allows for incompatible stochastic variables. The need for incompatible variables is well known in quantum mechanics, but the need seems natural also in other contexts as we try to explain by a questionary example. Keywords and phrases: Epistemic probability, Integration with respect to mea- sures and other set functions, Banach algebras of continuous functions, Set func- tions and measures on topological spaces, States, Logical foundations of quantum mechanics.
The fiducial argument of Fisher (1973) has been described as his biggest blunder, but the recent review of Hannig et al. (2016) demonstrates the current and increasing interest in this brilliant idea. This short note analyses an example introduced by Seidenfeld (1992) where the fiducial distribution is restricted to a string. Keywords and phrases: Bayesian and fiducial inference, Restrictions on parameters, Uncertainty quantification, Epistemic probability, Statistics on a manifold.
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.