Searcharxiv⌕ Search

arXiv subjects

Russell J. Bowater

Publications and source records attributed to Russell J. Bowater.

13 recordsLinked to original sources
arXiv2025

Analogy making as the basis of statistical inference

Standard statistical theory has arguably proved to be unsuitable as a basis for constructing a satisfactory completely general framework for performing statistical inference. For example, frequentist theory has never come close to providing such a general inferential framework, which is not only attributable to the question surrounding the soundness of this theory, but also to its focus on attempting to address the problem of how to perform statistical inference only in certain special cases. Also, theories of inference that are grounded in the idea of deducing sample-based inferences about populations of interest from a given set of universally acceptable axioms, e.g. many theories that aim to justify Bayesian inference and theories of imprecise probability, suffer from the difficulty of finding such axioms that are weak enough to be widely acceptable, but strong enough to lead to methods of inference that can be regarded as being efficient. These observations justify the need to look for an alternative means by which statistical inference may be performed, and in particular, to explore the one that is offered by analogy making. What is presented here goes down this path. To be clear, this is done in a way that does not simply endorse the common use of analogy making as a supplementary means of understanding how statistical methods work, but formally develops analogy making as the foundation of a general framework for performing statistical inference. In the latter part of the paper, the use of this framework is illustrated by applying some of the most important analogies contained within it to a relatively simple but arguably still unresolved problem of statistical inference, which naturally leads to an original way being put forward of addressing issues that relate to Bartlett's and Lindley's paradoxes.

stat.OT↗
arXiv2024

Probabilistic inference when the population space is open

In using observed data to make inferences about a population quantity, it is commonly assumed that the sampling distribution from which the data were drawn belongs to a given parametric family of distributions, or at least, a given finite set of such families, i.e. the population space is assumed to be closed. Here, we address the problem of how to determine an appropriate post-data distribution for a given population quantity when such an assumption about the underlying sampling distribution is not made, i.e. when the population space is open. The strategy used to address this problem is based on the fact that even though, due to an open population space being non-measurable, we are not able to place a post-data distribution over all the sampling distributions contained in such a population space, it is possible to partition this type of space into a finite, countable or uncountable number of subsets such that a distribution can be placed over a variable that simply indicates which of these subsets contains the true sampling distribution. Moreover, it is argued that, by using sampling distributions that belong to a number of parametric families, it is possible to adequately and elegantly represent the sampling distributions that belong to each of the subsets of such a partition. Since a statistical model is conceived as being a model of a population space rather than a model of a sampling distribution, it is also argued that neither the type of models that are put forward nor the expression of pre-data knowledge via such models can be directly brought into question by the data. Finally, the case is made that, as well as not being required in the modelling process that is proposed, the standard practice of using P values to measure the absolute compatibility of an individual or family of sampling distributions with observed data is neither meaningful nor useful.

stat.ME↗
arXiv2023

The fiducial-Bayes fusion: A general theory of statistical inference

An overview is presented of a general theory of statistical inference that is referred to as the fiducial-Bayes fusion. This theory combines organic fiducial inference and Bayesian inference. The aim is that the reader is given a clear summary of the conceptual framework of the fiducial-Bayes fusion as well as pointers to further reading about its more technical aspects. Particular attention is paid to the issue of how much importance should be attached to the role of Bayesian inference within this framework. The appendix contains a substantive example of the application of the theory of the fiducial-Bayes fusion, which supplements various other examples of the application of this theory that are referenced in the paper.

stat.OT↗
arXiv2023

Sharp hypotheses and organic fiducial inference

A fundamental class of inferential problems are those characterised by there having been a substantial degree of pre-data (or prior) belief that the value of a model parameter $θ_j$ was equal or lay close to a specified value $θ^{*}_j$, which may, for example, be the value that indicates the absence of a treatment effect or the lack of correlation between two variables. This paper puts forward a generally applicable 'push-button' solution to problems of this type that circumvents the severe difficulties that arise when attempting to apply standard methods of inference, including the Bayesian method, to such problems. Usually the only input of major note that is required from the user in implementing this solution is the assignment of a pre-data or prior probability to the hypothesis that the parameter $θ_j$ lies in a narrow interval $[θ_{j0},θ_{j1}]$ that is assumed to contain the value of interest $θ^{*}_j$. On the other hand, the end result that is achieved by applying this method is, conveniently, a joint post-data distribution over all the parameters $θ_1,θ_2,\ldots,θ_k$ of the model concerned. The proposed method is constructed by naturally combining a simple Bayesian argument with an approach to inference called organic fiducial inference that was developed in a number of earlier papers. To begin with, the main theoretical arguments underlying this combined Bayesian and fiducial method are presented and discussed in detail. Various applications and useful extensions of this methodology are then outlined in the latter part of the paper. The examples that are considered are made relevant to the analysis of clinical trial data where appropriate.

stat.ME↗
arXiv2022

Physical, subjective and analogical probability

The aim of this paper is to show that the concept of probability is best understood by dividing this concept into two different types of probability, namely physical probability and analogical probability. Loosely speaking, a physical probability is a probability that applies to the outcomes of an experiment that have been judged as being equally likely on the basis of physical symmetry. Physical probabilities are arguably in some sense 'objective' and possess all the standard properties of the concept of probability. On the other hand, an analogical probability is defined by making an analogy between the uncertainty surrounding an event of interest and the uncertainty surrounding an event that has a physical probability. Analogical probabilities are undeniably subjective probabilities and are not obliged to have all the standard mathematical properties possessed by physical probabilities, e.g. they may not have the property of additivity or obey the standard definition of conditional probability. Nevertheless, analogical probabilities have extra properties, which are not possessed by physical probabilities, that assist in their direct elicitation, general derivation, comparison and justification. More specifically, these properties facilitate the application of analogical probability to real-world problems that can not be adequately resolved by using only physical probability, e.g. probabilistic inference about hypotheses on the basis of observed data. Careful definitions are given of the concepts that are introduced and, where appropriate, examples of the application of these concepts are presented for additional clarity.

stat.OT↗
arXiv2022

On a generalised form of subjective probability

This paper is motivated by the questions of how to give the concept of probability an adequate real-world meaning, and how to explain a certain type of phenomenon that can be found, for instance, in Ellsberg's paradox. It attempts to answer these questions by constructing an alternative theory to one that was proposed in earlier papers on the basis of various important criticisms that were raised against this earlier theory. The conceptual principles of the corresponding definition of probability are laid out and explained in detail. In particular, what is required to fully specify a probability distribution under this definition is not just the distribution function of the variable concerned, but also an assessment of the internal and/or the external strength of this function relative to other distribution functions of interest. This way of defining probability is applied to various examples and problems including, perhaps most notably, to a long-running controversy concerning the distinction between Bayesian and fiducial inference. The characteristics of this definition of probability are carefully evaluated in terms of the issues that it sets out to address.

stat.OT↗
arXiv2021

A revision to the theory of organic fiducial inference

A principle is modified that underlies the theory of organic fiducial inference as this theory was presented in an earlier paper. This modification, which is arguably a natural one to make, allows Bayesian inference to sometimes have a minor role within the theory in question and, as a consequence, allows more information from the data to be incorporated into the way a full conditional fiducial density is defined in certain cases. The new version of the principle concerned is applied to examples that were analysed previously using the older version of this principle.

stat.OT↗
arXiv2021

Integrated organic inference (IOI): A reconciliation of statistical paradigms

It is recognised that the Bayesian approach to inference can not adequately cope with all the types of pre-data beliefs about population quantities of interest that are commonly held in practice. In particular, it generally encounters difficulty when there is a lack of such beliefs over some or all the parameters of a model, or within certain partitions of the parameter space concerned. To address this issue, a fairly comprehensive theory of inference is put forward called integrated organic inference that is based on a fusion of Fisherian and Bayesian reasoning. Depending on the pre-data knowledge that is held about any given model parameter, inferences are made about the parameter conditional on all other parameters using one of three methods of inference, namely organic fiducial inference, bispatial inference and Bayesian inference. The full conditional post-data densities that result from doing this are then combined using a framework that allows a joint post-data density for all the parameters to be sensibly formed without requiring these full conditional densities to be compatible. Various examples of the application of this theory are presented. Finally, the theory is defended against possible criticisms partially in terms of what was previously defined as generalised subjective probability.

stat.ME↗
arXiv2021

Organic fiducial inference

A substantial generalisation is put forward of the theory of subjective fiducial inference as it was outlined in earlier papers. In particular, this theory is extended to deal with cases where the data are discrete or categorical rather than continuous, and cases where there was important pre-data knowledge about some or all of the model parameters. The system for directly expressing and then handling this pre-data knowledge, which is via what are referred to as global and local pre-data functions for the parameters concerned, is distinct from that which involves attempting to directly represent this knowledge in the form of a prior distribution function over these parameters, and then using Bayes' theorem. In this regard, the individual attributes of what are identified as three separate types of fiducial argument, namely the strong, moderate and weak fiducial arguments, form an integral part of the theory that is developed. Various practical examples of the application of this theory are presented, including examples involving binomial, Poisson and multinomial data. The fiducial distribution functions for the parameters of the models in these examples are interpreted in terms of a generalised definition of subjective probability that was set out previously.

stat.OT↗
arXiv2021

Multivariate subjective fiducial inference

The aim of this paper is to firmly establish subjective fiducial inference as a rival to the more conventional schools of statistical inference, and to show that Fisher's intuition concerning the importance of the fiducial argument was correct. In this regard, methodology outlined in an earlier paper is modified, enhanced and extended to deal with general inferential problems in which various parameters are unknown. As a key part of what is put forward, the joint fiducial distribution of all the parameters of a given model is determined on the basis of the full conditional fiducial distributions of these parameters by using an analytical approach or a Gibbs sampling method, the latter of which does not require these conditional distributions to be compatible. Although the resulting theory is classified as being "subjective", this is essentially due to the argument that all probability statements made about fixed but unknown parameters must be inherently subjective. In particular, it is systematically argued that, in general, there is no need to place a great emphasis on the difference between the fiducial probabilities derived by using this theory of inference and objective probabilities. Some important examples of the application of this theory are presented.

math.ST↗
arXiv2021

Sharp hypotheses and bispatial inference

A fundamental class of inferential problems are those characterised by there having been a substantial degree of pre-data (or prior) belief that the value of a model parameter was equal or lay close to a specified value, which may, for example, be the value that indicates the absence of an effect. Standard ways of tackling problems of this type, including the Bayesian method, are often highly inadequate in practice. To address this issue, an inferential framework called bispatial inference is put forward, which can be viewed as both a generalisation and radical reinterpretation of existing approaches to inference that are based on P values. It is shown that to obtain an appropriate post-data density function for a given parameter, it is often convenient to combine a special type of bispatial inference, which is constructed around one-sided P values, with a previously outlined form of fiducial inference. Finally, by using what are called post-data opinion curves, this bispatial-fiducial theory is naturally extended to deal with the general scenario in which any number of parameters may be unknown. The application of the theory is illustrated in various examples, which are especially relevant to the analysis of clinical trial data.

stat.OT↗
arXiv2018

Bayesian, classical and hybrid methods of inference when one parameter value is special

This paper considers the problem of making statistical inferences about a parameter when a narrow interval centred at a given value of the parameter is considered special, which is interpreted as meaning that there is a substantial degree of prior belief that the true value of the parameter lies in this interval. A clear justification of the practical importance of this problem is provided. The main difficulty with the standard Bayesian solution to this problem is discussed and, as a result, a pseudo-Bayesian solution is put forward based on determining lower limits for the posterior probability of the parameter lying in the special interval by means of a sensitivity analysis. Since it is not assumed that prior beliefs necessarily need to be expressed in terms of prior probabilities, nor that post-data probabilities must be Bayesian posterior probabilities, hybrid methods of inference are also proposed that are based on specific ways of measuring and interpreting the classical concept of significance. The various methods that are outlined are compared and contrasted at both a foundational level, and from a practical viewpoint by applying them to real data from meta-analyses that appeared in a well-known medical article.

math.ST↗