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This paper seeks to provide a thorough account of the ubiquitous nature of the Bayesian paradigm in modern statistics, data science and artificial intelligence. Once maligned, on the one hand by those who philosophically hated the very idea…

Classical countably additive real-valued probabilities come at a philosophical cost: in many infinite situations, they assign the same probability value -- namely, zero -- to cases that are impossible as well as to cases that are possible.…

Decision-makers abhor uncertainty, and it is certainly true that the less there is of it the better. However, recognizing that uncertainty is part of the equation, particularly for deciding on environmental policy, is a prerequisite for…

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…

Many published research results are false, and controversy continues over the roles of replication and publication policy in improving the reliability of research. Addressing these problems is frustrated by the lack of a formal framework…

A novel way of defining limits in classical statistics is proposed. This is a natural extension of the original Neyman's method, and has the desirable property that only information relevant to the problem is used in making statistical…

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…

We derive inferential procedures for large sample sizes that remain valid under data-dependent significance levels (so-called "post-hoc valid inference"). Classical statistical tools require that the significance level -- the "type-I error"…

To estimate causal effects from observational data, an applied researcher must impose beliefs. The instrumental variables exclusion restriction, for example, represents the belief that the instrument has no direct effect on the outcome of…

Statistical schools-such as Bayesianism and Frequentism-are often presented as competing frameworks, each claiming technical rigour and superiority. Frequentism emphasizes objective inferences through repeated sampling, while Bayesianism…

Our interest is whether two binomial parameters differ, which parameter is larger, and by how much. This apparently simple problem was addressed by Fisher in the 1930's, and has been the subject of many review papers since then. Yet there…

There currently exists a gap between the theories proposed by the probability and uncertainty and the needs of Artificial Intelligence research. These theories primarily address the needs of expert systems, using knowledge structures which…

Statistical science (as opposed to mathematical statistics) involves far more than probability theory, for it requires realistic causal models of data generators - even for purely descriptive goals. Statistical decision theory requires more…

In this paper, we propose standard statistical tools as a solution to commonly highlighted problems in the explainability literature. Indeed, leveraging statistical estimators allows for a proper definition of explanations, enabling…

This paper offers a comprehensive introduction to Bayesian inference, combining historical context, theoretical foundations, and core analytical examples. Beginning with Bayes' theorem and the philosophical distinctions between Bayesian and…

Model inadequacy and measurement uncertainty are two of the most confounding aspects of inference and prediction in quantitative sciences. The process of scientific inference (the inverse problem) and prediction (the forward problem)…

Predictive inference is a fundamental task in statistics, traditionally addressed using parametric assumptions about the data distribution and detailed analyses of how models learn from data. In recent years, conformal prediction has…

The replication crisis has prompted many to call for statistical reform within the psychological sciences. Here we examine issues within Frequentist statistics that may have led to the replication crisis, and we examine the…

Exploration has been a crucial part of reinforcement learning, yet several important questions concerning exploration efficiency are still not answered satisfactorily by existing analytical frameworks. These questions include exploration…

A central focus of data science is the transformation of empirical evidence into knowledge. As such, the key insights and scientific attitudes of deep thinkers like Fisher, Popper, and Tukey are expected to inspire exciting new advances in…