Statistical Inference and the Plethora of Probability Paradigms: A Principled Pluralism (original) (raw)

Mutual Influence between Different Views of Probability and Statistical Inference

PARADIGMA

In this paper, we analyse the various meanings of probability and its different applications, and we focus especially on the classical, the frequentist, and the subjectivist view. We describe the different problems of how probability can be measured in each of the approaches, and how each of them can be well justified by a mathematical theory. We analyse the foundations of probability, where the scientific analysis of the theory that allows for a frequentist interpretation leads to unsolvable problems. Kolmogorov’s axiomatic theory does not suffice to establish statistical inference without further definitions and principles. Finally, we show how statistical inference essentially determines the meaning of probability and a shift emerges from purely objectivist views to a complementary conception of probability with frequentist and subjectivist constituents. For didactical purpose, the result of the present analyses explains basic problems of teaching, originating from a biased focus...

How Probabilities Reflect Evidence

Philosophical Perspectives, 2005

Many philosophers think of Bayesianism as a theory of practical rationality. This is not at all surprising given that the view's most striking successes have come in decision theory. Ramsey (1931), Savage (1972), and De Finetti (1964) showed how to interpret subjective degrees of belief in terms of betting behavior, and how to derive the central probabilistic requirement of coherence from reflections on the nature of rational choice. This focus on decision-making can obscure the fact that Bayesianism is also an epistemology. Indeed, the great statistician Harold Jeffries (1939), who did more than anyone else to further Bayesian methods, paid rather little heed to the work of Ramsey, de Finetti, and Savage. Jeffries, and those who followed him, saw Bayesianism as a theory of inductive evidence, whose primary role was not to help people make wise choices, but to facilitate sound scientific reasoning. 1 This paper seeks to promote a broadly Bayesian approach to epistemology by showing how certain central questions about the nature of evidence can be addressed using the apparatus of subjective probability theory. Epistemic Bayesianism, as understood here, is the view that evidential relationships are best represented probabilistically. It has three central components: Evidential Probability. At any time t, a rational believer's opinions can be faithfully modeled by a family of probability functions C t , hereafter called her credal state, 2 the members of which accurately reflect her total evidence at t. Learning as Bayesian Updating. Learning experiences can be modeled as shifts from one credal state to another that proceed in accordance with Bayes's Rule. Confirmational Relativity. A wide range of questions about evidential relationships can be answered on the basis of information about structural features credal states. The first of these three theses is most fundamental. Much of what Bayesians say about learning and confirmation only makes sense if probabilities in credal

The Role of Probability for Understanding Statistical Inference

Bridging the Gap: Empowering and Educating Today’s Learners in Statistics. Proceedings of the Eleventh International Conference on Teaching Statistics, 2022

Probability is the basis for intelligent actions and decisions in the face of uncertainty. That includes statistical inference as well as considerations of reliability, risk, and decision-making. Curricula have reduced approaches with respect to the nature of probability. With easy access to computer technology, simulation has become the predominant approach to teaching. Although simulation is an effective method to replace complicated mathematics, it reduces concepts to their frequentist part. This culminates in an approach to informal inference that makes probability and conditional probability redundant. However, the relevant properties of statistical inference require a comprehensive conception of probability to be shaped in the individual's cognitive system.

Evidential probability and objective Bayesian epistemology

Philosophy of statistics, handbook of …, 2011

Wheeler, Gregory and Williamson, Jon (2011) Evidential probability and objective Bayesian epistemology. In: Philosophy of statistics. Handbook of the Philosophy of Science . Elsevier, pp. 307-331. ... The full text of this publication is not available from this repository.

Bayesian Confirmation by Uncertain Evidence: A Reply to Huber [2005]

The British Journal for the Philosophy of Science, 2008

Bayesian epistemology postulates a probabilistic analysis of many sorts of ordinary and scientific reasoning. Huber ([2005]) has provided a novel criticism of Bayesianism, whose core argument involves a challenging issue: confirmation by uncertain evidence. In this paper, we argue that under a properly defined Bayesian account of confirmation by uncertain evidence, Huber's criticism fails. By contrast, our discussion will highlight what we take as some new and appealing features of Bayesian confirmation theory.

Likelihoodism, Bayesianism, and relational confirmation

Synthese, 2007

Likelihoodists and Bayesians seem to have a fundamental disagreement about the proper probabilistic explication of relational (or contrastive) conceptions of evidential support (or confirmation). In this paper, I will survey some recent arguments and results in this area, with an eye toward pinpointing the nexus of the dispute. This will lead, first, to an important shift in the way the debate has been couched, and, second, to an alternative explication of relational support, which is in some sense a “middle way” between Likelihoodism and Bayesianism. In the process, I will propose some new work for an old probability puzzle: the “Monty Hall” problem.

In defence of objective Bayesianism

How strongly should you believe the various propositions that you can express? That is the key question facing Bayesian epistemology. Subjective Bayesians hold that it is largely (though not entirely) up to the agent as to which degrees of belief to adopt. Objective Bayesians, on the other hand, maintain that appropriate degrees of belief are largely (though not entirely) determined by the agent's evidence. This book states and defends a version of objective Bayesian epistemology. According to this version, objective Bayesianism is characterized by three norms: · Probability - degrees of belief should be probabilities · Calibration - they should be calibrated with evidence · Equivocation - they should otherwise equivocate between basic outcomes Objective Bayesianism has been challenged on a number of different fronts. For example, some claim it is poorly motivated, or fails to handle qualitative evidence, or yields counter-intuitive degrees of belief after updating, or suffers from a failure to learn from experience. It has also been accused of being computationally intractable, susceptible to paradox, language dependent, and of not being objective enough. Especially suitable for graduates or researchers in philosophy of science, foundations of statistics and artificial intelligence, the book argues that these criticisms can be met and that objective Bayesianism is a promising theory with an exciting agenda for further research.

Bayesianism II: Applications and Criticisms

Philosophy Compass, 2011

In the first paper, I discussed the basic claims of Bayesianism (that degrees of belief are important, that they obey the axioms of probability theory, and that they are rationally updated by either standard or Jeffrey conditionalization) and the arguments that are often used to support them. In this paper, I will discuss some applications these ideas have had in confirmation theory, epistemology, and statistics, and criticisms of these applications.

Balkanization and Unification of Probabilistic Inferences

Online Submission, 2005

Many research-related classes in social sciences present probability as a unified approach based upon mathematical axioms, but neglect the diversity of various probability theories and their associated philosophical assumptions. Although currently the dominant statistical and probabilistic approach is the Fisherian tradition, the use of Fisherian significance testing of the null hypothesis and its probabilistic inference has been an ongoing debate. This paper attempts to explore the richness and complexity of the ideas of probability with the emphasis on the relationships between Fisherian and other probability theories. First, it clarifies the differences between Fisher and Jeffreys and explains the background history relating to Fisher's quest for certainty. Second, it explains the differences between Fisher and Pearson and explains the limitations of the Fisherian approach. In addition, it argues that although Fisher criticized the Bayesian school for its alleged lack of objectivity, Fisher's quest for certainty is driven by his subjective faith in experimental methods, eugenics and Darwinism. Last, it will briefly introduce the synthesized approaches by Berger and Pawitan, respectively, as a possible remedy.

Why Bayesianism? A Primer on a Probabilistic Philosophy of Science

Several attempts have been made both in the present and past to impose some a priori desiderata on statistical/inductive inference (Fitleson, 1999, Jeffreys, 1961, Zellner, 1996, Jaynes, 2003, Lele, 2004). Bringing this literature on desiderata to the fore, I argue that these attempts to understand inference could be controversial.