A Bayesian solution to the conflict of narrowness and precision in direct inference (original) (raw)
Related papers
Discussion of “Is Bayes Posterior just Quick and Dirty Confidence?” by D. A. S. Fraser
Statistical Science, 2011
We congratulate Professor Fraser for this very engaging article. It gives us an opportunity to gaze at the past and future of Bayes and confidence. It is well known that a Bayes posterior can only provide credible intervals and has no assurance of frequentist coverage (known as confidence). Professor Fraser's article provides a detailed and insightful exploration into the root of this issue. It turns out that the Bayes posterior is exactly a confidence in the linear case (a mathematics coincidence), and Professor Fraser's insightful and far-reaching examples demonstrate how the departure from linearity induces the departure of a posterior, in a proportionate way, from being a confidence. Of course, Bayesian inference is not bounded by frequestist criteria or geared to provide confidence statements, even though in some applications researchers have treated the Bayes credible intervals as confidence intervals on asymptotic grounds. It is debatable whether this departure of Bayesian inference from confidence should be a concern or not. But, nevertheless, the article provides us a powerful exploration and demonstration which can help us better comprehend the two statistical philosophies and the 250-year debate between Bayesians and frequentists. In the midst of the 250-year debate, Fisher's "fiducial distribution" played a prominent role, which, however, is now referred to as the "biggest blunder" of the father of modern statistical inference [1].
Bayesian Versus Frequentist Inference
Bayesian Evaluation of Informative Hypotheses, 2008
Throughout this book, the topic of order-restricted inference is dealt with almost exclusively from a Bayesian perspective. Some readers may wonder why the other main school for statistical inference -frequentist inferencehas received so little attention here. Isn't it true that in the field of psychology, almost all inference is frequentist inference?
Psychological Review, 1999
Bayesian reasoning can be improved by representing information in frequency formats rather than in probabilities. This thesis opens up applications in medicine, law, statistics education, and other fi elds. The benefi cial effect is no longer in dispute, but rather its cause and its boundary conditions. C. Lewis and G. argued that the effect of frequency formats is due to "joint statements" rather than to "frequency statements." However, they overlooked the fact that our thesis is about frequency formats, not just any kind of frequency statements. We show that joint statements alone cannot account for the effect. B. A. proposed a boundary condition under which the benefi cial effect is reduced. In a reanalysis of our original data, we found this reduction for the problem they used but not for any other problem. We conclude by summarizing results indicating that teaching frequency representations fosters insight into Bayesian reasoning.
Admissibility Troubles for Bayesian Direct Inference Principles
2018
Direct inferences identify certain probabilistic credences or confirmation-function-likelihoods with values of objective chances or relative frequencies. The best known version of a direct inference principle is David Lewis's Principal Principle. Certain kinds of statements undermine direct inferences. Lewis calls such statements inadmissible. We show that on any Bayesian account of direct inference several kinds of intuitively innocent statements turn out to be inadmissible. This may pose a significant challenge to Bayesian accounts of direct inference. We suggest some ways in which these challenges may be addressed.
Objections to Bayesian statistics
Bayesian inference is one of the more controversial approaches to statistics. The fundamental objections to Bayesian methods are twofold: on one hand, Bayesian methods are presented as an automatic inference engine, and this raises suspicion in anyone with applied experience. The second objection to Bayes comes from the opposite direction and addresses the subjective strand of Bayesian inference. This article presents a series of objections to Bayesian inference, written in the voice of a hypothetical anti-Bayesian statistician. The article is intended to elicit elaborations and extensions of these and other arguments from non-Bayesians and responses from Bayesians who might have different perspectives on these issues.
A Note on the Subtleties of Bayesian Inference
1997
The Bayesian approach plays a central role in economics, decision theory and game theory. Bayesianism is usually characterized as the philosophical view that probability can be interpreted subjectively and that the rational way to assimilate information into one’s structure of beliefs is by a process called “conditionalization”. Thus Bayesianism has a static part and a dynamic part. The former asserts that a coherent set of beliefs can be represented by a probability function over sentences or events (see De Finetti, 1937, Ramsey, 1931, Savage, 1954, and, for a recent survey, Hammond, in press). The dynamic part of Bayesian theory asserts that rational change of beliefs, in response to new evidence, goes by conditionalization: if the individual starts with a subjective probability distribution P o and observes E, where P o (E) > 0, then her new beliefs should be given by the probability distribution P n defined as follows: for every event A, P n (A) = P A E P E o o ( ) ( ) ∩ . Th...
How to Improve Bayesian Reasoning: Comment on Gigerenzer and Hoffrage (1995)
G. Gigerenzer and U. Hoffrage (1995) claimed that Bayesian inference problems, which have been notoriously difficult for laypeople to solve using base rates, hit rates, and false-alarm rates, become computationally simpler when information is presented with frequencies based on natural sampling. They made an evolutionary argument for the improved performance. The authors of the present article show that performance can improve with either probabilities or frequencies, depending on the rareness of the events and the type of information presented. When events are rare, probabilities are more difficult to understand than frequencies (i.e., 5 out of 1,000 vs. .005.). Furthermore, when the information is presented as joint and marginal events, nested sets become more apparent. Frequencies based on natural sampling have these desirable properties. The authors agree with Gigerenzer and Hoffrage that frequencies can improve Bayesian reasoning, but they attribute that improvement to the use of mental models that involve elements of nested sets.
Standards for Modest Bayesian Credences
Gordon Belot argues that Bayesian theory is epistemologically immodest. In response, we show that the topological conditions that underpin his criticisms of asymptotic Bayes-ian conditioning are self-defeating. They require extreme a priori credences regarding, for example, the limiting behavior of observed relative frequencies. We offer a different explication of Bayesian modesty using a goal of consensus: rival scientific opinions should be responsive to new facts as a way to resolve their disputes. Also we address Adam Elga's rebuttal to Belot's analysis, which focuses attention on the role that the assumption of countable additivity plays in Belot's criticisms. 1. Introduction. Consider the following compound result about asymp-totic statistical inference. A community of Bayesian investigators who begin an investigation with conflicting opinions about a common family of statistical hypotheses use shared evidence to achieve a consensus about which hypothesis is the true one. Specifically, suppose the investigators agree on a partition of statistical hypotheses and share observations of an increasing sequence of random samples with respect to whichever is the true statistical hypothesis from this partition. 1 Then, under various combinations of formal conditions that we review in this essay, ex ante (i.e., before accepting the new evidence) it is practically certain that each of the investigators' conditional probabilities approach 1 for the one true hypothesis in the partition. The result is compound: individual investigators achieve asymptotic certainty about the unknown, true statistical hypothesis. Second, the shared ev
Statistical Inference and the Plethora of Probability Paradigms: A Principled Pluralism
The major competing statistical paradigms share a common remarkable but unremarked thread: in many of their inferential applications, different probability interpretations are combined. How this plays out in different theories of inference depends on the type of question asked. We distinguish four question types: confirmation, evidence, decision, and prediction. We show that Bayesian confirmation theory mixes what are intuitively “subjective” and “objective” interpretations of probability, whereas the likelihood-based account of evidence melds three conceptions of what constitutes an “objective” probability