Reasoning to a Foregone Conclusion
Abstract
When can a Bayesian select an hypothesis H and design an experiment (or a sequence of experiments) to make certain that, given the experimental outcome(s), the posterior probability of H will be greater than its prior probability? We discuss an elementary result that establishes sufficient conditions under which this reasoning to a foregone conclusion cannot occur. We illustrate how when the sufficient conditions fail, because probability is finitely but not countably additive, it may be that a Bayesian can design an experiment to lead his/her posterior probability into a foregone conclusion. The problem has a decision theoretic version in which a Bayesian might rationally pay not to see the outcome of certain cost-free experiments, which we discuss from several perspectives. Also, we relate this issue in Bayesian hypothesis testing to various concerns about “optional stopping”.
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- "Recently, several researchers have argued for the use of Bayesian statistics in psychological science (for recent discussions of Bayesian methods, see Gallistel, 2009; Kruschke, 2010; Masson, 2011; Rouder et al., 2009; Wagenmakers, 2007; Wetzels et al., 2011). These arguments are relevant to the present discussion because Bayesian methods are often touted as robust to stopping rules (Bernardo & Smith, 1994; Edwards, Lindman, & Savage, 1963; Kadane, Schervish, & Seidenfeld, 1996). For example, Edwards et al. stated, " It is entirely appropriate to collect data until a point has been proven or disproven, or until the data collector runs out of time, money, or patience " (p. "
[Show abstract] [Hide abstract] ABSTRACT: The ongoing discussion among scientists about null-hypothesis significance testing and Bayesian data analysis has led to speculation about the practices and consequences of "researcher degrees of freedom." This article advances this debate by asking the broader questions that we, as scientists, should be asking: How do scientists make decisions in the course of doing research, and what is the impact of these decisions on scientific conclusions? We asked practicing scientists to collect data in a simulated research environment, and our findings show that some scientists use data collection heuristics that deviate from prescribed methodology. Monte Carlo simulations show that data collection heuristics based on p values lead to biases in estimated effect sizes and Bayes factors and to increases in both false-positive and false-negative rates, depending on the specific heuristic. We also show that using Bayesian data collection methods does not eliminate these biases. Thus, our study highlights the little appreciated fact that the process of doing science is a behavioral endeavor that can bias statistical description and inference in a manner that transcends adherence to any particular statistical framework.- "Smithson 1999). Partition-independence of prior probabilities (Walley 1991, 227-8; 1996). The handling of group decision problems (see e.g. "
[Show abstract] [Hide abstract] ABSTRACT: We argue that indeterminate probabilities are not only rationally permissible for a Bayesian agent, but they may even be rationally required. Our first argument begins by assuming a version of interpretivism: your mental state is the set of probability and utility functions that rationalize your behavioral dispositions as well as possible. This set may consist of multiple probability functions. Then according to interpretivism, this makes it the case that your credal state is indeterminate. Our second argument begins with our describing a world that plausibly has indeterminate chances. Rationality requires a certain alignment of your credences with corresponding hypotheses about the chances. Thus, if you hypothesize the chances to be indeterminate, your will inherit their indeterminacy in your corresponding credences. Our third argument is motivated by a dilemma. Epistemic rationality requires you to stay open-minded about contingent matters about which your evidence has not definitively legislated. Practical rationality requires you to be able to act decisively at least sometimes. These requirements can conflict with each other-for thanks to your open-mindedness, some of your options may have undefined expected utility, and if you are choosing among them, decision theory has no advice to give you. Such an option is playing Nover and Hajek's Pasadena Game, and indeed any option for which there is a positive probability of playing the Pasadena Game. You can serve both masters, epistemic rationality and practical rationality, with an indeterminate credence to the prospect of playing the Pasadena game. You serve epistemic rationality by making your upper probability positive-it ensures that you are open-minded. You serve practical rationality by making your lower probability 0-it provides guidance to your decision-making. No sharp credence could do both.- "Failures of conglomerability are problematic. As Kadane et al. [19] write, " the door to foregone conclusions is opened whenever P is not conglomerable " . But non-conglomerability of a probability measure P is not a decisive reason to reject it. "
[Show abstract] [Hide abstract] ABSTRACT: Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for two reasons. First, the failure is unusual in that the symmetry intuitions are demonstrably coherent, even though other constraints make it impossible to find a probability model for the scenario. Second, the best probability models have properties analogous to non-conglomerability, motivating a proposed extension of that concept (and corresponding limits on Bayesian conditionalization).
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