An important suggestion of objective Bayesians is that the maximum entropy principle can replace a principle which is known to get into paradoxical difficulties: the principle of indifference. No one has previously determined whether the maximum entropy principle is better able to solve Bertrand’s chord paradox than the principle of indifference. In this paper I show that it is not. Additionally, the course of the analysis brings to light a new paradox, a revenge paradox of the chords, that is unique (...) to the maximum entropy principle. (shrink)

In robustness analysis, hypotheses are supported to the extent that a result proves robust, and a result is robust to the extent that we detect it in diverse ways. But what precise sense of diversity is at work here? In this paper, I show that the formal explications of evidential diversity most often appealed to in work on robustness – which all draw in one way or another on probabilistic independence – fail to shed light on the notion of diversity (...) relevant to robustness analysis. I close by briefly outlining a promising alternative approach inspired by Horwich’s (1982) eliminative account of evidential diversity. (shrink)

The Borel–Kolmogorov Paradox is typically taken to highlight a tension between our intuition that certain conditional probabilities with respect to probability zero conditioning events are well defined and the mathematical definition of conditional probability by Bayes’ formula, which loses its meaning when the conditioning event has probability zero. We argue in this paper that the theory of conditional expectations is the proper mathematical device to conditionalize and that this theory allows conditionalization with respect to probability zero events. The conditional probabilities (...) on probability zero events in the Borel–Kolmogorov Paradox also can be calculated using conditional expectations. The alleged clash arising from the fact that one obtains different values for the conditional probabilities on probability zero events depending on what conditional expectation one uses to calculate them is resolved by showing that the different conditional probabilities obtained using different conditional expectations cannot be interpreted as calculating in different parametrizations of the conditional probabilities of the same event with respect to the same conditioning conditions. We conclude that there is no clash between the correct intuition about what the conditional probabilities with respect to probability zero events are and the technically proper concept of conditionalization via conditional expectations—the Borel–Kolmogorov Paradox is just a pseudo-paradox. (shrink)

In standard probability theory, probability zero is not the same as impossibility. But many have suggested that only impossible events should have probability zero. This can be arranged if we allow infinitesimal probabilities, but infinitesimals do not solve all of the problems. We will see that regular probabilities are not invariant over rigid transformations, even for simple, bounded, countable, constructive, and disjoint sets. Hence, regular chances cannot be determined by space-time invariant physical laws, and regular credences cannot satisfy seemingly reasonable (...) symmetry principles. Moreover, the examples here are immune to the objections against Williamson’s infinite coin flips. (shrink)

The Indifference Principle, its Paradoxes and Kolmogorov's Probability Space.

Many philosophers have claimed that Bayesianism can provide a simple justification for hypothetico-deductive inference, long regarded as a cornerstone of the scientific method. Following up a remark of van Fraassen, we analyze a problem for the putative Bayesian justification of H-D inference in the case where what we learn from observation is logically stronger than what our theory implies. Firstly, we demonstrate that in such cases the simple Bayesian justification does not necessarily apply. Secondly, we identify a set of sufficient (...) conditions for the mismatch in logical strength to be justifiably ignored as a "harmless idealization''. Thirdly, we argue, based upon scientific examples, that the pattern of H-D inference of which there is a ready Bayesian justification is only rarely the pattern that one actually finds at work in science. Whatever the other virtues of Bayesianism, the idea that it yields a simple justification of a pervasive pattern of scientific inference appears to have been oversold. (shrink)

Certain mathematical problems prove very hard to solve because some of their intuitive features have not been assimilated or cannot be assimilated by the available mathematical resources. This state of affairs triggers an interesting dynamic whereby the introduction of novel conceptual resources converts the intuitive features into further mathematical determinations in light of which a solution to the original problem is made accessible. I illustrate this phenomenon through a study of Bertrand’s paradox.

In probability textbooks, it is widely claimed that zero probability does not mean impossibility. But what stands behind this claim? In this paper I offer an explanation to this claim based on Kolmogorov's formalism. As such, this explanation is relevant to all interpretations of Kolmogorov's probability theory. I start by clarifying that this claim refers only to nonempty events, since empty events are always considered as impossible. Then, I offer the following three reasons for the claim that nonempty events with (...) zero probability are considered as possible: The main reason is simply because they are nonempty. Hence, they are considered as possible despite their zero probability. The second reason is that sometimes the zero probability is taken to be an approximation of some infinitesimal probability value. Such a value is strictly positive and as such does not imply impossibility in a strict sense. Finally, the third reason is that there are interpretations according to which the same event can have different probabilities. Specifically, it is assumed that an event with exactly zero probability can have strictly positive probabilities. This means that such an event can be possible which implies that its zero probability does not mean impossibility. (shrink)