8. Alan Hájek: The Extremely Improbable

The speaker begins by expressing gratitude to Bill, reflecting on their journey as a nervous graduate student in 1985. They recount their initial discomfort in the field of philosophy, having a background in mathematics and statistics, and how Bill's mentorship was pivotal. The speaker's first philosophy seminar with Bill on scientific reasoning sparked a lifelong interest in the philosophy of probability, leading to published work on near certainty and extreme improbability. The speaker invites the audience to share in the fascination with the improbable, setting the stage for a talk that will explore the significance of improbability in various fields.

This paragraph delves into the concept of improbability, its applications in statistical methodology, and its philosophical implications. The speaker discusses significance testing in classical statistics, where the rejection of a null hypothesis occurs if the probability of observing data as extreme as what has been collected is deemed too improbable. The speaker also touches on the idea that data fitting a hypothesis too well can be suspicious, referencing Fisher's critique of Mendel's experiments. The paragraph further explores the principle that events of small probability are often treated as if they are zero or one, highlighting the importance of considering improbability in both scientific and philosophical contexts.

The speaker presents the paradoxical nature of improbability in everyday life and its philosophical implications. They discuss the lottery paradox, which challenges the notion of rational binary belief and the closure of rational beliefs under conjunction. The paradox suggests that believing in the high probability of an event and simultaneously believing in the low probability of its opposite can lead to inconsistencies. The speaker also addresses skepticism about knowledge, using the example of the small probability of one's car being stolen to question the certainty of one's knowledge about the car's location. The paragraph emphasizes the philosophical importance of improbability and its role in shaping beliefs and understanding the world.

In this paragraph, the speaker provides a more precise definition of improbability, distinguishing between events with probability zero, those with small positive real number probabilities, and those with imprecise probabilities with an improbable upper limit. The speaker then explores the peculiar properties of low probabilities, such as their inability to be divided, their role as a multiplicative destroyer, and their paradoxical behavior in addition and subtraction. These properties lead to philosophical problems, such as issues with the canonical ratio analysis of conditional probability, which breaks down when dealing with events that have a probability of zero.

The speaker examines the impact of zero probability on the concepts of independence and decision theory. They argue that the orthodox definition of independence, which states that two events are independent if the probability of their conjunction is the product of their individual probabilities, leads to counterintuitive results when one of the events has a probability of zero. This is because any event with a zero probability is deemed independent of itself and all other events, including its negation. The speaker also discusses the implications of zero probability for decision theory, particularly the conflict between expected utility theory and dominance reasoning when applied to options with a zero probability of a highly favorable outcome.

The speaker challenges the orthodox Bayesian approach to probability and decision theory, highlighting the issues that arise from the treatment of zero and small probabilities. They argue that the traditional methods of handling conditional probabilities, independence, and decision-making are inadequate when confronted with cases involving improbability. The speaker suggests that a reevaluation of these theories is necessary to address the philosophical problems they create, such as the paradoxes of conditional probability and the inconsistencies in decision-making under uncertainty.

In this paragraph, the speaker delves into the nature of chance processes, using the example of a coin toss to illustrate the concept. They argue that it is a mistake to assume that there is a predetermined outcome for a chance event, as this misunderstands the very essence of chance. The speaker extends this argument to the broader context of indeterminism in the world, suggesting that even seemingly deterministic events are subtly influenced by quantum mechanics and therefore are not predetermined. This challenges the conventional understanding of counterfactuals, which assume that there is a fixed outcome for any given action.

The speaker discusses the philosophical implications of assigning infinite probabilities to certain events, using Pascal's wager as a central example. They argue that the traditional interpretation of Pascal's wager, which suggests that one should wager for God due to the potential infinite reward, is flawed when considering mixed strategies. The speaker points out that any action, no matter how mundane, could be part of a mixed strategy that results in an infinite expected utility, thus undermining the original argument for wagering for God. This leads to a broader discussion of the problems with using infinite utilities in decision-making.

The speaker emphasizes the importance of considering low probability, high impact events, such as the eruption of a volcano causing the closure of European airports. They argue that even when the probability of such events is small, the high stakes involved necessitate taking them seriously. The speaker uses this example to highlight the need for a more nuanced understanding of probability and risk, particularly in the context of decision-making and policy.

In this final paragraph, the speaker discusses the relevance of counterfactuals in the context of decision-making and the value of human life. They consider the implications of assigning infinite or large finite values to the loss of human life and how this affects the analysis of risk and decision-making. The speaker also addresses the concept of independence in probability, suggesting that there may be multiple senses of independence that need to be considered, particularly when dealing with conditional probabilities and the information they convey.

The speaker concludes the discussion by reflecting on the challenges that probability theory faces, particularly when dealing with idealized cases and the assignment of zero probabilities. They acknowledge the limitations of frequentist interpretations of probability and the need for a more general probability theory that can accommodate a wider range of scenarios. The speaker also addresses the potential for probability gaps, where certain events may not have assigned probabilities, and the philosophical implications of these gaps for concepts like conditional probability and independence.