The Weight of Evidence

Courts of law must weigh evidence to determine the likelihood of competing interpretations of past events, and different legal contexts require different standards of proof, but this falls short of a quantification of probability. Bayes’s theorem and the associated formula provide a way of combining multiple elements of evidence and using them to refine prior assessments of probability. The prosecutor’s fallacy involves an incorrect reversal of the logic of evidence. The ecological fallacy involves incorrectly attributing proportions derived from large groups to smaller groups or individuals.

Prediction and Assessment of the Occupational Risks in the Mining Industry Using the Bayess Theorem

The analysis of existing methods for assessing occupational risks is carried out, and the need for searchinga fundamentally new approach to the assessment and prediction of risks in the mining industry is substantiated. Based on the results of the analysis of modern methods and technologies, it is established that the development of the methodology for assessment and prediction of the occupational risks using Bayes's theorem has significant advantages: simplicity and accessibility for the occupational safety specialists, reproducibility considering many factors of working conditions, as well as the possibility of preventive measures prediction and development. The application of Bayes's theorem is promising in determining cause-and-effect relationships and predicting the occupational morbidity of the employees, which is also an advantage of this methodology for managing occupational risks in the mining industry. Bayes's approaches to modeling are characterized by high performance, intuitively clear in the form of a graph. The example is given concerning the application of Bayes's theorem to assess the risk of a fatal incident taking into account the statistics on the mining industry. Also, the simplest types of Bayes’s trust networks were developed reflecting the possibility of establishing cause-and-effect relationships (both for assessment and prediction), and are the basis for further modeling.

Sleeping Beauty on Monty Hall

Inspired by the Monty Hall Problem and a popular simple solution to it, we present a number of game-show puzzles that are analogous to the notorious Sleeping Beauty Problem (and variations on it), but much easier to solve. We replace the awakenings of Sleeping Beauty by contestants on a game show, like Monty Hall’s, and increase the number of awakenings/contestants in the same way that the number of doors in the Monty Hall Problem is increased to make it easier to see what the solution to the problem is. We show that these game-show proxies for the Sleeping Beauty Problem and variations on it can be solved through simple applications of Bayes’s theorem. This means that we will phrase our analysis in terms of credences or degrees of belief. We will also rephrase our analysis, however, in terms of relative frequencies. Overall, our paper is intended to showcase, in a simple yet non-trivial example, the efficacy of a tried-and-true strategy for addressing problems in philosophy of science, i.e., develop a simple model for the problem and vary its parameters. Given that the Sleeping Beauty Problem, much more so than the Monty Hall Problem, challenges the intuitions about probabilities of many when they first encounter it, the application of this strategy to this conundrum, we believe, is pedagogically useful.

Evidence and Probability Data

This chapter discusses evidence and probability data with particular attention on Bayesian estimation. The Protestant ethic slowed probability developments in the United States, but the idea of quantification continued apace in England and on the Continent. In particular, Thomas Bayes invented a simple but profound mathematical means to connect outcomes with causes with conditional probabilities and Bayesian estimation. The chapter explains conditional probabilities and Bayesian logic, giving several examples, including incidence of accurate cancer diagnosis with inexact diagnostics. The chapter also introduces Bayes’s magnum opus An Essay Toward Solving a Problem in the Doctrine of Chances and gives his example of rolling billiard balls on a billiard table to show Bayes’s theorem.

A Set-Theoretic Approach to Bayesian Process Tracing

This article develops a set-theoretic approach to Bayes’s theorem and Bayesian process tracing. In the approach, hypothesis testing is the procedure whereby one updates beliefs by narrowing the range of states of the world that are regarded as possible, thus diminishing the domain in which the actual world can reside. By explicitly connecting Bayesian analysis to its set-theoretic foundations, the approach makes process tracing more intuitive and thus easier to apply for qualitative researchers. Moreover, the set-theoretic approach provides new tools for assessing both the consequentialness and expectedness of evidence when conducting process tracing. It also provides a new way to classify and interpret process-tracing tests, such as hoop tests and smoking gun tests, by viewing them as zones in a continuous space whose dimensions reflect the magnitude of changes in sets. The article shows that Bayesian process tracing and set-theoretic process tracing are not alternatives to each other but rather two sides of the same coin.