What does the proof of Birnbaum's theorem prove?

Birnbaum's theorem, that the sufficiency and conditionality principles entail the likelihood principle, has engendered a great deal of controversy and discussion since the publication of the result in 1962. In particular, many have raised doubts as to the validity of this result. Typically these doubts are concerned with the validity of the principles of sufficiency and conditionality as expressed by Birnbaum. Technically it would seem, however, that the proof itself is sound. In this paper we use set theory to formalize the context in which the result is proved and show that in fact Birnbaum's theorem is incorrectly stated as a key hypothesis is left out of the statement. When this hypothesis is added, we see that sufficiency is irrelevant, and that the result is dependent on a well-known flaw in conditionality that renders the result almost vacuous

Incremental Construction of Large Specifications: Case Study and Techniques

The RODIN project is an EU-funded project concerned with the provision of methods and tools for rigorous development of complex software-based systems. Ultimately, through the development of open-source tools and techniques, the project aims to make formal methods more appealing and accessible to industry. The project is driven by a number of case studies, each of which is designed to exercise the technology being developed and create methodologies for the future. In this paper we focus on the methodologies being developed in one of the case studies (the CDIS subset). This case study is based on a commercial air traffic information system that was developed using formal methods 14 years ago, and it is still in operation today. The key goals of our approach are to improve the comprehensibility of large specifications and to achieve a complete mechanical proof of consistency

Trainee teachers' cognitive styles and notions of differentiation

Purpose – To compare the cognitive styles of trainee teachers with their notions of differentiation and perceptions of its place/location within their teaching and learning during a PGCE programme of ITE. Methodology – 80 trainee teachers completed the Cognitive Style Index (CSI) (Allinson & Hayes, 1996) at the beginning and at the end of their course. After completing the CSI measure trainees received instruction on cognitive styles. To assess their initial understanding and prior knowledge of differentiation, all trainees completed a questionnaire at the beginning at the end of their course. Findings – At the outset rudimentary understandings of differentiation were found to be held by the trainees, as well as stylistic differences between the four style groupings. Gains in understanding of differentiation and the use of cognitive style in school were evident in all trainees. Moderate changes in style were evident, with all trainees becoming more intuitive over the course of the programme. Research limitations – The sample size may be seen as a limitation in terms of generalisability. Practical implications –The predominant direction of cognitive style movement was from analytic to intuitive. The suggestion that cognitive style whilst relatively fixed is also something that can be developed, is a feature which should offer encouragement to those developing university courses through interventions such as this. Originality - Teaching sessions on how cognitive styles can be used in the classroom were used to enhance trainee understandings of individual learning differences and increase awareness of own style to facilitate understanding of differentiation

Inferences from prior-based loss functions

Inferences that arise from loss functions determined by the prior are considered and it is shown that these lead to limiting Bayes rules that are closely connected with likelihood. The procedures obtained via these loss functions are invariant under reparameterizations and are Bayesian unbiased or limits of Bayesian unbiased inferences. These inferences serve as well-supported alternatives to MAP-based inferences