Probabilistic inversion: a preliminary discussion

We continue the discussion on the possibility of interpreting probability as a logic, that we have started in the previous IMEKO TC1-TC7-TC13 Symposium. We show here how a probabilistic logic can be extended up to including direct and inverse functions. We also discuss the relationship between this framework and the Bayes-Laplace rule, showing how the latter can be formally interpreted as a probabilistic inversion device. We suggest that these findings open a new perspective in the evaluation of measurement uncertainty

Consciousness as Integrated Information: A Provisional Philosophical Critique

Giulio Tononi (2008) has offered his integrated information theory of consciousness (IITC) as a ‘provisional manifesto’. I critically examine how the approach fares. I point out some (relatively) internal concerns with the theory and then more broadly philosophical ones; finally I assess the prospects for IITC as a fundamental theory of consciousness. I argue that the IITC’s scientific promise does carry over to a significant extent to broader philosophical theorizing about qualia and consciousness, though not as directly as Tononi suggests, since the account is much more focused on the qualitative character of experience rather than on consciousness itself. I propose understanding it as ‘integrated information theory of qualia’(IITQ), rather than of consciousness

Exploring problems encountered in the teaching and learning of statistics in grade 11

The purpose of the study was to explore problems encountered in the teaching and learning of statistics in grade 11 and to offer ways of addressing them. A convenient sample of 100 grade 11 mathematics teachers and 448 grade 11 mathematics learners participated in the study. A descriptive survey design was used. Data were collected from the teachers using a teacher questionnaire, classroom observation schedule and teacher interview, and from learners through the use of a learner questionnaire, classroom observation schedule, diagnostic test and learner interview. The validity and reliability of all these instruments was established. Descriptive statistics and frequencies were used to analyse learner questionnaires, teacher questionnaires and learners‟ responses to a diagnostic test. Teacher and learner interviews were transcribed and classified according to themes. Classroom observations were analysed by using themes and checking for similarities and differences. The results showed that teachers had difficulty with the interpretation and calculation of measures of dispersion; representation and interpretation of data on graphs or plots; determining the five number summary; constructing and interpreting probability diagrams and tables; and interpreting probability terminology. Also, the results showed that learners experienced difficulties when using graphs to predict the results; interpreting and determining measures of dispersion; computation of quartiles when the total number of data values was even; representing data on graphs or plots; interpreting and determining measures of central tendency; constructing and interpreting probability graphs and tables; and interpreting probability terminology. The results found possible causes of the teachers‟ difficulties to be (1) their lack of statistics content knowledge; (2) inadequate textbooks; (3) in-service programmes which did not cover statistical topics, or which did not pay adequate attention to probability; and (4) teachers failure to attend these in-service teacher workshops. Further, the findings of the study were that the probable causes of learners‟ difficulties were (1) inadequate teaching of statistics topics in previous grades; (2) teachers‟ lack of content knowledge in statistics meant they had difficulty explaining concepts to learners; (3) inadequate learning material and learners‟ inability to use the statistics function mode on their calculators; and (4) learners‟ lack of conceptual knowledge of certain aspects of statistics. Lastly, the results of the study found that the possible ways to address the problems in the teaching and learning of statistics in grade 11 to be: (1) teachers should receive financial support from their schools/districts to attend in-service education and training programmes; (2) textbooks should be well written (provide thorough explanations) and contain all the information necessary to teach data handling and probability (i.e. formulae, more examples); (3) in-service teacher programmes should meet the needs of the teachers by offering topics that teachers find difficult to teach; and (4) more and longer inset programmes on probability, preferably five-day workshops, should be arranged.Mathematics EducationM. Sc. (Mathematics, Science and Technology Education

Empirical interpretation of imprecise probabilities

This paper investigates the possibility of a frequentist interpretation of imprecise probabilities, by generalizing the approach of Bernoulli’s Ars Conjectandi. That is, by studying, in the case of games of chance, under which assumptions imprecise probabilities can be satisfactorily estimated from data. In fact, estimability on the basis of finite amounts of data is a necessary condition for imprecise probabilities in order to have a clear empirical meaning. Unfortunately, imprecise probabilities can be estimated arbitrarily well from data only in very limited settings

Counting Steps: A Finitist Approach to Objective Probability in Physics

We propose a new interpretation of objective probability in statistical physics based on physical computational complexity. This notion applies to a single physical system (be it an experimental set-up in the lab, or a subsystem of the universe), and quantifies (1) the difficulty to realize a physical state given another, (2) the 'distance' (in terms of physical resources) between a physical state and another, and (3) the size of the set of time-complexity functions that are compatible with the physical resources required to reach a physical state from another. This view (a) exorcises 'ignorance' from statistical physics, and (b) underlies a new interpretation to non-relativistic quantum mechanics