Beginning inference in fourth grade : exploring variation in measurement

This paper addresses one of the foundational components of beginning infernce, namely variation, with 5 classes of Year 4 students undertaking a measurement activity using scaled instruments in two contexts: all students measuring one person's arm span and recording the values obtained, and each student having his/her own arm span measured and recorded. The results included documentation of students' explicit appreciation of the variety of ways in which varitation can occur, including outliers, and their ability to create and describe valid representations of their data.

Statistical literacy in the elementary school: Opportunities for problem posing

This chapter addresses opportunities for problem posing in developing young children’s statistical literacy, with a focus on student-directed investigations. Although the notion of problem posing has broadened in recent years, there nevertheless remains limited research on how problem posing can be integrated within the regular mathematics curriculum, especially in the areas of statistics and probability. The chapter first reviews briefly aspects of problem posing that have featured in the literature over the years. Consideration is next given to the importance of developing children’s statistical literacy in which problem posing is an inherent feature. Some findings from a school playground investigation conducted in four, fourth-grade classes illustrate the different ways in which children posed investigative questions, how they made predictions about their outcomes and compared these with their findings, and the ways in which they chose to represent their findings.

Paradigms in Statistical Inference for Finite Populations; Up to the 1950s

Modern sample surveys started to spread after statistician at the U.S. Bureau of the Census in the 1940s had developed a sampling design for the Current Population Survey (CPS). A significant factor was also that digital computers became available for statisticians. In the beginning of 1950s, the theory was documented in textbooks on survey sampling. This thesis is about the development of the statistical inference for sample surveys. For the first time the idea of statistical inference was enunciated by a French scientist, P. S. Laplace. In 1781, he published a plan for a partial investigation in which he determined the sample size needed to reach the desired accuracy in estimation. The plan was based on Laplace s Principle of Inverse Probability and on his derivation of the Central Limit Theorem. They were published in a memoir in 1774 which is one of the origins of statistical inference. Laplace s inference model was based on Bernoulli trials and binominal probabilities. He assumed that populations were changing constantly. It was depicted by assuming a priori distributions for parameters. Laplace s inference model dominated statistical thinking for a century. Sample selection in Laplace s investigations was purposive. In 1894 in the International Statistical Institute meeting, Norwegian Anders Kiaer presented the idea of the Representative Method to draw samples. Its idea was that the sample would be a miniature of the population. It is still prevailing. The virtues of random sampling were known but practical problems of sample selection and data collection hindered its use. Arhtur Bowley realized the potentials of Kiaer s method and in the beginning of the 20th century carried out several surveys in the UK. He also developed the theory of statistical inference for finite populations. It was based on Laplace s inference model. R. A. Fisher contributions in the 1920 s constitute a watershed in the statistical science He revolutionized the theory of statistics. In addition, he introduced a new statistical inference model which is still the prevailing paradigm. The essential idea is to draw repeatedly samples from the same population and the assumption that population parameters are constants. Fisher s theory did not include a priori probabilities. Jerzy Neyman adopted Fisher s inference model and applied it to finite populations with the difference that Neyman s inference model does not include any assumptions of the distributions of the study variables. Applying Fisher s fiducial argument he developed the theory for confidence intervals. Neyman s last contribution to survey sampling presented a theory for double sampling. This gave the central idea for statisticians at the U.S. Census Bureau to develop the complex survey design for the CPS. Important criterion was to have a method in which the costs of data collection were acceptable, and which provided approximately equal interviewer workloads, besides sufficient accuracy in estimation.

In the beginning...

Professor John Frazer and architect Gianni Botsford met as tutor and student in 1994 at the Architural Association. Two years of never talking about architecture and nearly burning the studio down led to a new way of approaching design.

Bayesian versus frequentist statistical inference for investi gating a one-off cancer cluster reported to a health department

Background The problem of silent multiple comparisons is one of the most difficult statistical problems faced by scientists. It is a particular problem for investigating a one-off cancer cluster reported to a health department because any one of hundreds, or possibly thousands, of neighbourhoods, schools, or workplaces could have reported a cluster, which could have been for any one of several types of cancer or any one of several time periods. Methods This paper contrasts the frequentist approach with a Bayesian approach for dealing with silent multiple comparisons in the context of a one-off cluster reported to a health department. Two published cluster investigations were re-analysed using the Dunn-Sidak method to adjust frequentist p-values and confidence intervals for silent multiple comparisons. Bayesian methods were based on the Gamma distribution. Results Bayesian analysis with non-informative priors produced results similar to the frequentist analysis, and suggested that both clusters represented a statistical excess. In the frequentist framework, the statistical significance of both clusters was extremely sensitive to the number of silent multiple comparisons, which can only ever be a subjective "guesstimate". The Bayesian approach is also subjective: whether there is an apparent statistical excess depends on the specified prior. Conclusion In cluster investigations, the frequentist approach is just as subjective as the Bayesian approach, but the Bayesian approach is less ambitious in that it treats the analysis as a synthesis of data and personal judgements (possibly poor ones), rather than objective reality. Bayesian analysis is (arguably) a useful tool to support complicated decision-making, because it makes the uncertainty associated with silent multiple comparisons explicit.

High-level concept annotation using ontology and probabilistic inference

Image annotation is a significant step towards semantic based image retrieval. Ontology is a popular approach for semantic representation and has been intensively studied for multimedia analysis. However, relations among concepts are seldom used to extract higher-level semantics. Moreover, the ontology inference is often crisp. This paper aims to enable sophisticated semantic querying of images, and thus contributes to 1) an ontology framework to contain both visual and contextual knowledge, and 2) a probabilistic inference approach to reason the high-level concepts based on different sources of information. The experiment on a natural scene database from LabelMe database shows encouraging results.

Mid-level concept learning with visual contextual ontologies and probabilistic inference for image annotation

To date, automatic recognition of semantic information such as salient objects and mid-level concepts from images is a challenging task. Since real-world objects tend to exist in a context within their environment, the computer vision researchers have increasingly incorporated contextual information for improving object recognition. In this paper, we present a method to build a visual contextual ontology from salient objects descriptions for image annotation. The ontologies include not only partOf/kindOf relations, but also spatial and co-occurrence relations. A two-step image annotation algorithm is also proposed based on ontology relations and probabilistic inference. Different from most of the existing work, we specially exploit how to combine representation of ontology, contextual knowledge and probabilistic inference. The experiments show that image annotation results are improved in the LabelMe dataset.

A fully Bayesian approach to inference for Coxian phase-type distributions with covariate dependent mean

Phase-type distributions represent the time to absorption for a finite state Markov chain in continuous time, generalising the exponential distribution and providing a flexible and useful modelling tool. We present a new reversible jump Markov chain Monte Carlo scheme for performing a fully Bayesian analysis of the popular Coxian subclass of phase-type models; the convenient Coxian representation involves fewer parameters than a more general phase-type model. The key novelty of our approach is that we model covariate dependence in the mean whilst using the Coxian phase-type model as a very general residual distribution. Such incorporation of covariates into the model has not previously been attempted in the Bayesian literature. A further novelty is that we also propose a reversible jump scheme for investigating structural changes to the model brought about by the introduction of Erlang phases. Our approach addresses more questions of inference than previous Bayesian treatments of this model and is automatic in nature. We analyse an example dataset comprising lengths of hospital stays of a sample of patients collected from two Australian hospitals to produce a model for a patient's expected length of stay which incorporates the effects of several covariates. This leads to interesting conclusions about what contributes to length of hospital stay with implications for hospital planning. We compare our results with an alternative classical analysis of these data.