Learning mixtures of structured distributions over discrete domains

Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n] = \{1,...,n\}.$ We show that if $\mathfrak{C}$ satisfies a rather general condition -- essentially, that each distribution in $\mathfrak{C}$ can be well-approximated by a variable-width histogram with few bins -- then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of $k$ unknown distributions from $\mathfrak{C}.$ We analyze several natural types of distributions over $[n]$, including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems.Comment: preliminary full version of soda'13 pape

Some Studies in the Chemistry of Substituted Azetinone and Pyrroloindole-Carboxylic Acids

This thesis is concerned with the chemistry of azetidinone carboxylic acids and pyrrolodine carboxylic acids and their derivatives. As such, the introduction surveys the general field of β-lactam biochemistry, the more specific area of carbapenem synthesis and the chemistry of pyrroloindole-2-carboxylic acids. The first main section of the thesis describes the free-radical chemistry of azetidin-2-one-4-carboxylic acids and the attempted elaboration of these materials into precursors to the carbapenem antibiotic methylthienamycin. The second section involves an investigation into both the one and two electron chemistry of pyrroloindole-2-carboxylic acids. Various derivatives of these acids were also synthesized in an attempt to elucidate the mechanism for the high stereoselective formation of the endo-isomer. The oxidative cleavage of the indole ring of tryptophan derivatives was investigated resulting in enantiospecific syntheses of R- and S-methyl aspartic acids. During the course of these studies, a crystalline mixed anhydride was isolated from reaction of a carboxylic acid with a chloroformate and its structure determined by single crystal X-ray analysis

Fear of crime among older persons : an exploratory qualitative study in different environments in Hong Kong

Fear of crime among various groups has long been studied in Western societies. Many studies have concluded that older persons tend to exhibit higher levels of fear of crime than other age groups even though they are generally at a lower risk of being victims of crime. However, there have been relatively few studies on fear of crime and associated reasons amongst older persons in Asian cities and Chinese societies. Moreover, most existing studies have generally utilized quantitative methods to examine the possibly causal relationships between fear of crime and its underlying factors, and subjective evaluations by older people themselves of factors related to the fear of crime are very few. This study aimed to investigate factors related to fear of crime by exploring older persons’ perspectives on their living environments and their own situations. A qualitative research design was used to explore how and why fear arises in spite of considerable objective evidence that older persons are at relatively low risk of falling victim of crime. The study employed eight focus group discussions (FGDs) and two individual interviews. To provide a rage of typical HK residential environment, participants were drawn from two main categories of housing (traditional village housing and purpose-built housing) and four different types of physical living environment in Hong Kong (a village, an island, old-town housing, and new towns). Environmental factors, individual factors and moderators of fear of crime have been identified in this study. First, the qualitative findings suggest that environmental factors can be categorized as three dimensions, which include vulnerability (defect of the living environment), defensibility (level of protection that provided by the environment) and supportability (availability of social support that older persons can get when they are at risk). Vulnerability appears to be positive associated with fear of crime, but defensibility and supportability appear to be negatively associated with fear of crime. Second, the findings on individual factors enrich the Vulnerability model proposed by previous researchers, in which physical, psychological and behavioural weakness of older persons can be discussed. Third, moderators of fear of crime (e.g. people who have adjusted to a dangerous place by knowing the latest local crime event or figure.) which concentrate on the cognitive and behavioural adjustment among older persons, have been identified. Finally, policy recommendations for the welfare of older persons in Hong Kong are suggested based on the findings of the research

How do teachers' beliefs affect the implementation of inquiry-based learning in the PGS Curriculum? A case study of two primary schools in Hong Kong.

In 2000, the Curriculum Development Council (CDC) of Hong Kong launched curriculum reform for all school subjects to equip students with generic skills perceived essential for the 21st Century. As part of this, the new Primary General Studies (PGS) programme implemented in 2004, adopted an inquiry-based learning approach. The literature shows that inquiry-based learning not only has origins linked to science inquiry and Dewey’s theory of inquiry, but also intersects with theories of constructivism. Similar to constructivism inquiry-based learning also incurs the controversies for its theoretical foundation. Moreover, the precedents of enacting inquiry-based learning in classrooms have alerted teachers to its practice-based challenges. Furthermore, teachers’ beliefs have been recognized as a major factor influencing teachers’ actions especially in the implementation of a new teaching method. Therefore, three years after its launch, a study was proposed to investigate the impacts of teachers’ beliefs on the implementation of inquiry-based learning in the new PGS curriculum. The research was in the form of qualitative case studies of two schools. Eight teachers were involved, while four of them were studied in more detail. The results show that different teachers held diverse beliefs about inquiry-based learning. Such variation in teachers’ beliefs was found to impact on teachers’ implementation of inquiry-based learning. Finally, recommendations about the importance of teachers’ reflection, arrangement of resources, preparation for teachers and students and in-services training, are made to teachers, school administration, and local authority. In this study, the “hypothetical components of belief” suggested by Sigel (1985) were adopted as the major theoretical framework and within such a framework contextual factors of individual school were found to have played crucial roles both in influencing teachers’ beliefs and teachers’ actions