Tight Probability Bounds with Pairwise Independence

Probability bounds on the sum of $n$ pairwise independent Bernoulli random variables exceeding an integer $k$ have been proposed in the literature. However, these bounds are not tight in general. In this paper, we provide three results towards finding tight probability bounds on the sum of pairwise independent Bernoulli random variables. Firstly, for $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided. Secondly, for $k \geq 2$, the tightest upper bound with identical marginals is provided. Lastly, for general pairwise independent Bernoulli random variables, new upper bounds are derived for $k \geq 2$, by ordering the probabilities. These bounds improve on existing bounds and are tight under certain conditions. The proofs of tightness are developed using techniques of linear optimization. Numerical examples are provided to quantify the improvement of the bounds over existing bounds.Comment: 33 pages, 4 figure

Allocating Students to Multidisciplinary Capstone Projects Using Discrete Optimization

We discuss an allocation mechanism of capstone projects to senior-year undergraduate students, which the recently established Singapore University of Technology and Design (SUTD) has implemented. A distinguishing feature of these projects is that they are multidisciplinary ; each project must involve students from at least two disciplines. This is an instance of a bipartite many-to-one matching problem with one-sided preferences and with additional lower and upper bounds on the number of students from the disciplines that must be matched to projects. This leads to challenges in applying many existing algorithms.We propose the use of discrete optimization to find an allocation that considers both efficiency and fairness. This provides flexibility in incorporating side constraints, which are often introduced in the final project allocation using inputs from the various stakeholders. Over a three-year period from 2015 to 2017, the average rank of the project allocated to the student is roughly halfway between their top two choices, with around 78 percent of the students assigned to projects in their top-three choices. We discuss practical design and optimization issues that arise in developing such an allocation

Analysis of Search on Clinical Narrative within the EHR

Electronic Health Records (EHRs) are used increasingly in the hospital and outpatient set- tings, and patients are amassing digitized clinical information. On one hand, aggregating all the patient's clinical information can greatly assist health care workers in making sound decisions. On the other hand, it can result in information overload, making it difficult to browse for information within the health record. Considering the time constraints clinicians face, one way to reduce information overload is through a search utility. However, traditional, free-text search engines within the EHR can potentially miss documents that do not contain the query but that are relevant to the clinical user's search. This dissertation aims at addressing this gap by analyzing within-patient search of the EHR and examining various semantic search approaches on clinical narrative. Our work consists of three studies where clinical users' search needs are examined, traditional string-matching is analyzed, and semantic search approaches on clinical narrative are evaluated. The first study applied a mixed method approach in order to provide a better understanding of clinical users' search needs within the EHR. It is comprised of a retrospective log analysis of search log files and a survey that was administered to clinical professionals within our institution. The log analysis attempts to categorize how users of a search system query for information, and the survey tries to understand users' search preferences. This study showed that clinical users were very interested in search functionality within the EHR and that various types of users utilize a search utility differently. Overall, most users searched for specific laboratory tests and diseases within the health record. The last two studies rely on a gold standard, which was developed specifically for this dissertation. The gold standard contained a document collection, a set of queries, and for each document/query pair, a relevance judgment. This gold standard was used to evaluate and compare different search models on clinical narrative. The second study conducted was an error analysis of the traditional, vector-space model search approach. The study examined the false positives and false negatives of this approach and categorized the errors in order to identify gaps that semantic approaches may fill. The last study was a systematic evaluation of five different semantic search approaches. These search methods consisted of distributional semantic approaches and an ontology-based approach. The study identified that a mixed topic modeling and vector-space model approach was the best performing search algorithm on our gold standard. All of these studies lay the foundation for us to gain a deeper understanding of information retrieval methods within the electronic health record. Ultimately, this will allow health care professionals to easily access pertinent patient information, which could result in better health care delivery

Prevalence of thyroid dysfunction in patients with polycystic ovarian syndrome: a cross sectional study

Background: Polycystic ovary syndrome (PCOS) and thyroid disorders are two of the most common endocrine disorders in the general population. Both of these endocrine disorders share common predisposing factors, gynaecological features and have profound effect on reproductive function in women. The aim of this study is to study the prevalence of thyroid dysfunction in patients with polycystic ovarian syndrome and to evaluate the relationship between polycystic ovarian syndrome and thyroid dysfunction.Methods: This is a cross sectional observational study done on 100 patients with Poly Cystic Ovarian Syndrome based on Rotterdam’s criteria. The exclusion criteria was hyperprolactinemia, congenital adrenal hyperplasia and virilising tumour. Thyroid function was evaluated by measurement of fasting serum thyroid stimulating hormone (TSH), free thyroxine levels (free T3 and free T4).Results: The mean age of the study patients was 26±4.2 years. Among the study patients, 11% of them had goitre. 18% of the patients with presented with subclinical hypothyroidism. The mean TSH levels in the study patients was 4.62±2.12 mIU/ml. The overall prevalence of thyroid dysfunction was 33% in the study patients with PCOS.Conclusions: This study concludes that the prevalence of hypothyroidism is increased in women with PCOS patients

Robustness to dependency in portfolio optimization using overlapping marginals

In this paper, we develop a distributionally robust portfolio optimization model where the robustness is across different dependency structures among the random losses. For a Fr´echet class of discrete distributions with overlapping marginals, we show that the distributionally robust portfolio optimization problem is efficiently solvable with linear programming. To guarantee the existence of a joint multivariate distribution consistent with the overlapping marginal information, we make use of a graph theoretic property known as the running intersection property. Building on this property, we develop a tight linear programming formulation to find the optimal portfolio that minimizes the worst-case Conditional Value-at-Risk measure. Lastly, we use a data-driven approach with financial return data to identify the Fr´echet class of distributions satisfying the running intersection property and then optimize the portfolio over this class of distributions. Numerical results in two different datasets show that the distributionally robust portfolio optimization model improves on the sample-based approac

Probability bounds for nn random events under (n−1)(n-1)-wise independence

A collection of $n$ random events is said to be $(n - 1)$-wise independent if any $n - 1$ events among them are mutually independent. We characterise all probability measures with respect to which $n$ random events are $(n - 1)$-wise independent. We provide sharp upper and lower bounds on the probability that at least $k$ out of $n$ events with given marginal probabilities occur over these probability measures. The bounds are shown to be computable in polynomial time.Comment: 18 pages, 2 table

Managing humanitarian operations: the impact of amount, schedules, and uncertainty in funding

Global health spending has increased manyfold in the last few decades reaching US $6.5 trillion in 2012. Despite these increases, humanitarian organizations from around the world, working on different diseases including Malaria and Tuberculosis, have warned about potential funding shortfalls in the near future. Facing a growing need for health services and commodities, resource-constrained organizations are constantly looking for ways to maximize health outcomes through efficient and effective use of available resources. In this dissertation, we develop approaches to make efficient operational decisions under variable and unpredictable donor funding, a situation that is commonly faced by many humanitarian organizations. In the first chapter, we study the problem of managing inventory of a nutritional product under variable funding constraints. Despite the complexities associated with funding, we show that the optimal replenishment policy is easy to compute and straightforward to implement. We also provide several insights into how the funding amount, funding schedules and uncertainty in funding impact operating costs in this setting. In chapter 2, we look at the problem of dynamically allocating a limited amount of donor funding to patients in different health states in a humanitarian health setting. We show that the optimal allocation policy is state-dependent and prove several structural properties of the optimal policy that would help simplify its computation. Due to the complexity involved in calculating the optimal policy, we develop two heuristics to handle real-size problems with longer planning horizons. Computational results suggest that both heuristics perform well in many cases but one of the heuristics is more robust across a wide variety of settings. In addition to the allocation policy, we also provide some interesting insights into the impact of funding level and funding uncertainty in the multiple health states setting. In the third chapter, we focus on the supply- vs. demand-side investment dilemma frequently faced by public health managers who have a limited budget at their disposal. First, we consider a centralized setting where a single entity, referred as the principal, makes both supply- and demand-side investment decisions. We determine the principal's optimal investment mix in this budget constrained environment and provide insights into how the investment mix varies with the different supply- and demand-side parameters. We then consider a decentralized setting where the principal invests in improving the supply chain while demand mobilization activities are contracted to an agent, who is a profit maximizer. For the decentralized setting, we identify two contracts that ensure that the coverage in the decentralized setting is at least as high as the centralized case.Doctor of Philosoph