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

Distributionally robust optimization through the lens of submodularity

Distributionally robust optimization is used to solve decision making problems under adversarial uncertainty where the distribution of the uncertainty is itself ambiguous. In this paper, we identify a class of these instances that is solvable in polynomial time by viewing it through the lens of submodularity. We show that the sharpest upper bound on the expectation of the maximum of affine functions of a random vector is computable in polynomial time if each random variable is discrete with finite support and upper bounds (respectively lower bounds) on the expected values of a finite set of submodular (respectively supermodular) functions of the random vector are specified. This adds to known polynomial time solvable instances of the multimarginal optimal transport problem and the generalized moment problem by bridging ideas from convexity in continuous optimization to submodularity in discrete optimization. In turn, we show that a class of distributionally robust optimization problems with discrete random variables is solvable in polynomial time using the ellipsoid method. When the submodular (respectively supermodular) functions are structured, the sharp bound is computable by solving a compact linear program. We illustrate this in two cases. The first is a multimarginal optimal transport problem with given univariate marginal distributions and bivariate marginals satisfying specific positive dependence orders along with an extension to incorporate higher order marginal information. The second is a discrete moment problem where a set of marginal moments of the random variables are given along with lower bounds on the cross moments of pairs of random variables. Numerical experiments show that the bounds improve by 2 to 8 percent over bounds that use only univariate information in the first case, and by 8 to 15 percent over bounds that use the first moment in the second case.Comment: 36 Pages, 6 Figure

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