scholarly journals Tree Bounds for Sums of Bernoulli Random Variables: A Linear Optimization Approach

2020 ◽  
pp. ijoo.2019.0038
Author(s):  
Divya Padmanabhan ◽  
Karthik Natarajan
Keyword(s):  
Conditional Independence ◽  
Graphical Model ◽  
Linear Optimization ◽  
Random Variables ◽  
Linear Program ◽  
Upper And Lower Bounds ◽  
Optimization Approach ◽  
Bivariate Distributions ◽  
Bernoulli Random Variables ◽  
Probability Estimates

We study the problem of computing the tightest upper and lower bounds on the probability that the sum of n dependent Bernoulli random variables exceeds an integer k. Under knowledge of all pairs of bivariate distributions denoted by a complete graph, the bounds are NP-hard to compute. When the bivariate distributions are specified on a tree graph, we show that tight bounds are computable in polynomial time using a compact linear program. These bounds provide robust probability estimates when the assumption of conditional independence in a tree-structured graphical model is violated. We demonstrate, through numericals, the computational advantage of our compact linear program over alternate approaches. A comparison of bounds under various knowledge assumptions, such as univariate information and conditional independence, is provided. An application is illustrated in the context of Chow–Liu trees, wherein our bounds distinguish between various trees that encode the maximum possible mutual information.

scholarly journals Approximation of continuous random variables for the evaluation of the reliability parameter of complex stress–strength models

2021 ◽  
Author(s):  
Alessandro Barbiero ◽  
Asmerilda Hitaj
Keyword(s):  
Order Approximation ◽  
Linear Optimization ◽  
Closed Form Solution ◽  
Random Variables ◽  
Discrete Distribution ◽  
Computation Time ◽  
Form Solution ◽  
Engineering Problem ◽  
Approximation Technique ◽  
Reliability Parameter

AbstractIn many management science or economic applications, it is common to represent the key uncertain inputs as continuous random variables. However, when analytic techniques fail to provide a closed-form solution to a problem or when one needs to reduce the computational load, it is often necessary to resort to some problem-specific approximation technique or approximate each given continuous probability distribution by a discrete distribution. Many discretization methods have been proposed so far; in this work, we revise the most popular techniques, highlighting their strengths and weaknesses, and empirically investigate their performance through a comparative study applied to a well-known engineering problem, formulated as a stress–strength model, with the aim of weighting up their feasibility and accuracy in recovering the value of the reliability parameter, also with reference to the number of discrete points. The results overall reward a recently introduced method as the best performer, which derives the discrete approximation as the numerical solution of a constrained non-linear optimization, preserving the first two moments of the original distribution. This method provides more accurate results than an ad-hoc first-order approximation technique. However, it is the most computationally demanding as well and the computation time can get even larger than that required by Monte Carlo approximation if the number of discrete points exceeds a certain threshold.

On improvements of the order of approximation in the Poisson limit theorem

1996 ◽  
Vol 33 (01) ◽  
pp. 146-155 ◽  
Author(s):  
K. Borovkov ◽  
D. Pfeifer
Keyword(s):  
Limit Theorem ◽  
Random Variables ◽  
Poisson Approximation ◽  
Order Of Approximation ◽  
Operator Semigroups ◽  
Bernoulli Random Variables ◽  
Usual Rate ◽  
Poisson Limit ◽  
Poisson Measures ◽  
Special Case

In this paper we consider improvements in the rate of approximation for the distribution of sums of independent Bernoulli random variables via convolutions of Poisson measures with signed measures of specific type. As a special case, the distribution of the number of records in an i.i.d. sequence of length n is investigated. For this particular example, it is shown that the usual rate of Poisson approximation of O(1/log n) can be lowered to O(1/n 2). The general case is discussed in terms of operator semigroups.

scholarly journals Vines--a new graphical model for dependent random variables

2002 ◽  
Vol 30 (4) ◽  
pp. 1031-1068 ◽  
Author(s):  
Tim Bedford ◽  
Roger M. Cooke
Keyword(s):  
Graphical Model ◽  
Random Variables ◽  
Dependent Random Variables

scholarly journals Association rule hiding using integer linear programming

2021 ◽  
Vol 11 (4) ◽  
pp. 3451
Author(s):  
Suma B. ◽  
Shobha G.
Keyword(s):  
Data Mining ◽  
Linear Programming ◽  
Integer Linear Programming ◽  
Association Rule ◽  
Real Life ◽  
Linear Program ◽  
Integer Linear Program ◽  
Optimization Approach ◽  
Integer Linear Program Formulation ◽  
The Impact

<span>Privacy preserving data mining has become the focus of attention of government statistical agencies and database security research community who are concerned with preventing privacy disclosure during data mining. Repositories of large datasets include sensitive rules that need to be concealed from unauthorized access. Hence, association rule hiding emerged as one of the powerful techniques for hiding sensitive knowledge that exists in data before it is published. In this paper, we present a constraint-based optimization approach for hiding a set of sensitive association rules, using a well-structured integer linear program formulation. The proposed approach reduces the database sanitization problem to an instance of the integer linear programming problem. The solution of the integer linear program determines the transactions that need to be sanitized in order to conceal the sensitive rules while minimizing the impact of sanitization on the non-sensitive rules. We also present a heuristic sanitization algorithm that performs hiding by reducing the support or the confidence of the sensitive rules. The results of the experimental evaluation of the proposed approach on real-life datasets indicate the promising performance of the approach in terms of side effects on the original database.</span>

scholarly journals Distribution of Geometrically Weighted Sum of Bernoulli Random Variables

2011 ◽  
Vol 02 (11) ◽  
pp. 1382-1386 ◽  
Author(s):  
Deepesh Bhati ◽  
Phazamile Kgosi ◽  
Ranganath Narayanacharya Rattihalli
Keyword(s):  
Random Variables ◽  
Weighted Sum ◽  
Bernoulli Random Variables
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