scholarly journals Bound Propagation

2003 ◽  
Vol 19 ◽  
pp. 139-154 ◽  
Author(s):  
M. Leisink ◽  
B. Kappen

In this article we present an algorithm to compute bounds on the marginals of a graphical model. For several small clusters of nodes upper and lower bounds on the marginal values are computed independently of the rest of the network. The range of allowed probability distributions over the surrounding nodes is restricted using earlier computed bounds. As we will show, this can be considered as a set of constraints in a linear programming problem of which the objective function is the marginal probability of the center nodes. In this way knowledge about the maginals of neighbouring clusters is passed to other clusters thereby tightening the bounds on their marginals. We show that sharp bounds can be obtained for undirected and directed graphs that are used for practical applications, but for which exact computations are infeasible.

Author(s):  
Alfred Galichon

This chapter considers the finite-dimensional case, which is the case when the marginal probability distributions are discrete with finite support. In this case, the Monge–Kantorovich problem becomes a finite-dimensional linear programming problem; the primal and the dual solutions are related by complementary slackness, which is interpreted in terms of stability. The solutions can be conveniently computed by linear programming solvers, and the chapter shows how this is done using some matrix algebra and Gurobi.


Mathematics ◽  
2020 ◽  
Vol 8 (11) ◽  
pp. 2044
Author(s):  
Florin Sofonea ◽  
Ioan Ţincu

This paper is concerned with the orthogonal polynomials. Upper and lower bounds of Legendre polynomials are obtained. Furthermore, entropies associated with discrete probability distributions is a topic considered in this paper. Bounds of the entropies which improve some previously known results are obtained in terms of inequalities. In order to illustrate the results obtained in this paper and to compare them with other results from the literature some graphs are provided.


2018 ◽  
Vol 34 (S1) ◽  
pp. 20-20
Author(s):  
Conor Teljeur ◽  
Patricia Harrington ◽  
Máirín Ryan

Introduction:Fully probabilistic analyses are now standard for economic models, with all parameters varied according to probability distributions. Using univariate sensitivity analyses to explore the influence of different parameters on the model results are also standard. Although there are several approaches available, there has been little discussion of the merits of each or justification for the method used in any given analysis. The aim of this study was to compare three approaches to univariate sensitivity analysis using a case study.Methods:We considered three univariate sensitivity analysis approaches: (i) set one parameter at its upper and lower bounds while all others are set at their mean value; (ii) analysis of variance; and (iii) set one parameter at its mean and vary all others. We compared these approaches using an economic model of mechanical thrombectomy for the treatment of acute ischemic stroke, considering outcomes of incremental costs, incremental quality-adjusted life-years (QALYs), and net monetary benefit (NMB).Results:For incremental costs and QALYs the correlation between the approaches was moderate to high, with correlation coefficients between 0.46 and 0.94. For NMB the correlation between approaches was also high (range 0.89 to 0.98), but some of the most influential parameters were ranked differently. Setting one parameter at its upper and lower bounds was the only method that facilitated an analysis of direction of influence.Conclusions:The three approaches addressed different but relevant questions. Setting individual parameters at their bounds is effectively a systematic scenario analysis and may be misleading to decision makers. Analysis of variance may be more easily interpreted, but it has disadvantages. Setting a parameter at its mean, while varying other parameters, is similar to value of information analysis. As with any sensitivity analysis, it is imperative that the uncertainty associated with each parameter is adequately captured in the model.


1972 ◽  
Vol 6 (3) ◽  
pp. 479-482 ◽  
Author(s):  
Donald A. Mcquarrie ◽  
Ronald T. Jamieson ◽  
Mitchel Shen

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Jia-Bao Liu ◽  
Jing Zhao ◽  
Zhi-Yu Shi ◽  
Jinde Cao ◽  
Fuad E. Alsaadi

The hypercube Qn is one of the most admirable and efficient interconnection network due to its excellent performance for some practical applications. The Kirchhoff index KfG is equal to the sum of resistance distances between any pairs of vertices in networks. In this paper, we deduce some bounds with respect to Kirchhoff index of hypercube network Qn.


2006 ◽  
Vol 16 (04) ◽  
pp. 333-343
Author(s):  
MOSHE DROR ◽  
YUSIN LEE ◽  
JAMES B. ORLIN ◽  
VALENTIN POLISHCHUK

This paper introduces a new notion related to the traveling salesperson problem (TSP) — the notion of the TSP ratio. The TSP ratio of a TSP instance I is the sum of the marginal values of the nodes of I divided by the length of the optimal TSP tour on I, where the marginal value of a node i ∈ I is the difference between the length of the optimal tour on I and the length of the optimal tour on I\i. We consider the problem of establishing exact upper and lower bounds on the TSP ratio. To our knowledge, this problem has not been studied previously. We present a number of cases for which the ratio is never greater than 1. We establish a tight upper bound of 2 on the TSP ratio of any metric TSP. For the TSP on six nodes, we determine the maximum ratio of 1.5 in general, 1.2 for the case of metric TSP, and 10/9 for the geometric TSP in the L1 metric. We also compute the TSP ratio experimentally for a large number of random TSP instances on small number of points.


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