Pandora's Box with Correlations: Learning and Approximation

In 1979, Weitzman introduced Pandora's box problem as a framework for sequential search with costly inspections. Recently, there has been a surge of interest in Pandora's box problem, particularly among researchers working at the intersection of economics and computation. This survey provides an overview of the recent literature on Pandora's box problem, including its latest extensions and applications in areas such as market design, decision theory, and machine learning.

Pandora's Box Problem with Sequential Inspections

Copulas have become an important analytic tool for characterizing multivariate distributions and dependence. One is often interested in simulating data from copula estimates. The process can be analytically and computationally complex and usually involves steps that are unique to a given parametric copula. We describe an alternative approach that uses \probability{proportional{to{size" (PPS) random sampling with weights formed from the copula likelihood function. The method is flexible and can be applied to parametric and nonparametric marginal density estimates. The precision of the simulation can be calibrated by adjusting the density of the multidimensional grid used in the simulation process. The approach is fully transparent to any copula function with continuous random variables. An example evaluates a number of goodness-of-fit criteria and provides strong support for the validity and practicality of the method.

We consider in this paper the convergence of abstract random variables taken in the sense of definition (1). If the random variables converge in some classical sense or in the extended sense of Frechet and Doss, then they will also converge in the sense of definition (1). The classical relations between different types of convergence, generally speaking, are preserved. Furthermore, for a single parameter system of abstract random variables the joint distributions are defined (distributions on cylinder sets) and it is shown that convergence of the abstract random variables implies convergence of the corresponding joint distributions, andd for abstract random variables with values from a separable, locally compact space, the collection of joint distributions is fully defined by the probability measure on this space. In conclusion, a classification of abstract random variables is carried out and certain special results are obtained.

Capacity is an important property for QoS support in Mobile Ad Hoc Networks (MANETs) and has recently attracted considerable attention in research. However, most approaches rely on simplified models (infinite network, isotropic radio propagation, unidirectional links, perfect scheduling, etc.) and provide only asymptotic bounds or integer linear programming equations. In this paper we take a probabilistic approach and model capacity as a random variable that depends on node distribution, communication pattern, radio propagation and channel assignment. Expected values for the random variable at a given point within the parameter space are then computed using Monte-Carlo simulation. The modularity of the approach allows for capacity analysis under more realistic network models. We demonstrate the potential of such a model by showing that for acknowledgement-based networks throughput capacity increases in the presence of shadowing, even above the critical node density.

The movements of the prostate relative to the pelvic lymph nodes during intensity-modulated radiation therapy treatment can limit margin reduction and affect the protection of the organs at risk (OAR). In this study, the authors performed an analysis of three adaptive treatment strategies that combine information from both bony and gold marker registrations. The robustness of those treatments against the interfraction prostate movements was evaluated. A retrospective study was conducted on five prostate cancer patients with 7-13 daily cone-beam CTs (CBCTs). The clinical target volumes (CTVs) consisting of pelvic lymph nodes, prostate, and seminal vesicles as well as the OARs were delineated on each CBCT and the initial CT. Three adaptive strategies were analyzed. Two of these methods relied on a two-step patient positioning at each fraction. First step: a bony registration was used to deliver the nodal CTV prescription. Second step: a gold marker registration was then used either to (1) complete the dose delivered to the prostate (complement); (2) or give almost the entire prescription to the prostate with a weak dose gradient between the targets to compensate for possible motions (gradient). The third method (COR) used a pool of precalculated plans based on images acquired at previous treatment fractions. At each new fraction, a plan is selected from that pool based on the daily position of prostate center-of-mass. The dosimetric comparison was conducted and results are presented with and without the systematic shift in the prostate position on the CT planning. The adaptive strategies were compared to the current clinical standard where all fractions are treated with the initial nonadaptive plan. The minimum daily prostate D95% is improved by 2%, 9%, and 6% for the complement, the gradient, and the COR approaches, respectively, compared to the nonadaptive method. The average nodal CTV D95% remains constant across the strategies, except for the gradient approach where a reduction of 7% is observed. However, a correction of the systematic shift reduced the problem, and the adaptive strategies remain robust against the prostate movement across the fraction. The bladder V55Gy is reduced by 35% on average for the adaptive strategies. Because they offer increased CTV coverage and OAR sparing, adaptive methods may be suitable candidates for simple and efficient adaptive treatment strategies for prostate cancer. Margin reduction and systematic error correction in the prostate position improve the protection of the OAR and the dose coverage. A cumulative dose to simulate a complete treatment would show real effects and allow a better comparison between each method.

We describe efficient constructions of small probability spaces that approximate the joint distribution of general random variables. Previous work on efficient constructions concentrate on approximations of the joint distribution for the special case of identical, uniformly distributed random variables. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 1–16, 1998

Opinions about the value of biological control are often extreme. Colloquially, biological control most often refers to classical biological control, in which one species is introduced from another region to control pests such as arthropod herbivores in agricultural systems, or weeds in managed and natural systems.1 As such, biological control has the potential to be a low-cost, chemical free, means to control pests. Numerous biological control programs have been unqualified successes (Bellows 2001), such as the control of cacti in Australia with the moth Cactoblastis cactorum (Raghu and Walton 2007), of cottony-cushion scale (Icerya purchasi) in California with the vedalia lady beetle, Rodolia cardinalis (Caltagirone and Doutt 1989), and of glassy-winged sharpshooters in French Polynesia with the egg parasitoid Gonatocerus ashmeadi (Grandgirard et al. 2009). Yet, classical biological control, as with any introduction of a species into a new area, necessarily involves the unknown and therefore carries some inherent risk (Simberloff and Stiling 1996) – what will these organisms actually do in a novel ecosystem? The most unpredictable element in biological control is the extent to which the realized niche is modified in the new environment. This effect has been responsible for some disastrous outcomes of classical biological control, many of which occurred during an era when vertebrates were being introduced around the world by Europeans for a variety of reasons (e.g., introducing the birds of Shakespeare to America, Mirsky 2008), including for biological control (Howarth 1991). The introductions as biological control agents of cane toad to Australia (Crossland et al. 2000) and mongoose to Hawaii (Hays and Conant 2007) are notorious. Introductions of generalist invertebrate agents also have had dire consequences, such as the introduction of predatory snails to French Polynesia (Murray et al. 1988; Coote 2007). In retrospect, some of the unintended consequences of biological control could have been avoided with more ecological knowledge (McEvoy and Coombs 2000) or more societal appreciation for native species (which has developed with time, Henneman and Memmott 2001), but with other introductions, it would have been impossible to know ahead of time what the risks would be (e.g., gall fly agents of knapweeds providing supplementary food to mice that harbor hantavirus, Pearson and Callaway 2006). Many of the unknown outcomes of biological control are purely ecological – what is the risk that a wasp, introduced to parasitize an agricultural pest, will also be able to feed on a native insect? Other unknowns involve evolution – will a herbivore adapt over time to be able to feed on a new nontarget host or hybridize with a closely related species? This volume explores the evolutionary aspects of biological control. Although often overlooked, evolutionary considerations are critical to all stages of classical biological control, from agent selection, to quarantine, release, establishment, and ultimately success in pest control (Ehler et al. 2004). Many questions are unresolved. For example, should agents be chosen that have a long history with the host or are ‘new associations’ more likely to succeed (Hokkanen and Pimentel 1989)? Can one improve effectiveness through artificial selection (Hopper et al. 1993)? Will postcolonization adaptation of the agent increase the likelihood of success, and/or are hosts equally likely to evolve resistance over time (Roderick 1992; Holt and Hochberg 1997; Hufbauer 2001)? Are generalist consumers more likely to survive in novel environments or are specialists more effective (Murdoch et al. 1985; Waage 1990; Brodeur 2012)? More recently, concern for the environment, as well as theory examining the reasons for success of generalist predators, prompted a shift to the release of specialized consumers typically preceded by extensive testing aimed at delimiting the host range of candidate biological control agents. While this approach has clearly made biological control more predictive ecologically, research focused on host range currently lacks measures of genetic variation in host use and responses of those hosts, and thus evolutionary uncertainties remain.

Many random variables occurring in nature are circular random variables, i.e., its probability density function has period 2 π and its support is the unit circle. The support of a linear random variable is a subset of the real line. When one is interested in the relation between a circular random variable and a linear random variable it is necessary to construct their joint distribution. The support of the joint distribution of a circular and a linear random variable is a cylinder. In this paper, we use copulas and circular distributions based on non‐negative trigonometric sums to construct the joint distribution of a circular and a linear random variable. As an application of the proposed methodology the hourly quantile curves of ground‐level ozone concentration for a monitoring station in Mexico City are analyzed. In this case the circular random variable has a uniform distribution if we have the same number of observations in each hour during the day and, the linear random variable is the ground‐level ozone concentration.

A probabilistic graphical model approach to stochastic multiscale partial differential equations

There are various definitions of mutual information. Essentially, these definitions can be divided into two classes: (1) definitions with random variables and (2) definitions with ensembles. However, there are some mathematical flaws in these definitions. For instance, Class 1 definitions either neglect the probability spaces or assume the two random variables have the same probability space. Class 2 definitions redefine marginal probabilities from the joint probabilities. In fact, the marginal probabilities are given from the ensembles and should not be redefined from the joint probabilities. Both Class 1 and Class 2 definitions assume a joint distribution exists. Yet, they all ignore an important fact that the joint or the joint probability measure is not unique. In this paper, we first present a new unified definition of mutual information to cover all the various definitions and to fix their mathematical flaws. Our idea is to define the joint distribution of two random variables by taking the marginal probabilities into consideration. Next, we establish some properties of the newly defined mutual information. We then propose a method to calculate mutual information in machine learning. Finally, we apply our newly defined mutual information to credit scoring.

A methodology to solve the Order Batching Problem

Opposition-based Magnetic Optimization Algorithm with parameter adaptation strategy

For reliability analysis, there may be several potential distributions for a random variable due to limited samples available. For the same reason, a distribution may not be available. Simply assuming a normal distribution may result in a large error for the reliability prediction. Moment-based methods use only moments of random variables for reliability analysis and can effectively address the problem of multiple distributions or lack of distributions. The existing moment-based methods, however, may produce large errors or may result in instability in the analysis process. This study extends the high-moment method for higher accuracy of the reliability prediction by including the parameters ignored by the existing high-moment method. The proposed method derives the reliability index from the first four moments of random input variables based on the statistical properties of the standard normal random variable. Compared with the existing method, the proposed method is more accurate and stable for problems for which the distributions of input random variables are unknown. Numerical examples show the improved accuracy from the proposed method.

Conventional Particle Swarm Optimization (PSO) algorithms often suffer premature convergences. Hybrid algorithms, for instance, the Simulated Annealing-based PSO, present low convergence speeds. In this paper, we develop an Adaptive Particle Swarm Optimization (APSO) algorithm to solve unconstrained global optimization problems with highly multimodal functions, in which two adaptive strategies (including an adaptive inertia weight strategy with hybrid time-varying dynamics and an adaptive random mutation strategy) are merged into the basic PSO algorithm to guarantee the algorithm performance. The proposed algorithm is numerically verified by fifteen classical multimodal functions which include Ackley, Bukin f6, Cross-in-Tray, Drop-Wave, Eggholder, Griewank, Holder Table, Langermann, Levy, Levy f13, Rastrigin, Schaffer f2, Schaffer f4, Schwefel, and Shubert. Numerical experiments demonstrate that the proposed algorithm has a potential to achieve better solutions with acceptable computational time, especially for high-dimensional optimization problems.