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Causal Concepts and Graphical Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
Traditional statistical regression models for an outcome Y and an exposure X specify some aspect of the conditional distribution p(y|x), or in words: the distribution of Y when we observeX=x $ X=x $ . This describes an association in the sense of how seeing different values of X helps us to predict possibly different (expected) values of Y. A regression-based analysis would typically be accompanied by a warning that ‘association is not causation’. The warning seems necessary so that people do not expect the predicted change in Y to actually materialize when they intervene in a system to manipulate the value of X. In other words, the warning is given so that individuals or policy makers do not base decisions about their actions on a finding that is ‘just associational’. Intervening can be understood as actively doing something to the system instead of passively observing it. To make this explicit, Pearl [49] introduced the so-called do(·) $ \text{ do}(\cdot ) $ -notation defined below.
Probability and Statistics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
The conditional distribution of a certain subset of the random variables is the joint distribution of this subset under the condition that the remaining variables are given certain values. The conditional distribution of X1, X2, . . ., Xp, given Xp+1, Xp+2, ⋯, Xk, is h(x1,…,xp|xp+1,…,xk)=f(x1,x2,…,xk)g(xp+1,xp+2…,xk) $$ h(x_{1} ,~ \ldots ,~x_{p} |x_{{p + 1}} ,~ \ldots ,~x_{k} ) = \frac{{f(x_{1} ,x_{2} , \ldots ,x_{k} )}}{{g(x_{{p + 1}} ,x_{{p + 2}} \ldots ,x_{k} )}} $$
Basic Stochastic Mathematics
Published in Ning Zhang, Chongqing Kang, Ershun Du, Yi Wang, Analytics and Optimization for Renewable Energy Integration, 2019
Ning Zhang, Chongqing Kang, Ershun Du, Yi Wang
For two random variables, the conditional distribution describes the probability distribution of one random variable when the other variable is known to be a particular value. The concept can be extended to multiple random variables.
Data-driven risk-averse newsvendor problems: developing the CVaR criteria and support vector machines
Published in International Journal of Production Research, 2023
The overstock and understock losses are denoted by and , respectively. Because of the use of the feature vector , the probability distribution is replaced by the conditional distribution in the following. The thresholds for the two losses at the probability levels and are represented by where is the probability of the loss not larger than the threshold , and is the probability of the loss not larger than the threshold .
Impact analysis of behavior of front-line managers on employee safety behavior by integrating interpretive structural modeling and Bayesian network
Published in International Journal of Occupational Safety and Ergonomics, 2022
Su-Xia Liu, Hua-Zhong Chen, Qiang Mei, Ying Zhou, Nkrumah Nana Kwame Edmund
The interpretative structural model (ISM) is a simple method that can handle complex relationships among various elements and transform them into an intuitive hierarchical structure [15]. A BN combines sample data with prior knowledge and is an effective tool to achieve knowledge representation and inferred prediction in uncertain situations. It combines graph theory, probability theory, computer science and statistics; it considers the conditional distribution of objective entities and reveals their causal relationship [16]. The integration of the ISM and a BN has demonstrated strong analytical capabilities in some areas. Wu et al. [17] proposed combining the ISM with a BN to assess operational risks in a marine environment. The mixed approach of a BN and the ISM identified a threat assessment model for helping practical policing decisions [18]. In the present study, we have determined the structural relationship between FLM behavior factors using the ISM, and then applied a BN to quantitatively analyze this structural relationship. Such a hybrid approach not only overcomes the limitation imposed by a lack of a theoretical basis of the BN structures, but also makes full use of a BN in explaining the uncertainties.
Correlation-based dynamic sampling for online high dimensional process monitoring
Published in Journal of Quality Technology, 2021
Mohammad Nabhan, Yajun Mei, Jianjun Shi
The novelty of our proposed dynamic sampling method lies in exploiting the spatial correlation structure to provide an upper confidence bound of post-change parameter estimation and is therefore named Correlation-based Dynamic Sampling (CDS). The procedure is dynamic in the sampling process of the variables to be observed at each acquisition period, as well as in providing compensation for the unobserved variables. The dynamic behavior is achieved by combining the correlation structure with the information obtained from the observed partition of the data streams. The dynamic compensation we propose is constructed from the upper confidence bound of the marginal conditional distribution of the unobserved variables given the observed variables. When a well-structured framework such as multivariate normal distribution is assumed, the marginal conditional distribution is very well defined to be another Gaussian distribution. The marginal distribution is tractable even in high dimension when the spatial structure is readily available. This sensor assignment procedure allows for a pseudo-random sampling strategy when the process is in-control, as well as fast localization of faulty variables when the process is out-of-control. We use the term “pseudo-random” here because, although the sampling procedure tends to select a cluster of variables to be observed at any given time based on the spatial structure, the clusters themselves are randomly constructed. Furthermore, these clusters are formed from variables that are correlated. This feature of cluster formation will be illustrated further in the simulation and case studies.