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Search for Causal Models
Published in Marloes Maathuis, Mathias Drton, Steffen Lauritzen, Martin Wainwright, Handbook of Graphical Models, 2018
In constraint-based searches, under the Causal Markov and Causal Faithfulness Assumptions, the constraints on the DAG are that all and only the conditional independence relations that hold in the population are entailed (via d-separation) by the true causal DAG. In practice, whether a conditional independence relation holds in the population or not is judged by performing a statistical test of conditional independence. However, the number of conditional independence and dependence relations entailed by a DAG (under the Causal Markov and Causal Faithfulness Assumptions) grows exponentially with the number of variables in the DAG, which presents a computational problem. It also raises a statistical problem since it is generally the case that if all of the conditional independence relations are tested on finite data sets, statistical tests will make some errors. If the constraints on the DAG are taken to be that the true causal DAG entails all and only the conditional independence relations that pass a statistical test on finite data, then given the number of such constraints, it is highly probable that no DAG satisfies all of the constraints. Various solutions to this problem are discussed in more detail in Section 1.5.1.2.
Swarm Intelligence and Machine Learning Algorithms for Cancer Diagnosis
Published in Shikha Agrawal, Manish Gupta, Jitendra Agrawal, Dac-Nhuong Le, Kamlesh Kumar Gupta, Swarm Intelligence and Machine Learning, 2022
Pankaj Sharma, Vinay Jain, Mukul Tailang
Bayesian Networks are pictorial constructions that permit an undetermined subject to be represented and reasoned about. BNs are a mix of probability theory and pattern recognition. By itemising a collection of conditional suppositions of independence and also a collection of likelihood function; BNs describe the probabilistic model regulating a number of indicators. The network’s network comprises discrete or continuous parameters, with arcs illustrating their interdependencies. In Bayesian Networks [22], conditional independence is a crucial concept.
Brain Connectivity Assessed with Functional MRI
Published in Troy Farncombe, Krzysztof Iniewski, Medical Imaging, 2017
Aiping Liu, Junning Li, Martin J. McKeown, Z. Jane Wang
Instead of simply referring to different brain regions that are covarying, partial correlation can be employed to estimate if one brain region has direct influence over another [29], as it measures the normalized correlation with the effect of all other variables removed. The application of partial correlation in inferring the relationship between two variables is based on the conditional-independence test. The definition of conditional independence is as follows: X and Y are conditional-independent given Z if and only if P(XY|Z) = P(X|Z) P(Y|Z). It is similar to the pair-wise independence definition P(XY) = P(X) P(Y), but conditional on a third variable Z. Note that pair-wise independence does not imply conditional independence, and vice versa (Figure 21.2). For instance, the activities of two brain regions A and B are commonly driven by that of a third region C, then the activities of A and B maybe correlated in pair-wise fashion, but if the influence from C is removed, then their activities will become independent, as shown in Figure 2.12b. On the other hand, if the activities of two brain regions are correlated even after all possible influence from other regions are removed (as shown in Figure 21.2c), then very likely there is a direct connection between them (i.e., the two regions are conditionally dependent). Therefore, the conditional dependence implies that two brain regions are directly connected. More importantly, conditional independence is a key concept in multivariate analyses such as graphical modeling, where two nodes are connected if and only if the corresponding variables are not conditionally independent, which we will discussed in the next section.
Multi-scale geotechnical features of dredger fills and subsidence risk evaluation in reclaimed land using BN
Published in Marine Georesources & Geotechnology, 2020
Linbo Wu, Jianxiu Wang, Jie Zhou, Tianliang Yang, Xuexin Yan, Yu Zhao, Zhenhua Ye, Na Xu
Based on the conditional independence and the chain rule (Pearl 1988), the joint probability distribution of a set of random variables U= {B1,…,Bn} can be defined in BN as follows: where P(U) is the joint probability distribution of U, and Pa(Bi) is the parent set of variable Bi. Probability of variable Bi, i.e., P(Bi) can be calculated as: where U\Bi means the summation is done using all the variables in U except Bi. For a basic BN acyclic graph constituted of three variables B1, B2, B3, information flow in BN can be modeled based on the following rules: (1) information may flow through a serial (B1→B2→B3)/diverging (B1←B2→B3) connection unless the evidence for the intermediate variable B2 is given; (2) information may flow through a converging (B1→B2←B3) connection whenever the state of the intermediate variable B2 or one of its descendants is given (Sagrado et al. 2016). BN determines the relevant variables to a given target variable by using these rules.
Fuzzy belief propagation in constrained Bayesian networks with application to maintenance decisions
Published in International Journal of Production Research, 2020
Ke Wang, Yan Yang, Jian Zhou, Mark Goh
Recall that the joint probability distribution of the variables can be explored compactly using the chain rule and the conditional independence assumption (Heckerman, Geiger, and Chickering 1995): where represents the set of parent nodes of . Meanwhile, the marginal probability for , can be represented as Based on Equations (10)–(12), the posterior probabilities in Model (7) can be represented by the parameters given in the FBN. The model can be solved using some well-developed optimisation software.
Analysis of Repeatability and Reproducibility Studies With Ordinal Measurements
Published in Technometrics, 2018
Stacey L. Culp, Kenneth J. Ryan, Juan Chen, Michael S. Hamada
To fit model (1), de Mast and van Wieringen (2010) transform the response {Yijk} into {Rijh} such that Rijh = |{k|Yijk = h}|, that is, the number of repetitions out of K for which the jth rater assigns category h to part i. Note that ∑Hh = 1qj(h|x) = 1, so if Rij = (Rij1, …, RijH), then given the assumption of conditional independence. If the same arbitrary scalar is added to the Xi and the cut points , the multinomial probabilities defined in (1) do not change. That is, the model is unidentifiable. This identifiability problem is circumvented by assuming the latent variables are a random sample from the standard normal distribution, that is, Assuming complete data Rijh = rijh and Xi = xi are observed for all i = 1, …, I, j = 1, …, J, and h = 1, …, H, the resulting likelihood is where φ( · ) is the probability density function of the standard normal distribution. de Mast and van Wieringen (2010) used maximum likelihood estimation for parameters αj and δjm by integrating out the latent variables Xi with numerical integration.