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We also examined the conditional distribution of a special molecular phenotype, called triple-negative, in which ER, PR and NEU were absent. We confirmed that ER is a key marker and we found that it was able to define subpopulations of patients characterized by different conditional independence relations among biomarkers. Oncological patients with similar clinical features could have heterogeneous dynamics of disease recurrence and response to therapies.

Such a behaviour is prevalently associated to biological features [ 1 ]. For this reason, molecular oncology is actually focused on the identification of cancer phenotypes with homogeneous biological profile that could explain the differential response of the disease to the therapies and associable to specific biomarkers. In such a perspective, according to the pharmacogenomic paradigm, biomolecular cancer staging systems have been proposed as alternative to classical systems [ 2 ]. However, the use of "omic" techniques that have initially opened potential developments it has been followed by frustrations and doubts about the real clinical usefulness of results [ 3 ].

Two typical questions involve biomarkers: the current patients state, referred to disease profiling or subtyping and the patients outcome for example, the five-year horizon since surgery , referred to disease dynamics. As regard the disease profiling, a shared interest by biomedical scientists is about the associations existing among the different biomarkers that characterize specific metabolic assets of the disease according to its natural history, genetic and environmental determinants.

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However, such conventional analysis tools could be limiting since binary associations, although related, provide few insights about features of the joint multivariate distribution. Graphical models are aimed to face such a problem and among them, Bayesian Networks BNs have interesting properties that allow to address the above arose questions in a full probabilistic setting.

Indeed, among the leading technologies, BNs are able to describe and derive the conditional independence CI relations existing in highly structured stochastic systems [ 5 ], and references therein. Besides simplifying communication among experts of different fields, BNs support causal reasoning [ 6 ] and the development of efficient algorithms for conditioning and marginalization through exact local computation [ 6 ]. Although the increasing hype surrounding BNs [ 7 ] partially comes from records of success in processing biomedical [ 8 ] and molecular data [ 9 ], Directed Acyclic Graphs DAGs are a valuable tool for reasoning on the design of a study [ 10 ].

To better understand the information provided by biomarkers, in the present paper a probabilistic interpretation of DAGs was applied in a case study previously published by Ambrogi et al. In particular, the possible heterogeneity of CI relationships for certain classes of patients was addressed by learning the structure of MultiDAGs, a generalization of DAGs developed to represent some context-specific conditional independence relationships.

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Finally, to challenge the tool we examined a peculiar breast cancer molecular phenotype, the so-called triple-negative group, in which ER, PR and HER2 are absent. Since, in triple-negative tumors p53 protein appears heterogeneously expressed [ 13 ], suggesting that it may be associated with specific subgroups, we investigated the distribution of p53 conditioned to such a phenotype. The first structural learning task was performed on the full set of molecular profiles made by five biomarkers. Several run of a greedy search one edge change and score and of a simulated annealing algorithms were performed with equivalent sample size N equal to 9.

The selected structures are members of an equivalence class defined by the same set of conditional independence relations, a statement derived from the application of the Directed Markov Factorization DMF theorem. Variable ER separates all other pairs of variables, therefore pairs of variables separated by ER are conditionally independent given ER. The BDe log-score is equal to Learned structures without AGE. Belief bars represent estimated marginal probabiliti values at each node.

A more compact representation would be obtained by removing edge orientation from all arrows, although this would hide the different causal information carried by the above DAGs even if equivalent as regards CI relationships. In Figure 1 e , one of the learned BNs is displayed using belief bars representing marginal probability values at each node after parameter learning based on the instances of the case study.

From the learned structures we judged that pairs of variables are typically correlated despite the fact that conditioning on ER make them independent. This is a reasonable but not trivial finding, because the usual practice of inspecting the empirical distribution of pairs of variables is not prone to reveal such CI relations. It is well known that the metabolism of oestrogens and of progesterone is related to patient's AGE, even though AGE might also affect other markers as regards their conditional distribution. Therefore, the second structural learning task was performed on the full set of molecular profiles composed of five biomarkers including variable AGE among nodes in V.

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Several runs of greedy search and simulated annealing algorithms were performed with equivalent sample size equal to 9 and a uniform prior distribution for Z. The inspection of top scored networks allowed to find the equivalence class of structures and to inspect CI relationships. The DBe log-score is equal to It was chosen by exploiting the a-priori information of molecular pathologists.

All the other members not shown of the equivalence class are characterized by an arrow from ER to AGE. While such edge orientation is probabilistically sound it is counterintuitive on causal ground because AGE is likely to cause changes in the distribution of five markers and not vice-versa probably due to physiological variation in the biologic behaviour associated with aging see the discussion. Learning with AGE.

Naive Bayes 2: Conditional Independence

The role played by age on estrogen receptors is well documented in the literature, therefore it is very interesting to investigate about the presence of heterogeneous CI relations within groups of patients belonging to different age classes. In other words, CI relationships among markers could be different within the three age classes. Therefore, we performed an exhaustive search by scoring structures under the BDe metric for each age class separately. The equivalent sample size N was always equal to 9.

Equivalence class of structures within patients with AGE equal to High.

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Node labels are shown only in DAG a. Variable ER have been found to play a key-role in making all the other pairs of variables conditional independent. Moreover, from a theoretical point of view, the absence of estrogen receptors could modify the pattern of association among other biomarkers. Given ER, CI holds in general among biomarkers but with some exceptions.

DAGs belonging to equivalence class of structures for each value of the distinguished variable ER. The log-BF is greater than 6. This aggressiveness is best illustrated by the fact that the peak risk of recurrence is between the first and third years of follow up and the majority of deaths occur in the first five years following therapy [ 16 ]. From a biological point of view, salient features of triple-negative breast cancers include overexpression of EGFR and c KIT, high proliferative rates, frequent genomic alterations, phenotypic similarity to BRCA1-associated cancers and frequent mutations of Tp53 [ 14 , 15 ].

In particular, p53 appears heterogeneously expressed, suggesting that it may be associated with specific subgroups. Therefore, we examined the p53 marginal distribution conditioned on the subset of triple-negative cancers included in the case series analyzed by Ambrogi et al.

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It is of some interest to show belief bars representing marginal distributions at each node calculated from the point estimates of :. The multiDAG with ER as distinguished variable is the best structure to be exploited while obtaining the above mentioned point estimates. The marginal distribution of p53 is for about one half on the null value 0. The probability of a null value for P53 decreases to 0. Heterogeneous patients dynamics might depend on such pretty much substantial difference which is worth to be further characterized. A point estimate of typically neglects the uncertainty left after learning.

The more disperse curve on the left dashed line refers to the prior belief for such event before learning structure and parameters when marginal independence is assumed among biomarkers, thus P53 is disconnected. Finally, the continuous line in the middle is the final marginal distribution obtained after structural and parameter learning given a triplo-negative profile under a multiDAG ER model. The comparison reveals that the initial prior distribution is quite disperse while posterior distributions based on two different DAGs substantially differ.

Probabilistic Conditional Independence Structures (Hardcover, 2005 ed.)

The Bayes factor is strongly in favour of the learned multiDAG structure therefore we can also estimate the bias induced by a poorly supported model, like a marginal independence model as the difference between point estimates under the two mentioned models. Probability Distribution of P New Releases. Description Probabilistic Conditional Independence Structures provides the mathematical description of probabilistic conditional independence structures; the author uses non-graphical methods of their description, and takes an algebraic approach.

The monograph presents the methods of structural imsets and supermodular functions, and deals with independence implication and equivalence of structural imsets. Motivation, mathematical foundations and areas of application are included, and a rough overview of graphical methods is also given. In particular, the author has been careful to use suitable terminology, and presents the work so that it will be understood by both statisticians, and by researchers in artificial intelligence. The necessary elementary mathematical notions are recalled in an appendix.

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Probabilistic Conditional Independence Structures (Hardcover, 2005 ed.)

Back cover copy Conditional independence is a topic that lies between statistics and artificial intelligence. Probabilistic Conditional Independence Structures provides the mathematical description of probabilistic conditional independence structures; the author uses non-graphical methods of their description, and takes an algebraic approach. Probabilistic Conditional Independence Structures will be a valuable new addition to the literature, and will interest applied mathematicians, statisticians, informaticians, computer scientists and probabilists with an interest in artificial intelligence.

The book may also interest pure mathematicians as open problems are included. Table of contents Basic Concepts. Review Text From the reviews: "This monograph aims to present methods of structural imsets and supermodel functions and considers the independence implication and equivalence of structural imsets. Graphical Methods. Structural Imsets: Fundamentals. Description of Probabilistic Models.