Conjunctive Bayesian Networks Research Articles

Latent conjunctive Bayesian network: Unify attribute hierarchy and Bayesian network for cognitive diagnosis

Latent conjunctive Bayesian network: Unify attribute hierarchy and Bayesian network for cognitive diagnosis

Discrete max-linear Bayesian networks

Discrete max-linear Bayesian networks are directed graphical models specified by the same recursive structural equations as max-linear models but with discrete innovations. When all of the random variables in the model are binary, these models are isomorphic to the conjunctive Bayesian network (CBN) models of Beerenwinkel, Eriksson, and Sturmfels. Many of the techniques used to study CBN models can be extended to discrete max-linear models and similar results can be obtained. In particular, we extend the fact that CBN models are toric varieties after linear change of coordinates to all discrete max-linear models.

Comparing mutational pathways to lopinavir resistance in HIV-1 subtypes B versus C.

Although combination antiretroviral therapies seem to be effective at controlling HIV-1 infections regardless of the viral subtype, there is increasing evidence for subtype-specific drug resistance mutations. The order and rates at which resistance mutations accumulate in different subtypes also remain poorly understood. Most of this knowledge is derived from studies of subtype B genotypes, despite not being the most abundant subtype worldwide. Here, we present a methodology for the comparison of mutational networks in different HIV-1 subtypes, based on Hidden Conjunctive Bayesian Networks (H-CBN), a probabilistic model for inferring mutational networks from cross-sectional genotype data. We introduce a Monte Carlo sampling scheme for learning H-CBN models for a larger number of resistance mutations and develop a statistical test to assess differences in the inferred mutational networks between two groups. We apply this method to infer the temporal progression of mutations conferring resistance to the protease inhibitor lopinavir in a large cross-sectional cohort of HIV-1 subtype C genotypes from South Africa, as well as to a data set of subtype B genotypes obtained from the Stanford HIV Drug Resistance Database and the Swiss HIV Cohort Study. We find strong support for different initial mutational events in the protease, namely at residue 46 in subtype B and at residue 82 in subtype C. The inferred mutational networks for subtype B versus C are significantly different sharing only five constraints on the order of accumulating mutations with mutation at residue 54 as the parental event. The results also suggest that mutations can accumulate along various alternative paths within subtypes, as opposed to a unique total temporal ordering. Beyond HIV drug resistance, the statistical methodology is applicable more generally for the comparison of inferred mutational networks between any two groups.

Estimating the predictability of cancer evolution.

MotivationHow predictable is the evolution of cancer? This fundamental question is of immense relevance for the diagnosis, prognosis and treatment of cancer. Evolutionary biologists have approached the question of predictability based on the underlying fitness landscape. However, empirical fitness landscapes of tumor cells are impossible to determine in vivo. Thus, in order to quantify the predictability of cancer evolution, alternative approaches are required that circumvent the need for fitness landscapes.ResultsWe developed a computational method based on conjunctive Bayesian networks (CBNs) to quantify the predictability of cancer evolution directly from mutational data, without the need for measuring or estimating fitness. Using simulated data derived from >200 different fitness landscapes, we show that our CBN-based notion of evolutionary predictability strongly correlates with the classical notion of predictability based on fitness landscapes under the strong selection weak mutation assumption. The statistical framework enables robust and scalable quantification of evolutionary predictability. We applied our approach to driver mutation data from the TCGA and the MSK-IMPACT clinical cohorts to systematically compare the predictability of 15 different cancer types. We found that cancer evolution is remarkably predictable as only a small fraction of evolutionary trajectories are feasible during cancer progression.Availability and implementation https://github.com/cbg-ethz/predictability\\_of\\_cancer\\_evolution Supplementary information Supplementary data are available at Bioinformatics online.

Large-scale inference of conjunctive Bayesian networks.

The continuous time conjunctive Bayesian network (CT-CBN) is a graphical model for analyzing the waiting time process of the accumulation of genetic changes (mutations). CT-CBN models have been successfully used in several biological applications such as HIV drug resistance development and genetic progression of cancer. However, current approaches for parameter estimation and network structure learning of CBNs can only deal with a small number of mutations (<20). Here, we address this limitation by presenting an efficient and accurate approximate inference algorithm using a Monte Carlo expectation-maximization algorithm based on importance sampling. The new method can now be used for a large number of mutations, up to one thousand, an increase by two orders of magnitude. In simulation studies, we present the accuracy as well as the running time efficiency of the new inference method and compare it with a MLE method, expectation-maximization, and discrete time CBN model, i.e. a first-order approximation of the CT-CBN model. We also study the application of the new model on HIV drug resistance datasets for the combination therapy with zidovudine plus lamivudine (AZT + 3TC) as well as under no treatment, both extracted from the Swiss HIV Cohort Study database. The proposed method is implemented as an R package available at https://github.com/cbg-ethz/MC-CBN CONTACT: niko.beerenwinkel@bsse.ethz.ch Supplementary data are available at Bioinformatics online.

Estimating the dynamics and dependencies of accumulating mutations with applications to HIV drug resistance.

We introduce a new model called the observed time conjunctive Bayesian network (OT-CBN) that describes the accumulation of genetic events (mutations) under partial temporal ordering constraints. Unlike other CBN models, the OT-CBN model uses sampling time points of genotypes in addition to genotypes themselves to estimate model parameters. We developed an expectation-maximization algorithm to obtain approximate maximum likelihood estimates by accounting for this additional information. In a simulation study, we show that the OT-CBN model outperforms the continuous time CBN (CT-CBN) (Beerenwinkel and Sullivant, 2009. Markov models for accumulating mutations. Biometrika 96: (3), 645-661), which does not take into account individual sampling times for parameter estimation. We also show superiority of the OT-CBN model on several datasets of HIV drug resistance mutations extracted from the Swiss HIV Cohort Study database.

Identifying restrictions in the order of accumulation of mutations during tumor progression: effects of passengers, evolutionary models, and sampling.

BackgroundCancer progression is caused by the sequential accumulation of mutations, but not all orders of accumulation are equally likely. When the fixation of some mutations depends on the presence of previous ones, identifying restrictions in the order of accumulation of mutations can lead to the discovery of therapeutic targets and diagnostic markers. The purpose of this study is to conduct a comprehensive comparison of the performance of all available methods to identify these restrictions from cross-sectional data. I used simulated data sets (where the true restrictions are known) but, in contrast to previous work, I embedded restrictions within evolutionary models of tumor progression that included passengers (mutations not responsible for the development of cancer, known to be very common). This allowed me to assess, for the first time, the effects of having to filter out passengers, of sampling schemes (when, how, and how many samples), and of deviations from order restrictions.ResultsPoor choices of method, filtering, and sampling lead to large errors in all performance measures. Having to filter passengers lead to decreased performance, especially because true restrictions were missed. Overall, the best method for identifying order restrictions were Oncogenetic Trees, a fast and easy to use method that, although unable to recover dependencies of mutations on more than one mutation, showed good performance in most scenarios, superior to Conjunctive Bayesian Networks and Progression Networks. Single cell sampling provided no advantage, but sampling in the final stages of the disease vs. sampling at different stages had severe effects. Evolutionary model and deviations from order restrictions had major, and sometimes counterintuitive, interactions with other factors that affected performance.ConclusionsThis paper provides practical recommendations for using these methods with experimental data. It also identifies key areas of future methodological work and, in particular, it shows that it is both possible and necessary to embed assumptions about order restrictions and the nature of driver status within evolutionary models of cancer progression to evaluate the performance of inferential approaches.Electronic supplementary materialThe online version of this article (doi:10.1186/s12859-015-0466-7) contains supplementary material, which is available to authorized users.

The Individualized Genetic Barrier Predicts Treatment Response in a Large Cohort of HIV-1 Infected Patients

The success of combination antiretroviral therapy is limited by the evolutionary escape dynamics of HIV-1. We used Isotonic Conjunctive Bayesian Networks (I-CBNs), a class of probabilistic graphical models, to describe this process. We employed partial order constraints among viral resistance mutations, which give rise to a limited set of mutational pathways, and we modeled phenotypic drug resistance as monotonically increasing along any escape pathway. Using this model, the individualized genetic barrier (IGB) to each drug is derived as the probability of the virus not acquiring additional mutations that confer resistance. Drug-specific IGBs were combined to obtain the IGB to an entire regimen, which quantifies the virus' genetic potential for developing drug resistance under combination therapy. The IGB was tested as a predictor of therapeutic outcome using between 2,185 and 2,631 treatment change episodes of subtype B infected patients from the Swiss HIV Cohort Study Database, a large observational cohort. Using logistic regression, significant univariate predictors included most of the 18 drugs and single-drug IGBs, the IGB to the entire regimen, the expert rules-based genotypic susceptibility score (GSS), several individual mutations, and the peak viral load before treatment change. In the multivariate analysis, the only genotype-derived variables that remained significantly associated with virological success were GSS and, with 10-fold stronger association, IGB to regimen. When predicting suppression of viral load below 400 cps/ml, IGB outperformed GSS and also improved GSS-containing predictors significantly, but the difference was not significant for suppression below 50 cps/ml. Thus, the IGB to regimen is a novel data-derived predictor of treatment outcome that has potential to improve the interpretation of genotypic drug resistance tests.

Learning Oncogenetic Networks by Reducing to Mixed Integer Linear Programming

Cancer can be a result of accumulation of different types of genetic mutations such as copy number aberrations. The data from tumors are cross-sectional and do not contain the temporal order of the genetic events. Finding the order in which the genetic events have occurred and progression pathways are of vital importance in understanding the disease. In order to model cancer progression, we propose Progression Networks, a special case of Bayesian networks, that are tailored to model disease progression. Progression networks have similarities with Conjunctive Bayesian Networks (CBNs) [1],a variation of Bayesian networks also proposed for modeling disease progression. We also describe a learning algorithm for learning Bayesian networks in general and progression networks in particular. We reduce the hard problem of learning the Bayesian and progression networks to Mixed Integer Linear Programming (MILP). MILP is a Non-deterministic Polynomial-time complete (NP-complete) problem for which very good heuristics exists. We tested our algorithm on synthetic and real cytogenetic data from renal cell carcinoma. We also compared our learned progression networks with the networks proposed in earlier publications. The software is available on the website https://bitbucket.org/farahani/diprog.

Cumulative disease progression models for cross‐sectional data: A review and comparison

A better understanding of disease progression is beneficial for early diagnosis and appropriate individual therapy. Many different approaches for statistical modelling of cumulative disease progression have been proposed in the literature, including simple path models up to complex restricted Bayesian networks. Important fields of application are diseases such as cancer and HIV. Tumour progression is measured by means of chromosome aberrations, whereas people infected with HIV develop drug resistances because of genetic changes of the HI-virus. These two very different diseases have typical courses of disease progression, which can be modelled partly by consecutive and partly by independent steps. This paper gives an overview of the different progression models and points out their advantages and drawbacks. Different models are compared via simulations to analyse how they work if some of their assumptions are violated. In a simulation study, we evaluate how models perform in terms of fitting induced multivariate probability distributions and topological relationships. We often find that the true model class used for generating data is outperformed by either a less or a more complex model class. The more flexible conjunctive Bayesian networks can be used to fit oncogenetic trees, whereas mixtures of oncogenetic trees with three tree components can be well fitted by mixture models with only two tree components.

Efficient sampling for Bayesian inference of conjunctive Bayesian networks

Cancer development is driven by the accumulation of advantageous mutations and subsequent clonal expansion of cells harbouring these mutations, but the order in which mutations occur remains poorly understood. Advances in genome sequencing and the soon-arriving flood of cancer genome data produced by large cancer sequencing consortia hold the promise to elucidate cancer progression. However, new computational methods are needed to analyse these large datasets. We present a Bayesian inference scheme for Conjunctive Bayesian Networks, a probabilistic graphical model in which mutations accumulate according to partial order constraints and cancer genotypes are observed subject to measurement noise. We develop an efficient MCMC sampling scheme specifically designed to overcome local optima induced by dependency structures. We demonstrate the performance advantage of our sampler over traditional approaches on simulated data and show the advantages of adopting a Bayesian perspective when reanalyzing cancer datasets and comparing our results to previous maximum-likelihood-based approaches. An R package including the sampler and examples is available at http://www.cbg.ethz.ch/software/bayes-cbn. niko.beerenwinkel@bsse.ethz.ch.

The Temporal Order of Genetic and Pathway Alterations in Tumorigenesis

Cancer evolves through the accumulation of mutations, but the order in which mutations occur is poorly understood. Inference of a temporal ordering on the level of genes is challenging because clinically and histologically identical tumors often have few mutated genes in common. This heterogeneity may at least in part be due to mutations in different genes having similar phenotypic effects by acting in the same functional pathway. We estimate the constraints on the order in which alterations accumulate during cancer progression from cross-sectional mutation data using a probabilistic graphical model termed Hidden Conjunctive Bayesian Network (H-CBN). The possible orders are analyzed on the level of genes and, after mapping genes to functional pathways, also on the pathway level. We find stronger evidence for pathway order constraints than for gene order constraints, indicating that temporal ordering results from selective pressure acting at the pathway level. The accumulation of changes in core pathways differs among cancer types, yet a common feature is that progression appears to begin with mutations in genes that regulate apoptosis pathways and to conclude with mutations in genes involved in invasion pathways. H-CBN models provide a quantitative and intuitive model of tumorigenesis showing that the genetic events can be linked to the phenotypic progression on the level of pathways.

Learning Monotonic Genotype-Phenotype Maps

Evolutionary escape of pathogens from the selective pressure of immune responses and from medical interventions is driven by the accumulation of mutations. We introduce a statistical model for jointly estimating the dynamics and dependencies among genetic alterations and the associated phenotypic changes. The model integrates conjunctive Bayesian networks, which define a partial order on the occurrences of genetic events, with isotonic regression. The resulting genotype-phenotype map is non-decreasing in the lattice of genotypes. It describes evolutionary escape as a directed process following a phenotypic gradient, such as a monotonic fitness landscape. We present efficient algorithms for parameter estimation and model selection. The model is validated using simulated data and applied to HIV drug resistance data. We find that the effect of many resistance mutations is non-linear and depends on the genetic background in which they occur.

Waiting Time Models of Cancer Progression

Cancer progression is an evolutionary process driven by mutation and selection in a population of tumor cells. In multistage models of cancer progression, each stage is associated with the occurrence of genetic alterations and their fixation in the population. The accumulation of mutations is described using conjunctive Bayesian networks, an exponential family of waiting time models in which the occurrence of mutations is constrained by a partial temporal order. Two opposing limit cases arise if mutations either follow a linear order or occur independently. Exact analytical expressions for the waiting time until a specific number of mutations have accumulated are derived in these limit cases as well as for the general conjunctive Bayesian network. In a stochastic population genetics model that accounts for mutation and selection, waves of clonal expansions sweep through the population at equidistant intervals. An approximate analytical expression for the waiting time is compared to the results obtained with conjunctive Bayesian networks.

Quantifying cancer progression with conjunctive Bayesian networks

Motivation: Cancer is an evolutionary process characterized by accumulating mutations. However, the precise timing and the order of genetic alterations that drive tumor progression remain enigmatic.Results: We present a specific probabilistic graphical model for the accumulation of mutations and their interdependencies. The Bayesian network models cancer progression by an explicit unobservable accumulation process in time that is separated from the observable but error-prone detection of mutations. Model parameters are estimated by an Expectation-Maximization algorithm and the underlying interaction graph is obtained by a simulated annealing procedure. Applying this method to cytogenetic data for different cancer types, we find multiple complex oncogenetic pathways deviating substantially from simplified models, such as linear pathways or trees. We further demonstrate how the inferred progression dynamics can be used to improve genetics-based survival predictions which could support diagnostics and prognosis.Availability: The software package ct-cbn is available under a GPL license on the web site cbg.ethz.ch/software/ct-cbnContact: moritz.gerstung@bsse.ethz.ch

Markov models for accumulating mutations

Summary We introduce and analyze a waiting time model for the accumulation of genetic changes. The continuous-time conjunctive Bayesian network is defined by a partially ordered set of mutations and by the rate of fixation of each mutation. The partial order encodes constraints on the order in which mutations can fixate in the population, shedding light on the mutational pathways underly ing the evolutionary process. We study a censored version of the model and derive equations for an em algorithm to perform maximum likelihood estimation of the model parameters. We also show how to select the maximum likelihood partially ordered set. The model is applied to ge netic data from cancer cells and from drug resistant human immunodeficiency viruses, indicating implications for diagnosis and treatment.

Conjunctive Bayesian networks

Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gröbner basis.