Likelihood analysis of phylogenetic networks using directed graphical models.

Rooted phylogenetic networks, or more generally, directed acyclic graphs (DAGs), are widely used to model species or gene relationships that traditional rooted trees cannot fully capture, especially in the presence of reticulate processes or horizontal gene transfers. Such networks or DAGs are typically inferred from observable data (e.g., genomic sequences of extant species), providing only an estimate of the true evolutionary history. However, these inferred DAGs are often complex and difficult to interpret. In particular, many contain vertices that do not serve as least common ancestors (LCAs) for any subset of the underlying genes or species, thus may lack direct support from the observable data. In contrast, LCA vertices are witnessed by historical traces justifying their existence and thus represent ancestral states substantiated by the data. To reduce unnecessary complexity and eliminate unsupported vertices, we aim to simplify a DAG to retain only LCA vertices while preserving essential evolutionary information. In this paper, we characterize LCA-relevant and lca-relevant DAGs, defined as those in which every vertex serves as an LCA (or unique LCA) for some subset of taxa. We introduce methods to identify LCAs in DAGs and efficiently transform any DAG into an LCA-relevant or lca-relevant one while preserving key structural properties of the original DAG or network. This transformation is achieved using a simple operator “⊖” that mimics vertex suppression.

Evolutionary trees have the assumption that evolution and phylogeny can be represented in a strictly bifurcating manner. Firmly speaking, from one ancestral taxon, two descendant taxa emerge. Nevertheless, hybridization, recombination and horizontal gene transfer is in conflict with this straightforward concept. In such cases, evolutionary lines do not only separate from each other, but have the possibility of melting again and are called reticulations. Consequently, networks can represent evolutionary events more realistically than phylogenetic trees. Networks can display alternative topologies and co-existence of ancestors and descendants, which are otherwise not obvious when a comparison is done on several single trees or a consensus tree. Therefore, networks have the ability to visualize the conflicting information in a given data set. Moreover, the distribution, frequencies and arrangement of haplotypes in populations can reveal the phylogenetic histories of the taxa, regarding predictions from the coalescent theory. This review aims to: (1) give a brief comparison between phylogenetic trees and networks, (2) provide the overall concept of the coalescent theory, (3) clarify how phylogenetic networks can be used to display conflict data and evaluate phylogenetic histories, and (4) offer a useful starting point and guide for sequence analysis, with the aim to discover population dynamics. Key words: Phylogenetic networks, reticulation, coalescent theory, population history, character conflict.

Phylogenetic networks model the evolutionary history of sets of organisms when events such as hybrid speciation and horizontal gene transfer occur. In spite of their widely acknowledged importance in evolutionary biology, phylogenetic networks have so far been studied mostly for specific data sets. We present a general definition of phylogenetic networks in terms of directed acyclic graphs (DAGs) and a set of conditions. Further, we distinguish between model networks and reconstructible ones and characterize the effect of extinction and taxon sampling on the reconstructibility of the network. Simulation studies are a standard technique for assessing the performance of phylogenetic methods. A main step in such studies entails quantifying the topological error between the model and inferred phylogenies. While many measures of tree topological accuracy have been proposed, none exist for phylogenetic networks. Previously, we proposed the first such measure, which applied only to a restricted class of networks. In this paper, we extend that measure to apply to all networks, and prove that it is a metric on the space of phylogenetic networks. Our results allow for the systematic study of existing network methods, and for the design of new accurate ones.

Directed acyclic graphs (DAGs) play a large role in the modern approach to causal inference. DAGs describe the relationship between measurements taken at various discrete times including the effect of interventions. The causal mechanisms, on the other hand, would naturally be assumed to be a continuous process operating over time in a cause–effect fashion. How does such immediate causation, that is causation occurring over very short time intervals, relate to DAGs constructed from discrete observations? We introduce a time-continuous model and simulate discrete observations in order to judge the relationship between the DAG and the immediate causal model. We find that there is no clear relationship; indeed the Bayesian network described by the DAG may not relate to the causal model. Typically, discrete observations of a process will obscure the conditional dependencies that are represented in the underlying mechanistic model of the process. It is therefore doubtful whether DAGs are always suited to describe causal relationships unless time is explicitly considered in the model. We relate the issues to mechanistic modeling by using the concept of local (in)dependence. An example using data from the Swiss HIV Cohort Study is presented.

Chapter 11 Bayesian Networks

The key goals in learning Bayesian networks (BNs) from data are to identify significant statistical relationships between variables and to build a Directed Acyclic Graph (DAG) that represents these relationships through Joint Probability Distributions. Most research relies on score-based or conditional test methods for model selection. However, when using real-world data, it can be challenging to identify whether the learned DAG represents the underlying relations inherent in the limited datasets, particularly when evaluating data obtained from multiple independent sources. This study presents a methodology to assess the credible interval for both the existence and direction of each edge within Bayesian networks derived from data. Furthermore, it explores the fusion of models acquired from distinct and independent datasets. This approach enables the Bayesian learning of Bayesian Networks (BNs) from data by treating the uncertainty associated with the existence and orientation of each edge as a random variable. By evaluating the probability of the orientation of each edge, it is possible to suggest the existence of a potential latent variable within the dataset. If an edge exhibits equiprobable directions and is verified to exist, it becomes a plausible hypothesis for a latent variable. The Fast Causal Algorithm, originally introduced by [1], is the foundation of this approach. Finally, by employing a maximum a posteriori estimation, the most prominent edges and their respective orientations are identified and employed to create the leading DAG. We present our findings in simulated datasets with different length sizes. By comparing the structure of the learned DAGs with existing structures and evaluating the inference capabilities of the final BN, we establish that our approach achieves results comparable to the most recent studies in the field, while offering insights into the model’s reliability and improving the use of the data.

We propose a self-stabilizing algorithm to construct a minimal weakly $\mathcal{ST}$-reachable directed acyclic graph (DAG), which is suited for routing messages on wireless networks. Given an arbitrary, simple, connected, and undirected graph $G=(V, E)$ and two sets of nodes, senders $\mathcal{S} (\subset V)$ and targets $\mathcal{T} (\subset V)$, a directed subgraph $\vec{G}$ of $G$ is a weakly $\mathcal{ST}$-reachable DAG on $G$, if $\vec{G}$ is a DAG and every sender can reach at least one target, and every target is reachable from at least one sender in $\vec{G}$. We say that a weakly $\mathcal{ST}$-reachable DAG $\vec{G}$ on $G$ is minimal if any proper subgraph of $\vec{G}$ is no longer a weakly $\mathcal{ST}$-reachable DAG. This DAG is a relaxed version of the original (or strongly) $\mathcal{ST}$-reachable DAG, where every target is reachable from every sender. This is because a strongly $\mathcal{ST}$-reachable DAG $G$ does not always exist; some graph has no strongly $\mathcal{ST}$-reachable DAG even in the case $|\mathcal{S}|=|\mathcal{T}|=2$. On the other hand, the proposed algorithm always constructs a weakly $\mathcal{ST}$-reachable DAG for any $|\mathcal{S}|$ and $|\mathcal{T}|$. Furthermore, the proposed algorithm is self-stabilizing; even if the constructed DAG deviates from the reachability requirement by a breakdown or exhausting the battery of a node having an arc in the DAG, this algorithm automatically reconstructs the DAG to satisfy the requirement again. The convergence time of the algorithm is $O(D)$ asynchronous rounds, where $D$ is the diameter of a given graph. We conduct small simulations to evaluate the performance of the proposed algorithm. The simulation result indicates that its execution time decreases when the number of sender nodes or target nodes is large.

BackgroundDiagnosing urinary tract infections (UTIs) in children in the emergency department (ED) is challenging due to the variable clinical presentations and difficulties in obtaining a urine sample free from contamination. Clinicians need to weigh a range of observations to make timely diagnostic and management decisions, a difficult task to achieve without support due to the complex interactions among relevant factors. Directed acyclic graphs (DAG) and causal Bayesian networks (BN) offer a way to explicitly outline the underlying disease, contamination and diagnostic processes, and to further make quantitative inference on the event of interest thus serving as a tool for decision support.MethodsWe prospectively collected data on children present to ED with suspected UTIs. Through knowledge elicitation workshops and one-on-one meetings, a DAG was co-developed with clinical domain experts (the Expert DAG) to describe the causal relationships among variables relevant to paediatric UTIs. The Expert DAG was combined with prospective data and further domain knowledge to inform the development of an application-oriented BN (the Applied BN), designed to support the diagnosis of UTI. We assessed the performance of the Applied BN using quantitative and qualitative methods.ResultsWe summarised patient background, clinical and laboratory characteristics of 431 episodes of suspected UTIs enrolled from May 2019 to November 2020. The Expert DAG was presented with a narrative description, elucidating how infection, specimen contamination and management pathways causally interact to form the complex picture of paediatric UTIs. Parameterised using prospective data and expert-elicited parameters, the Applied BN achieved an excellent and stable performance in predicting Escherichia coli culture results, with a mean area under the receiver operating characteristic curve of 0.86 and a mean log loss of 0.48 based on 10-fold cross-validation. The BN predictions were reviewed via a validation workshop, and we illustrate how they can be presented for decision support using three hypothetical clinical scenarios.ConclusionCausal BNs created from both expert knowledge and data can integrate case-specific information to provide individual decision support during the diagnosis of paediatric UTIs in ED. The model aids the interpretation of culture results and the diagnosis of UTIs, promising the prospect of improved patient care and judicious use of antibiotics.

The accuracy of the Gibbs sampling Markov chain monte carlo procedure was examined for estimating item and person ( .) parameters in the one-parameter logistic model. Four datasets were analyzed using the Gibbs sampling method, conditional maximum likelihood, marginal maximum likelihood, and joint maximum likelihood. Maximum likelihood and expected a posteriori. estimation methods were used with marginal maximum likelihood estimation of item parameters. Item parameter estimates from the four methods were almost identical;. estimates from Gibbs sampling were similar to those obtained from the expected a posteriori method.

Restricted Boltzmann Machines and Deep Belief Networks have been successfully used in a wide variety of applications including image classification and speech recognition. Inference and learning in these algorithms uses a Markov Chain Monte Carlo procedure called Gibbs sampling. A sigmoidal function forms the kernel of this sampler which can be realized from the firing statistics of noisy integrate-and-fire neurons on a neuromorphic VLSI substrate. This paper demonstrates such an implementation on an array of digital spiking neurons with stochastic leak and threshold properties for inference tasks and presents some key performance metrics for such a hardware-based sampler in both the generative and discriminative contexts.

Spectral Bayesian network theory

In this paper, we address the challenges of random label ordering and limited interpretability associated with Ensemble Classifier Chains (ECC) by introducing a novel ECC method, ECC-MOO&BN, which integrates Bayesian Networks (BN) and Multi-Objective Optimization (MOO). This approach is designed to concurrently overcome these ECC limitations. The ECC-MOO&BN method focuses on extracting diverse and interpretable label orderings for the ECC classifier. We initiated this process by employing mutual information to investigate label relationships and establish the initial structures of the BN. Subsequently, an enhanced NSGA-II algorithm was applied to develop a series of Directed Acyclic Graphs (DAGs) that effectively balance the likelihood and complexity of the BN structure. The rationale behind using the MOO method lies in its ability to optimize both complexity and likelihood simultaneously, which not only diversifies DAG generation but also helps avoid overfitting during the production of label orderings. The DAGs, once sorted topologically, yielded a series of label orderings, which were then seamlessly integrated into the ECC framework for addressing multi-label classification (MLC) problems. Experimental results show that when benchmarked against eleven leading-edge MLC algorithms, our proposed method achieves the highest average ranking across seven evaluation criteria on nine out of thirteen MLC datasets. The results of the Friedman test and Nemenyi test also indicate that the performance of the proposed method has a significant advantage compared to other algorithms.

A Bayesian network (BN) Charniak (1991) Pearl (1988) Jensen (1996) Neapolitan (1990) is a directed acyclic graph (DAG) consisting of nodes and arrows, in which node represents random variables, and arrow represents dependence relationship between connected nodes in the sense of the probabilistic, deterministic, or functional. Each node in BN has a specified conditional probability distribution (CPD), where all CPDs together parameterize the model. BNs have been used as powerful probabilistic knowledge models for decision support under uncertainty over a few decades, with numerous applications such as classification, medical diagnosis, bioinformatics, speech recognition, etc. One of the most important features BN has is the factorization of the joint probability space, so that conditional independence can be exploited to simplify modeling and save computations. However, BN model is only useful when combined with efficient algorithms for inference.

Cancer progression models such as Oncogenetic Tress [1], Conjunctive Bayesian Networks [2], or CAPRI [3], try to infer possible restrictions in the order of accumulation of mutations using cancer cross-sectional data. These restrictions would arise from epistatic interactions between mutations. The Directed Acyclic Graphs (DAGs) returned by these methods predict the genotypes that can and cannot be observed. Since fitness landscapes are shaped by those same epistatics interactions and also specify what genotypes are accessible and inaccessible [4] we would expect a close correspondence between a fitness landscape and the DAGs inferred from evolutionary processes in that landscape. I will show here, however, that there is a many-to- many mapping between fitness landscapes and DAGs inferred from cancer progression models. I show that many fitness landscapes cannot be represented by a cancer progression DAG in the sense that no DAG can codify the genotypes that are accessible under that landscape and I propose two measures of “DAG-non-representability”. Next, using 500 different random fitness lanscapes (with 100 begin perfectly DAG-representable) I simulate cancer progression using different mutation rates and sampling regimes; I analyze 20 replicates (of 1000 genotypes) under each condition with CAPRI[3] and CBN[2]. I show that, even under constant mutation and sampling regime conditions, data obtained from the same landscape lead to inferring widely different DAGs (e.g., median Gini-Simpson index of 0.5); this one-landscape-to-many-DAGs phenomenon increases with DAG-non-representability. I also show the one-DAG-to-many-landscapes phenomenon where the same DAG is frequently inferred from very different fitness landscapes (e.g., > 50% difference in accessible genotypes). Finally, I “reverse landscape engineer” the pancreatic cancer data used in [6]: using 190 random fitness landscapes, I show that genotype frequencies similar to the empirically observed one can be obtained under fitness landscapes that are very different from each other and that produce data that lead to the inference of DAGs that are also widely different among themselves (e.g., pairwise difference in edges of DAGs > 50%). I discuss the implications of these results for the interpretation and applied use of cancer progression models. Novelty and significance This paper explicitly establishes a connection between the inference of cancer progression models and the fitness landscapes that condition that cancer progression. It shows there is a many-to- many mapping between them, even under large sample sizes, that can limit their interpretability.

One of the major limitation of using the back-propagation neural network and its variants in real applications are that the number of hidden nodes is unknown. It is usually estimated by trial-and-error and thus it is inefficient. The paper proposes an algorithm to determine the number of hidden nodes based on the input data. The dynamic expansion and contraction approach (DECA), which comes from dynamic programming, is used to determine the optimal number of hidden nodes. The object function minimises the number of hidden nodes while the constraints are a pre-defined error. A short interval of train/test interleaving is used to minimise the learning time and avoid over-training the network. The algorithm is applicable to the neural network used for function approximation as well as pattern classification. >