Manuele Leonelli

Manuele Leonelli

Bayesian networks (BNs) are widely used for modeling complex systems with uncertainty, yet repositories of pre-built BNs remain limited. This paper introduces bnRep, an open-source R package offering a comprehensive collection of documented BNs, facilitating benchmarking, replicability, and education. With over 200 networks from academic publications, bnRep integrates seamlessly with bnlearn and other R packages, providing users with interactive tools for network exploration.

Manuele Leonelli, Gherardo Varando

Bayesian networks faithfully represent the symmetric conditional independences existing between the components of a random vector. Staged trees are an extension of Bayesian networks for categorical random vectors whose graph represents non-symmetric conditional independences via vertex coloring. However, since they are based on a tree representation of the sample space, the underlying graph becomes cluttered and difficult to visualize as the number of variables increases. Here we introduce the first structural learning algorithms for the class of simple staged trees, entertaining a compact coalescence of the underlying tree from which non-symmetric independences can be easily read. We show that data-learned simple staged trees often outperform Bayesian networks in model fit and illustrate how the coalesced graph is used to identify non-symmetric conditional independences.

Manuele Leonelli, Gherardo Varando

Several structural learning algorithms for staged tree models, an asymmetric extension of Bayesian networks, have been defined. However, they do not scale efficiently as the number of variables considered increases. Here we introduce the first scalable structural learning algorithm for staged trees, which searches over a space of models where only a small number of dependencies can be imposed. A simulation study as well as a real-world application illustrate our routines and the practical use of such data-learned staged trees.

Rafael Ballester-Ripoll, Manuele Leonelli

Sensitivity analysis measures the influence of a Bayesian network's parameters on a quantity of interest defined by the network, such as the probability of a variable taking a specific value. In particular, the so-called sensitivity value measures the quantity of interest's partial derivative with respect to the network's conditional probabilities. However, finding such values in large networks with thousands of parameters can become computationally very expensive. We propose to use automatic differentiation combined with exact inference to obtain all sensitivity values in a single pass. Our method first marginalizes the whole network once using e.g. variable elimination and then backpropagates this operation to obtain the gradient with respect to all input parameters. We demonstrate our routines by ranking all parameters by importance on a Bayesian network modeling humanitarian crises and disasters, and then show the method's efficiency by scaling it to huge networks with up to 100'000 parameters. An implementation of the methods using the popular machine learning library PyTorch is freely available.

Fabio Crimaldi, Manuele Leonelli

This study explores the concept of creativity and artificial intelligence (AI) and their recent integration. While AI has traditionally been perceived as incapable of generating new ideas or creating art, the development of more sophisticated AI models and the proliferation of human-computer interaction tools have opened up new possibilities for AI in artistic creation. This study investigates the various applications of AI in a creative context, differentiating between the type of art, language, and algorithms used. It also considers the philosophical implications of AI and creativity, questioning whether consciousness can be researched in machines and AI's potential interests and decision-making capabilities. Overall, we aim to stimulate a reflection on AI's use and ethical implications in creative contexts.

Rafael Ballester-Ripoll, Manuele Leonelli

Sensitivity analysis measures the influence of a Bayesian network's parameters on a quantity of interest defined by the network, such as the probability of a variable taking a specific value. Various sensitivity measures have been defined to quantify such influence, most commonly some function of the quantity of interest's partial derivative with respect to the network's conditional probabilities. However, computing these measures in large networks with thousands of parameters can become computationally very expensive. We propose an algorithm combining automatic differentiation and exact inference to efficiently calculate the sensitivity measures in a single pass. It first marginalizes the whole network once, using e.g. variable elimination, and then backpropagates this operation to obtain the gradient with respect to all input parameters. Our method can be used for one-way and multi-way sensitivity analysis and the derivation of admissible regions. Simulation studies highlight the efficiency of our algorithm by scaling it to massive networks with up to 100'000 parameters and investigate the feasibility of generic multi-way analyses. Our routines are also showcased over two medium-sized Bayesian networks: the first modeling the country-risks of a humanitarian crisis, the second studying the relationship between the use of technology and the psychological effects of forced social isolation during the COVID-19 pandemic. An implementation of the methods using the popular machine learning library PyTorch is freely available.

Manuele Leonelli, Gherardo Varando

Bayesian networks are widely used to learn and reason about the dependence structure of discrete variables. However, they are only capable of formally encoding symmetric conditional independence, which in practice is often too strict to hold. Asymmetry-labeled DAGs have been recently proposed to both extend the class of Bayesian networks by relaxing the symmetric assumption of independence and denote the type of dependence existing between the variables of interest. Here, we introduce novel structural learning algorithms for this class of models which, whilst being efficient, allow for a straightforward interpretation of the underlying dependence structure. A comprehensive computational study highlights the efficiency of the algorithms. A real-world data application using data from the Fear of COVID-19 Scale collected in Italy showcases their use in practice.

Manuele Leonelli, Jim Q. Smith, Sophia K. Wright

Bayesian networks are one of the most widely used classes of probabilistic models for risk management and decision support because of their interpretability and flexibility in including heterogeneous pieces of information. In any applied modelling, it is critical to assess how robust the inferences on certain target variables are to changes in the model. In Bayesian networks, these analyses fall under the umbrella of sensitivity analysis, which is most commonly carried out by quantifying dissimilarities using Kullback-Leibler information measures. In this paper, we argue that robustness methods based instead on the familiar total variation distance provide simple and more valuable bounds on robustness to misspecification, which are both formally justifiable and transparent. We introduce a novel measure of dependence in conditional probability tables called the diameter to derive such bounds. This measure quantifies the strength of dependence between a variable and its parents. We demonstrate how such formal robustness considerations can be embedded in building a Bayesian network.

Muhammad Shoaib, Eva Riccomagno, Manuele Leonelli et al.

Staged tree models enhance Bayesian networks by incorporating context-specific dependencies through a stage-based structure. In this study, we present a new framework for estimating staged trees using hierarchical clustering on the probability simplex, utilizing simplex basesd divergences. We conduct a thorough evaluation of several distance and divergence metrics including Total Variation, Hellinger, Fisher, and Kaniadakis; alongside various linkage methods such as Ward.D2, average, complete, and McQuitty. We conducted the simulation experiments that reveals Total Variation, especially when combined with Ward.D2 linkage, consistently produces staged trees with better model fit, structure recovery, and computational efficiency. We assess performance by utilizing relative Bayesian Information Criterion (BIC), and Hamming distance. Our findings indicate that although Backward Hill Climbing (BHC) delivers competitive outcomes, it incurs a significantly higher computational cost. On the other, Total Variation divergence with Ward.D2 linkage, achieves similar performance while providing significantly better computational efficiency, making it a more viable option for large-scale or time sensitive tasks.

Rafael Ballester-Ripoll, Manuele Leonelli

Traditionally, the sensitivity analysis of a Bayesian network studies the impact of individually modifying the entries of its conditional probability tables in a one-at-a-time (OAT) fashion. However, this approach fails to give a comprehensive account of each inputs' relevance, since simultaneous perturbations in two or more parameters often entail higher-order effects that cannot be captured by an OAT analysis. We propose to conduct global variance-based sensitivity analysis instead, whereby $n$ parameters are viewed as uncertain at once and their importance is assessed jointly. Our method works by encoding the uncertainties as $n$ additional variables of the network. To prevent the curse of dimensionality while adding these dimensions, we use low-rank tensor decomposition to break down the new potentials into smaller factors. Last, we apply the method of Sobol to the resulting network to obtain $n$ global sensitivity indices. Using a benchmark array of both expert-elicited and learned Bayesian networks, we demonstrate that the Sobol indices can significantly differ from the OAT indices, thus revealing the true influence of uncertain parameters and their interactions.

Rafael Ballester-Ripoll, Manuele Leonelli

We show how to apply Sobol's method of global sensitivity analysis to measure the influence exerted by a set of nodes' evidence on a quantity of interest expressed by a Bayesian network. Our method exploits the network structure so as to transform the problem of Sobol index estimation into that of marginalization inference. This way, we can efficiently compute indices for networks where brute-force or Monte Carlo based estimators for variance-based sensitivity analysis would require millions of costly samples. Moreover, our method gives exact results when exact inference is used, and also supports the case of correlated inputs. The proposed algorithm is inspired by the field of tensor networks, and generalizes earlier tensor sensitivity techniques from the acyclic to the cyclic case. We demonstrate the method on three medium to large Bayesian networks that cover the areas of project risk management and reliability engineering.

Gherardo Varando, Federico Carli, Manuele Leonelli

Bayesian networks are a widely-used class of probabilistic graphical models capable of representing symmetric conditional independence between variables of interest using the topology of the underlying graph. For categorical variables, they can be seen as a special case of the much more general class of models called staged trees, which can represent any type of non-symmetric conditional independence. Here we formalize the relationship between these two models and introduce a minimal Bayesian network representation of the staged tree, which can be used to read conditional independences in an intutitive way. A new labeled graph termed asymmetry-labeled directed acyclic graph is defined, whose edges are labeled to denote the type of dependence existing between any two random variables. We also present a novel algorithm to learn staged trees which only enforces a specific subset of non-symmetric independences. Various datasets are used to illustrate the methodology, highlighting the need to construct models which more flexibly encode and represent non-symmetric structures.

Manuele Leonelli, Ramsiya Ramanathan, Rachel L. Wilkerson

Bayesian networks are a class of models that are widely used for risk assessment of complex operational systems. There are now multiple approaches, as well as implemented software, that guide their construction via data learning or expert elicitation. However, a constructed Bayesian network needs to be validated before it can be used for practical risk assessment. Here, we illustrate the usage of the bnmonitor R package: the first comprehensive software for the validation of a Bayesian network. An applied data analysis using bnmonitor is carried out over a medical dataset to illustrate the use of its wide array of functions.

Manuele Leonelli, Gherardo Varando

Causal discovery algorithms aim at untangling complex causal relationships from data. Here, we study causal discovery and inference methods based on staged tree models, which can represent complex and asymmetric causal relationships between categorical variables. We provide a first graphical representation of the equivalence class of a staged tree, by looking only at a specific subset of its underlying independences. We further define a new pre-metric, inspired by the widely used structural intervention distance, to quantify the closeness between two staged trees in terms of their corresponding causal inference statements. A simulation study highlights the efficacy of staged trees in uncovering complexes, asymmetric causal relationships from data, and real-world data applications illustrate their use in practical causal analysis.

Federico Carli, Manuele Leonelli, Gherardo Varando

Generative models for classification use the joint probability distribution of the class variable and the features to construct a decision rule. Among generative models, Bayesian networks and naive Bayes classifiers are the most commonly used and provide a clear graphical representation of the relationship among all variables. However, these have the disadvantage of highly restricting the type of relationships that could exist, by not allowing for context-specific independences. Here we introduce a new class of generative classifiers, called staged tree classifiers, which formally account for context-specific independence. They are constructed by a partitioning of the vertices of an event tree from which conditional independence can be formally read. The naive staged tree classifier is also defined, which extends the classic naive Bayes classifier whilst retaining the same complexity. An extensive simulation study shows that the classification accuracy of staged tree classifiers is competitive with that of state-of-the-art classifiers and an example showcases their use in practice.

Federico Carli, Manuele Leonelli, Eva Riccomagno et al.

stagedtrees is an R package which includes several algorithms for learning the structure of staged trees and chain event graphs from data. Score-based and clustering-based algorithms are implemented, as well as various functionalities to plot the models and perform inference. The capabilities of stagedtrees are illustrated using mainly two datasets both included in the package or bundled in R.

Manuele Leonelli, Eva Riccomagno

Sensitivity analysis in probabilistic discrete graphical models is usually conducted by varying one probability value at a time and observing how this affects output probabilities of interest. When one probability is varied then others are proportionally covaried to respect the sum-to-one condition of probability laws. The choice of proportional covariation is justified by a variety of optimality conditions, under which the original and the varied distributions are as close as possible under different measures of closeness. For variations of more than one parameter at a time proportional covariation is justified in some special cases only. In this work, for the large class of discrete statistical models entertaining a regular monomial parametrisation, we demonstrate the optimality of newly defined proportional multi-way schemes with respect to an optimality criterion based on the notion of I-divergence. We demonstrate that there are varying parameters choices for which proportional covariation is not optimal and identify the sub-family of model distributions where the distance between the original distribution and the one where probabilities are covaried proportionally is minimum. This is shown by adopting a new formal, geometric characterization of sensitivity analysis in monomial models, which include a wide array of probabilistic graphical models. We also demonstrate the optimality of proportional covariation for multi-way analyses in Naive Bayes classifiers.

Christiane Goergen, Manuele Leonelli

The accuracy of probability distributions inferred using machine-learning algorithms heavily depends on data availability and quality. In practical applications it is therefore fundamental to investigate the robustness of a statistical model to misspecification of some of its underlying probabilities. In the context of graphical models, investigations of robustness fall under the notion of sensitivity analyses. These analyses consist in varying some of the model's probabilities or parameters and then assessing how far apart the original and the varied distributions are. However, for Gaussian graphical models, such variations usually make the original graph an incoherent representation of the model's conditional independence structure. Here we develop an approach to sensitivity analysis which guarantees the original graph remains valid after any probability variation and we quantify the effect of such variations using different measures. To achieve this we take advantage of algebraic techniques to both concisely represent conditional independence and to provide a straightforward way of checking the validity of such relationships. Our methods are demonstrated to be robust and comparable to standard ones, which break the conditional independence structure of the model, using an artificial example and a medical real-world application.

Manuele Leonelli, Jim Q. Smith

A variety of statistical graphical models have been defined to represent the conditional independences underlying a random vector of interest. Similarly, many different graphs embedding various types of preferential independences, as for example conditional utility independence and generalized additive independence, have more recently started to appear. In this paper we define a new graphical model, called a directed expected utility network, whose edges depict both probabilistic and utility conditional independences. These embed a very flexible class of utility models, much larger than those usually conceived in standard influence diagrams. Our graphical representation, and various transformations of the original graph into a tree structure, are then used to guide fast routines for the computation of a decision problem's expected utilities. We show that our routines generalize those usually utilized in standard influence diagrams' evaluations under much more restrictive conditions. We then proceed with the construction of a directed expected utility network to support decision makers in the domain of household food security.

Manuele Leonelli, Eva Riccomagno, Jim Q. Smith

Influence diagrams provide a compact graphical representation of decision problems. Several algorithms for the quick computation of their associated expected utilities are available in the literature. However, often they rely on a full quantification of both probabilistic uncertainties and utility values. For problems where all random variables and decision spaces are finite and discrete, here we develop a symbolic way to calculate the expected utilities of influence diagrams that does not require a full numerical representation. Within this approach expected utilities correspond to families of polynomials. After characterizing their polynomial structure, we develop an efficient symbolic algorithm for the propagation of expected utilities through the diagram and provide an implementation of this algorithm using a computer algebra system. We then characterize many of the standard manipulations of influence diagrams as transformations of polynomials. We also generalize the decision analytic framework of these diagrams by defining asymmetries as operations over the expected utility polynomials.

Manuele Leonelli, Christiane Görgen, Jim Q. Smith

Sensitivity methods for the analysis of the outputs of discrete Bayesian networks have been extensively studied and implemented in different software packages. These methods usually focus on the study of sensitivity functions and on the impact of a parameter change to the Chan-Darwiche distance. Although not fully recognized, the majority of these results heavily rely on the multilinear structure of atomic probabilities in terms of the conditional probability parameters associated with this type of network. By defining a statistical model through the polynomial expression of its associated defining conditional probabilities, we develop a unifying approach to sensitivity methods applicable to a large suite of models including extensions of Bayesian networks, for instance context-specific and dynamic ones, and chain event graphs. By then focusing on models whose defining polynomial is multilinear, our algebraic approach enables us to prove that the Chan-Darwiche distance is minimized for a certain class of multi-parameter contemporaneous variations when parameters are proportionally covaried.