Benchmarks play a central role in machine learning (ML), offering standardized datasets and metrics that enable comparison and drive progress. In fairness-aware ML (fairML), however, benchmarks pose distinctive challenges. Fairness is not a purely technical property but a socio-technical concept, shaped by normative choices and institutional context. Benchmarks strip away this context: they reduce fairness to intrinsic metrics, obscure what is comparable, and collapse distinct notions of justice—from distributive allocation in credit scoring to basic rights in criminal justice—into a single optimization task. Moreover, when used as measures of progress, benchmarks risk enshrining oversimplified metrics as community standards, assuming an exception to Goodhart’s law. We argue that while benchmarking has value for building baselines and organizing competition, responsible evaluation of fairML requires complementary frameworks: ones that combine intrinsic with extrinsic, context-sensitive assessments, and that make explicit the normative assumptions underlying fairness interventions.

Bayesian networks (BN) are versatile probabilistic graphical models that enable efficient knowledge representation and inference. These models have proven effective across diverse domains, including healthcare, bioinformatics, economics, law, and image processing. The structure and parameters of a BN can be obtained by domain experts or directly learned from available data. However, as privacy concerns escalate, it becomes increasingly critical for publicly released models to safeguard sensitive information in training data. Typically, released models do not prioritize privacy by design, and the issue equally affects BNs. In particular, tracing attacks from adversaries can combine the released BN with auxiliary data to determine whether specific individuals belong to the data from which the BN was learned. The current approach to addressing this privacy issue involves introducing noise into the learned parameters. While this method offers robust protection against tracing attacks, it also significantly impacts the model’s utility, in terms of both the significance and accuracy of the resulting inferences. Hence, high privacy may be attained, but at the cost of releasing a possibly ineffective model. This paper introduces credal networks (CN) as a novel and practical solution for balancing the model’s privacy and utility. Specifically, after adapting the notion of tracing attacks, we demonstrate that a CN enables the masking of the learned BN, thereby reducing the probability of successful tracing attacks. As CNs are obfuscated but not noisy versions of BNs, they can achieve meaningful inferences while safeguarding the privacy of the released model. Moreover, we identify key learning information that must be concealed to prevent attackers from recovering the BN underlying the released CN. Finally, we conduct a set of numerical experiments to analyze how privacy gains can be modulated by tuning the CN hyperparameters. Our results confirm that CNs provide a principled, practical, and effective approach towards the development of privacy-aware probabilistic graphical models.

Probabilistic Circuits (PCs) are prominent tractable probabilistic models, allowing for a wide range of exact inferences. This paper focuses on a main algorithm for training PCs, LearnSPN, arguably a gold standard due to its efficiency, performance, and ease of use, in particular for tabular data. We show that LearnSPN is a greedy likelihood maximizer under mild assumptions. While inferences in PCs may use the entire circuit structure for processing queries, LearnSPN applies a hard method for learning PCs, propagating at each sum node a data point through one and only one of the children/edges as in a hard clustering process. We propose a new learning procedure named SoftLearn, that induces a PC using a soft clustering process. We investigate the effect of this learning-inference compatibility in PCs. Our experiments show that SoftLearn outperforms LearnSPN in many situations, yielding better likelihoods and arguably better samples. We also analyze comparable tractable models to highlight the differences between soft/hard learning and model querying.

Continuous latent variables (LVs) are a key ingredient of many generative models, as they allow modelling expressive mixtures with an uncountable number of components. In contrast, probabilistic circuits (PCs) are hierarchical discrete mixtures represented as computational graphs composed of input, sum and product units. Unlike continuous LV models, PCs provide tractable inference but are limited to discrete LVs with categorical (i.e. unordered) states. We bridge these model classes by introducing probabilistic integral circuits (PICs), a new language of computational graphs that extends PCs with integral units representing continuous LVs. In the first place, PICs are symbolic computational graphs and are fully tractable in simple cases where analytical integration is possible. In practice, we parameterise PICs with light-weight neural nets delivering an intractable hierarchical continuous mixture that can be approximated arbitrarily well with large PCs using numerical quadrature. On several distribution estimation benchmarks, we show that such PIC-approximating PCs systematically outperform PCs commonly learned via expectation-maximization or SGD.

Probabilistic integral circuits (PICs) have been recently introduced as probabilistic models enjoying the key ingredient behind expressive generative models: continuous latent variables (LVs). PICs are symbolic computational graphs defining continuous LV models as hierarchies of functions that are summed and multiplied together, or integrated over some LVs. They are tractable if LVs can be analytically integrated out, otherwise they can be approximated by tractable probabilistic circuits (PC) encoding a hierarchical numerical quadrature process, called QPCs. So far, only tree-shaped PICs have been explored, and training them via numerical quadrature requires memory-intensive processing at scale. In this paper, we address these issues, and present: (i) a pipeline for building DAG-shaped PICs out of arbitrary variable decompositions, (ii) a procedure for training PICs using tensorized circuit architectures, and (iii) neural functional sharing techniques to allow scalable training. In extensive experiments, we showcase the effectiveness of functional sharing and the superiority of QPCs over traditional PCs.

Abstract Bayesian Networks (BN) are robust probabilistic graphical models mainly used with discrete random variables requiring discretization and quantization of continuous data. Quantization is known to affect model accuracy, speed and interpretability, and there are various quantization methods and performance comparisons proposed in literature. Therefore, this paper introduces a novel approach called CPT limit-based quantization (CLBQ) aimed to address the trade-off among model quality, data fidelity and structure score. CLBQ sets CPT size limitation based on how large the dataset is so as to optimize the balance between the structure score of BNs and mean squared error. For such a purpose, a range of quantization values for each variable was evaluated and a Pareto set was designed considering structure score and mean squared error (MSE). A quantization value was selected from the Pareto set in order to balance MSE and structure score, and the method’s effectiveness was tested using different datasets, such as discrete variables with added noise, continuous variables and real continuous data. In all tests, CLBQ was compared to another quantization method known as Dynamic Discretization. Moreover, this study assesses the suitability of CLBQ for the search and score of BN structure learning, in addition to examining the landscape of BN structures while varying dataset sizes and confirming its consistency. It was sought to find the expected structure location through a landscape analysis and optimal BNs on it so as to confirm whether the expected results were actually achieved in the search and score of BN structure learning. Results demonstrate that CLBQ is quite capable of striking a balance between model quality, data fidelity and structure score, in addition to evidencing its potential application in the search and score of BN structure learning, thus further research should explore different structure scores and quantization methods through CLBQ. Furthermore, its code and used datasets have all been made available.

Probabilistic circuits are a class of probabilistic generative models that allow us to compute different types of probabilistic queries in polynomial time. Unlike many of the mainstream approaches for generative modeling, they can compute exact likelihoods, marginals, and expectations. Yet, assessing the reliability of their inferences is not straightforward. Credal probabilistic circuits are the imprecise counterpart of probabilistic circuits allowing, among other queries, computations of cautious inferences and sensitivity analyses. In this work, we propose an efficient algorithm to compute the lower and upper expectations for factorizing functions using a credal probabilistic circuit. We discuss under what structural assumptions and types of factorizing functions the algorithm works. We prove that such algorithm has polynomial time complexity in the input size. In the general case, we prove that computing cautious inferences using credal probabilistic circuits is an NP-hard problem, yet the proposed algorithm can be used as an approximation. Some experiments show how the approximation degrades with the complexity of the model structure.

Probabilistic models based on continuous latent spaces, such as variational autoencoders, can be understood as uncountable mixture models where components depend continuously on the latent code. They have proven to be expressive tools for generative and probabilistic modelling, but are at odds with tractable probabilistic inference, that is, computing marginals and conditionals of the represented probability distribution. Meanwhile, tractable probabilistic models such as probabilistic circuits (PCs) can be understood as hierarchical discrete mixture models, and thus are capable of performing exact inference efficiently but often show subpar performance in comparison to continuous latent-space models. In this paper, we investigate a hybrid approach, namely continuous mixtures of tractable models with a small latent dimension. While these models are analytically intractable, they are well amenable to numerical integration schemes based on a finite set of integration points. With a large enough number of integration points the approximation becomes de-facto exact. Moreover, for a finite set of integration points, the integration method effectively compiles the continuous mixture into a standard PC. In experiments, we show that this simple scheme proves remarkably effective, as PCs learnt this way set new state of the art for tractable models on many standard density estimation benchmarks.

Multi-dimensional classification (MDC) can be employed in a range of applications where one needs to predict multiple class variables for each given instance. Many existing MDC methods suffer from at least one of inaccuracy, scalability, limited use to certain types of data, hardness of interpretation or lack of probabilistic (uncertainty) estimations. This paper is an attempt to address all these disadvantages simultaneously. We propose a formal framework for probabilistic MDC in which learning an optimal multi-dimensional classifier can be decomposed, without loss of generality, into learning a set of (smaller) single-variable multi-class probabilistic classifiers and a directed acyclic graph. Current and future developments of both probabilistic classification and graphical model learning can directly enhance our framework, which is flexible and provably optimal. A collection of experiments is conducted to highlight the usefulness of this MDC framework.

Data integration is an important component of Big Data analytics. One of the key challenges in data integration is record linkage, that is, matching records that represent the same real-world entity. Because of computational costs, methods referred to as blocking are employed as a part of the record linkage pipeline in order to reduce the number of comparisons among records. In the past decade, a range of blocking techniques have been proposed. Real-world applications require approaches that can handle heterogeneous data sources and do not rely on labelled data. We propose high-value token-blocking (HVTB), a simple and efficient approach for blocking that is unsupervised and schema-agnostic, based on a crafted use of Term Frequency-Inverse Document Frequency. We compare HVTB with multiple methods and over a range of datasets, including a novel unstructured dataset composed of titles and abstracts of scientific papers. We thoroughly discuss results in terms of accuracy, use of computational resources, and different characteristics of datasets and records. The simplicity of HVTB yields fast computations and does not harm its accuracy when compared with existing approaches. It is shown to be significantly superior to other methods, suggesting that simpler methods for blocking should be considered before resorting to more sophisticated methods.

A fundamental task in AI is to assess (in)dependence between mixed-type variables (text, image, sound). We propose a Bayesian kernelised correlation test of (in)dependence using a Dirichlet process model. The new measure of (in)dependence allows us to answer some fundamental questions: Based on data, are (mixed-type) variables independent? How likely is dependence/independence to hold? How high is the probability that two mixed-type variables are more than just weakly dependent? We theoretically show the properties of the approach, as well as algorithms for fast computation with it. We empirically demonstrate the effectiveness of the proposed method by analysing its performance and by comparing it with other frequentist and Bayesian approaches on a range of datasets and tasks with mixed-type variables.

Many algorithms for score-based Bayesian network structure learning (BNSL), in particular exact ones, take as input a collection of potentially optimal parent sets for each variable in the data. Constructing such collections naively is computationally intensive since the number of parent sets grows exponentially with the number of variables. Thus, pruning techniques are not only desirable but essential. While good pruning rules exist for the Bayesian Information Criterion (BIC), current results for the Bayesian Dirichlet equivalent uniform (BDeu) score reduce the search space very modestly, hampering the use of the (often preferred) BDeu. We derive new non-trivial theoretical upper bounds for the BDeu score that considerably improve on the state-of-the-art. Since the new bounds are mathematically proven to be tighter than previous ones and at little extra computational cost, they are a promising addition to BNSL methods.

Decision Trees (DTs) and Random Forests (RFs) are powerful discriminative learners and tools of central importance to the everyday machine learning practitioner and data scientist. Due to their discriminative nature, however, they lack principled methods to process inputs with missing features or to detect outliers, which requires pairing them with imputation techniques or a separate generative model. In this paper, we demonstrate that DTs and RFs can naturally be interpreted as generative models, by drawing a connection to Probabilistic Circuits, a prominent class of tractable probabilistic models. This reinterpretation equips them with a full joint distribution over the feature space and leads to Generative Decision Trees (GeDTs) and Generative Forests (GeFs), a family of novel hybrid generative-discriminative models. This family of models retains the overall characteristics of DTs and RFs while additionally being able to handle missing features by means of marginalisation. Under certain assumptions, frequently made for Bayes consistency results, we show that consistency in GeDTs and GeFs extend to any pattern of missing input features, if missing at random. Empirically, we show that our models often outperform common routines to treat missing data, such as K-nearest neighbour imputation, and moreover, that our models can naturally detect outliers by monitoring the marginal probability of input features.

We present new algorithms for learning Bayesian networks from data with missing values using a data augmentation approach. An exact Bayesian network learning algorithm is obtained by recasting the problem into a standard Bayesian network learning problem without missing data. As expected, the exact algorithm does not scale to large domains. We build on the exact method to create an approximate algorithm using a hill-climbing technique. This algorithm scales to large domains so long as a suitable standard structure learning method for complete data is available. We perform a wide range of experiments to demonstrate the benefits of learning Bayesian networks with such new approach.

Bounding the tree-width of a Bayesian network can reduce the chance of overfitting, and allows exact inference to be performed efficiently. Several existing algorithms tackle the problem of learning bounded tree-width Bayesian networks by learning from k-trees as super-structures, but they do not scale to large domains and/or large tree-width. We propose a guided search algorithm to find k-trees with maximum Informative scores, which is a measure of quality for the k-tree in yielding good Bayesian networks. The algorithm achieves close to optimal performance compared to exact solutions in small domains, and can discover better networks than existing approximate methods can in large domains. It also provides an optimal elimination order of variables that guarantees small complexity for later runs of exact inference. Comparisons with well-known approaches in terms of learning and inference accuracy illustrate its capabilities.

We present a method for learning treewidth-bounded Bayesian networks from data sets containing thousands of variables. Bounding the treewidth of a Bayesian network greatly reduces the complexity of inferences. Yet, being a global property of the graph, it considerably increases the difficulty of the learning process. Our novel algorithm accomplishes this task, scaling both to large domains and to large treewidths. Our novel approach consistently outperforms the state of the art on experiments with up to thousands of variables.

We present a method for learning Bayesian networks from data sets containingthousands of variables without the need for structure constraints. Our approachis made of two parts. The first is a novel algorithm that effectively explores thespace of possible parent sets of a node. It guides the exploration towards themost promising parent sets on the basis of an approximated score function thatis computed in constant time. The second part is an improvement of an existingordering-based algorithm for structure optimization. The new algorithm provablyachieves a higher score compared to its original formulation. On very large datasets containing up to ten thousand nodes, our novel approach consistently outper-forms the state of the art.

Credal networks are graph-based statistical models whose parameters take values in a set, instead of being sharply specified as in traditional statistical models (e.g., Bayesian networks). The computational complexity of inferences on such models depends on the irrelevance/independence concept adopted. In this paper, we study inferential complexity under the concepts of epistemic irrelevance and strong independence. We show that inferences under strong independence are NP-hard even in trees with binary variables except for a single ternary one. We prove that under epistemic irrelevance the polynomial-time complexity of inferences in credal trees is not likely to extend to more general models (e.g., singly connected topologies). These results clearly distinguish networks that admit efficient inferences and those where inferences are most likely hard, and settle several open questions regarding their computational complexity. We show that these results remain valid even if we disallow the use of zero probabilities. We also show that the computation of bounds on the probability of the future state in a hidden Markov model is the same whether we assume epistemic irrelevance or strong independence, and we prove a similar result for inference in naive Bayes structures. These inferential equivalences are important for practitioners, as hidden Markov models and naive Bayes structures are used in real applications of imprecise probability.

Semi-qualitative probabilistic networks (SQPNs) merge two important graphical model formalisms: Bayesian networks and qualitative probabilistic networks. They provide a very general modeling framework by allowing the combination of numeric and qualitative assessments over a discrete domain, and can be compactly encoded by exploiting the same factorization of joint probability distributions that are behind the Bayesian networks. This paper explores the computational complexity of semi-qualitative probabilistic networks, and takes the polytree-shaped networks as its main target. We show that the inference problem is coNP-Complete for binary polytrees with multiple observed nodes. We also show that inferences can be performed in time linear in the number of nodes if there is a single observed node. Because our proof is constructive, we obtain an efficient linear time algorithm for SQPNs under such assumptions. To the best of our knowledge, this is the first exact polynomial-time algorithm for SQPNs. Together these results provide a clear picture of the inferential complexity in polytree-shaped SQPNs.

We present a new algorithm for exactly solving decision making problems represented as influence diagrams. We do not require the usual assumptions of no forgetting and regularity; this allows us to solve problems with simultaneous decisions and limited information. The algorithm is empirically shown to outperform a state-of-the-art algorithm on randomly generated problems of up to 150 variables and 10^64 solutions. We show that these problems are NP-hard even if the underlying graph structure of the problem has low treewidth and the variables take on a bounded number of states, and that they admit no provably good approximation if variables can take on an arbitrary number of states.

This paper presents new results for the (partial) maximum a posteriori (MAP) problem in Bayesian networks, which is the problem of querying the most probable state configuration of some of the network variables given evidence. It is demonstrated that the problem remains hard even in networks with very simple topology, such as binary polytrees and simple trees (including the Naive Bayes structure), which extends previous complexity results. Furthermore, a Fully Polynomial Time Approximation Scheme for MAP in networks with bounded treewidth and bounded number of states per variable is developed. Approximation schemes were thought to be impossible, but here it is shown otherwise under the assumptions just mentioned, which are adopted in most applications.

This paper considers inference from multinomial data and addresses the problem of choosing the strength of the Dirichlet prior under a mean-squared error criterion. We compare the Maxi-mum Likelihood Estimator (MLE) and the most commonly used Bayesian estimators obtained by assuming a prior Dirichlet distribution with non-informative prior parameters, that is, the parameters of the Dirichlet are equal and altogether sum up to the so called strength of the prior. Under this criterion, MLE becomes more preferable than the Bayesian estimators at the increase of the number of categories k of the multinomial, because non-informative Bayesian estimators induce a region where they are dominant that quickly shrinks with the increase of k. This can be avoided if the strength of the prior is not kept constant but decreased with the number of categories. We argue that the strength should decrease at least k times faster than usual estimators do.

We present a new algorithm for exactly solving decision-making problems represented as an influence diagram. We do not require the usual assumptions of no forgetting and regularity, which allows us to solve problems with limited information. The algorithm, which implements a sophisticated variable elimination procedure, is empirically shown to outperform a state-of-the-art algorithm in randomly generated problems of up to 150 variables and $10^{64}$ strategies.

This paper addresses exact learning of Bayesian network structure from data based on the Bayesian Dirichlet score function and its derivations. We describe useful properties that strongly reduce the computational costs of many known methods without losing global optimality guarantees. We show empirically the advantages of the properties in terms of time and memory consumptions, demonstrating that state-of-the-art methods, with the use of such properties, might handle larger data sets than those currently possible.

This paper addresses the problem of learning structure of Bayesian and Dynamic Bayesian networks from incomplete data based on the Bayesian Information Criterion. We describe a procedure to map the problem of the dynamic case into a corresponding augmented Bayesian network through the use of structural constraints. Because the algorithm is exact and anytime, it is well suitable for a structural Expectation-Maximization (EM) method where the only source of approximation is due to the EM itself. We show empirically that the use a global maximizer inside the structural EM is computationally feasible and leads to more accurate models.

This paper addresses exact learning of Bayesian network structure from data and expert's knowledge based on score functions that are decomposable. First, it describes useful properties that strongly reduce the time and memory costs of many known methods such as hill-climbing, dynamic programming and sampling variable orderings. Secondly, a branch and bound algorithm is presented that integrates parameter and structural constraints with data in a way to guarantee global optimality with respect to the score function. It is an any-time procedure because, if stopped, it provides the best current solution and an estimation about how far it is from the global solution. We show empirically the advantages of the properties and the constraints, and the applicability of the algorithm to large data sets (up to one hundred variables) that cannot be handled by other current methods (limited to around 30 variables).

This paper describes a new approach to unify constraints on parameters with training data to perform parameter estimation in Bayesian networks of known structure. The method is general in the sense that any convex constraint is allowed, which includes many proposals in the literature. Driven by a maximum entropy criterion and the Imprecise Dirichlet Model, we present a constrained convex optimization formulation to combine priors, constraints and data. Experiments indicate benefits of this framework.