On the Complexity of Strong and Epistemic Credal Networks

Credal networks are graph-based statistical models whose parameters take values on a set, instead of being sharply specified as in traditional statistical models (e.g., Bayesian networks). The result of inferences with such models depends on the irrelevance/independence concept adopted. In this paper, we study the computational complexity of inferences under the concepts of epistemic irrelevance and strong independence. We strengthen complexity results by showing that inferences with strong independence are NP-hard even in credal trees with ternary variables, which indicates that tractable algorithms, including the existing one for epistemic trees, cannot be used for strong independence. We prove that the polynomial time of inferences in credal trees under epistemic irrelevance is not likely to extend to more general models, because the problem becomes NP-hard even in simple polytrees. These results draw a definite line between networks with efficient inferences and those where inferences are hard, and close several open questions regarding the computational complexity of such models.

Complexity of Inferences in Polytree-shaped Semi-Qualitative Probabilistic Networks

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 linear time if there is a single observed node, which is a relevant practical case. 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.

Transform both sides model: A parametric approach

A parametric regression model for right-censored data with a log-linear median regression function and a transformation in both response and regression parts, named parametric Transform-Both-Sides (TBS) model, is presented. The TBS model has a parameter that handles data asymmetry while allowing various different distributions for the error, as long as they are unimodal symmetric distributions centered at zero. The discussion is focused on the estimation procedure with five important error distributions (normal, double-exponential, Student's t, Cauchy and logistic) and presents properties, associated functions (that is, survival and hazard functions) and estimation methods based on maximum likelihood and on the Bayesian paradigm. These procedures are implemented in TBSSurvival, an open-source fully documented R package. The use of the package is illustrated and the performance of the model is analyzed using both simulated and real data sets.

Prognostic Impact of Monocyte Count at Presentation in Mantle Cell Lymphoma

Increased number of circulating monocytes at presentation has been recently associated with shorter survival in Hodgkin lymphoma, follicular lymphoma and diffuse large B cell lymphoma. This study aimed to assess the prognostic impact of the absolute monocyte count (AMC) at diagnosis in mantle cell lymphoma (MCL). From the series of MCL cases recorded on the databases of the Oncology Institute of Southern Switzerland in Bellinzona (Switzerland) and the Division of Haematology of the Amedeo Avogadro University of Eastern Piedmont in Novara (Italy), the AMC at diagnosis was available in 97 cases. Cox regression was used for both univariate and multivariate analysis. With a median follow up of 7 years, the 5-year overall survival (OS) was 29% for patients with AMC >500/ul and 62% for patients with AMC

Efficient Structure Learning of Bayesian Networks using Constraints

This paper addresses the problem of learning Bayesian network structures from data based on score functions that are decomposable. It describes properties that strongly reduce the time and memory costs of many known methods without losing global optimality guarantees. These properties are derived for different score criteria such as Minimum Description Length (or Bayesian Information Criterion), Akaike Information Criterion and Bayesian Dirichlet Criterion. Then a branch-and-bound algorithm is presented that integrates structural constraints with data in a way to guarantee global optimality. As an example, structural constraints are used to map the problem of structure learning in Dynamic Bayesian networks into a corresponding augmented Bayesian network. Finally, we show empirically the benefits of using the properties with state-of-the-art methods and with the new algorithm, which is able to handle larger data sets than before.

IDS: A Divide-and-Conquer Algorithm for Inference in Polytree-Shaped Credal Networks

A credal network is a graph-theoretic model that represents imprecision in joint probability distributions. An inference in a credal net aims at computing an interval for the probability of an event of interest. Algorithms for inference in credal networks can be divided into exact and approximate. The selection of an algorithm is based on a trade off that ponders how much time someone wants to spend in a particular calculation against the quality of the computed values. This paper presents an algorithm, called IDS, that combines exact and approximate methods for computing inferences in polytree-shaped credal networks. The algorithm provides an approach to trade time and precision when making inferences in credal nets

Solving Decision Problems with Limited Information

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.

Updating Credal Networks is Approximable in Polynomial Time

Credal networks relax the precise probability requirement of Bayesian networks, enabling a richer representation of uncertainty in the form of closed convex sets of probability measures. The increase in expressiveness comes at the expense of higher computational costs. In this paper, we present a new variable elimination algorithm for exactly computing posterior inferences in extensively specified credal networks, which is empirically shown to outperform a state-of-the-art algorithm. The algorithm is then turned into a provably good approximation scheme, that is, a procedure that for any input is guaranteed to return a solution not worse than the optimum by a given factor. Remarkably, we show that when the networks have bounded treewidth and bounded number of states per variable the approximation algorithm runs in time polynomial in the input size and in the inverse of the error factor, thus being the first known fully polynomial-time approximation scheme for inference in credal networks.

A tree augmented classifier based on Extreme Imprecise Dirichlet Model

We present TANC, a TAN classifier (tree-augmented naive) based on imprecise probabilities. TANC models prior near-ignorance via the Extreme Imprecise Dirichlet Model (EDM). A first contribution of this paper is the experimental comparison between EDM and the global Imprecise Dirichlet Model using the naive credal classifier (NCC), with the aim of showing that EDM is a sensible approximation of the global IDM. TANC is able to deal with missing data in a conservative manner by considering all possible completions (without assuming them to be missing-at-random), but avoiding an exponential increase of the computational time. By experiments on real data sets, we show that TANC is more reliable than the Bayesian TAN and that it provides better performance compared to previous TANs based on imprecise probabilities. Yet, TANC is sometimes outperformed by NCC because the learned TAN structures are too complex; this calls for novel algorithms for learning the TAN structures, better suited for an imprecise probability classifier.

Generalized loopy 2U: A new algorithm for approximate inference in credal networks

Credal networks generalize Bayesian networks by relaxing the requirement of precision of probabilities. Credal networks are considerably more expressive than Bayesian networks, but this makes belief updating NP-hard even on polytrees. We develop a new efficient algorithm for approximate belief updating in credal networks. The algorithm is based on an important representation result we prove for general credal networks: that any credal network can be equivalently reformulated as a credal network with binary variables; moreover, the transformation, which is considerably more complex than in the Bayesian case, can be implemented in polynomial time. The equivalent binary credal network is then updated by L2U, a loopy approximate algorithm for binary credal networks. Overall, we generalize L2U to non-binary credal networks, obtaining a scalable algorithm for the general case, which is approximate only because of its loopy nature. The accuracy of the inferences with respect to other state-of-the-art algorithms is evaluated by extensive numerical tests.

Semi-Qualitative Probabilistic Networks in Computer Vision Problems

This paper explores the application of semi-qualitative probabilistic networks (SQPNs) that combine numeric and qualitative information to computer vision problems. Our version of SQPN allows qualitative influences and imprecise probability measures using intervals. We describe an Imprecise Dirichlet model for parameter learning and an iterative algorithm for evaluating posterior probabilities, maximum a posteriori and most probable explanations. Experiments on facial expression recognition and image segmentation problems are performed using real data.

Assembling a consistent set of sentences in relational probabilistic logic with stochastic independence

We examine the representation of judgements of stochastic independence in probabilistic logics. We focus on a relational logic where (i) judgements of stochastic independence are encoded by directed acyclic graphs, and (ii) probabilistic assessments are flexible in the sense that they are not required to specify a single probability measure. We discuss issues of knowledge representation and inference that arise from our particular combination of graphs, stochastic independence, logical formulas and probabilistic assessments.

Probabilistic logic with independence

This paper investigates probabilistic logics endowed with independence relations. We review propositional probabilistic languages without and with independence. We then consider graph-theoretic representations for propositional probabilistic logic with independence; complexity is analyzed, algorithms are derived, and examples are discussed. Finally, we examine a restricted first-order probabilistic logic that generalizes relational Bayesian networks.

Computing lower and upper expectations under epistemic independence

This paper investigates the computation of lower/upper expectations that must cohere with a collection of probabilistic assessments and a collection of judgements of epistemic independence. New algorithms, based on multilinear programming, are presented, both for independence among events and among random variables. Separation properties of graphical models are also investigated.