26 research outputs found
Algebraic model counting
Weighted model counting (WMC) is a well-known inference task on knowledge bases, and the basis for some of the most efficient techniques for probabilistic inference in graphical models. We introduce algebraic model counting (AMC), a generalization of WMC to a semiring structure that provides a unified view on a range of tasks and existing results. We show that AMC generalizes many well-known tasks in a variety of domains such as probabilistic inference, soft constraints and network and database analysis. Furthermore, we investigate AMC from a knowledge compilation perspective and show that all AMC tasks can be evaluated using sd-DNNF circuits, which are strictly more succinct, and thus more efficient to evaluate, than direct representations of sets of models. We identify further characteristics of AMC instances that allow for evaluation on even more succinct circuits
Probabilistic Inference in Piecewise Graphical Models
In many applications of probabilistic inference the models
contain piecewise densities that are differentiable except at
partition boundaries. For instance, (1) some models may
intrinsically have finite support, being constrained to some
regions; (2) arbitrary density functions may be approximated by
mixtures of piecewise functions such as piecewise polynomials or
piecewise exponentials; (3) distributions derived from other
distributions (via random variable transformations) may be highly
piecewise; (4) in applications of Bayesian inference such as
Bayesian discrete classification and preference learning, the
likelihood functions may be piecewise; (5) context-specific
conditional probability density functions (tree-CPDs) are
intrinsically piecewise; (6) influence diagrams (generalizations
of Bayesian networks in which along with probabilistic inference,
decision making problems are modeled) are in many applications
piecewise; (7) in probabilistic programming, conditional
statements lead to piecewise models. As we will show, exact
inference on piecewise models is not often scalable (if
applicable) and the performance of the existing approximate
inference techniques on such models is usually quite poor.
This thesis fills this gap by presenting scalable and accurate
algorithms for inference in piecewise probabilistic graphical
models. Our first contribution is to present a variation of Gibbs
sampling algorithm that achieves an exponential sampling speedup
on a large class of models (including Bayesian models with
piecewise likelihood functions). As a second contribution, we
show that for a large range of models, the time-consuming Gibbs
sampling computations that are traditionally carried out per
sample, can be computed symbolically, once and prior to the
sampling process. Among many potential applications, the
resulting symbolic Gibbs sampler can be used for fully automated
reasoning in the presence of deterministic constraints among
random variables. As a third contribution, we are motivated by
the behavior of Hamiltonian dynamics in optics âin particular,
the reflection and refraction of light on the refractive
surfacesâ to present a new Hamiltonian Monte Carlo method that
demonstrates a significantly improved performance on piecewise
models.
Hopefully, the present work represents a step towards scalable
and accurate inference in an important class of probabilistic
models that has largely been overlooked in the literature
Carte de compilation des diagrammes de décision ordonnés a valeurs réelles
National audienceValued decision diagrams (VDDs) are data structures that represent functions mapping variable-value assignments to non-negative real numbers. They prove useful to compile cost functions, utility functions, or probability distributions. While the complexity of some queries (notably optimization) and transformations (notably conditioning) on VDD languages has been known for some time, there remain many significant queries and transformations, such as the various kinds of cuts, marginalizations, and combinations, the complexity of which has not been identified so far. This paper contributes to filling this gap and completing previous results about the time and space efficiency of VDD languages, thus leading to a knowledge compilation map for real-valued functions. Our results show that many tasks that are hard on valued CSPs are actually tractable on VDDs.Les diagrammes de dĂ©cision valuĂ©s (VDDs) sont des structures de donnĂ©es reprĂ©sentant des fonctions Ă valeurs rĂ©elles positives. Ces structures sont utiles pour la compilation de fonctions de coĂ»t ou d'utilitĂ©, ou encore de distributions de probabilitĂ©s. Si la complexitĂ© de certaines requĂȘtes (comme l'optimisation) et de certaines transformations (comme le conditionnement) sur de tels langages est bien connue, il reste de nombreuses requĂȘtes et transformations importantes dont la complexitĂ© n'a pas encore Ă©tĂ© identifiĂ©e ; figurent parmi elles diffĂ©rents types de coupes, marginalisations, ou encore combinaisons. En Ă©tablissant une carte de compilation des diagrammes de dĂ©cision ordonnĂ©s Ă valeurs rĂ©elles, cet article contribue Ă combler ce manque. Nos rĂ©sultats montrent que beaucoup de tĂąches difficiles Ă partir de CSPs valuĂ©s sont traitables Ă partir de VDDs
Interactive Cost Configuration Over Decision Diagrams
Abstract In many AI domains such as product configuration, a user should interactively specify a solution that must satisfy a set of constraints. In such scenarios, offline compilation of feasible solutions into a tractable representation is an important approach to delivering efficient backtrack-free user interaction online. In particular, binary decision diagrams (BDDs) have been successfully used as a compilation target for product and service configuration. In this paper we discuss how to extend BDD-based configuration to scenarios involving cost functions which express user preferences. We first show that an efficient, robust and easy to implement extension is possible if the cost function is additive, and feasible solutions are represented using multi-valued decision diagrams (MDDs). We also discuss the effect on MDD size if the cost function is non-additive or if it is encoded explicitly into MDD. We then discuss interactive configuration in the presence of multiple cost functions. We prove that even in its simplest form, multiple-cost configuration is NP-hard in the input MDD. However, for solving two-cost configuration we develop a pseudo-polynomial scheme and a fully polynomial approximation scheme. The applicability of our approach is demonstrated through experiments over real-world configuration models and product-catalogue datasets. Response times are generally within a fraction of a second even for very large instances
Valueâbased potentials: Exploiting quantitative information regularity patterns in probabilistic graphical models
This study was jointly supported by the Spanish Ministry of Education and Science under projects PID2019-106758GB-C31 and TIN2016-77902-C3-2-P, and the European Regional Development Fund (FEDER). Funding for open access charge from Universidad de Granada/CBUA.When dealing with complex models (i.e., models with
many variables, a high degree of dependency between
variables, or many states per variable), the efficient representation
of quantitative information in probabilistic
graphical models (PGMs) is a challenging task. To address
this problem, this study introduces several new structures,
aptly named valueâbased potentials (VBPs), which are
based exclusively on the values. VBPs leverage repeated
values to reduce memory requirements. In the present
paper, they are compared with some common structures,
like standard tables or unidimensional arrays, and probability
trees (PT). Like VBPs, PTs are designed to reduce
the memory space, but this is achieved only if value repetitions
correspond to contextâspecific independence
patterns (i.e., repeated values are related to consecutive
indices or configurations). VBPs are devised to overcome
this limitation. The goal of this study is to analyze the
properties of VBPs. We provide a theoretical analysis of
VBPs and use them to encode the quantitative information
of a set of wellâknown Bayesian networks, measuring
the access time to their content and the computational
time required to perform some inference tasks.Spanish Government PID2019-106758GB-C31
TIN2016-77902-C3-2-PEuropean Commissio
On the Complexity of Optimization Problems based on Compiled NNF Representations
Optimization is a key task in a number of applications. When the set of
feasible solutions under consideration is of combinatorial nature and described
in an implicit way as a set of constraints, optimization is typically NP-hard.
Fortunately, in many problems, the set of feasible solutions does not often
change and is independent from the user's request. In such cases, compiling the
set of constraints describing the set of feasible solutions during an off-line
phase makes sense, if this compilation step renders computationally easier the
generation of a non-dominated, yet feasible solution matching the user's
requirements and preferences (which are only known at the on-line step). In
this article, we focus on propositional constraints. The subsets L of the NNF
language analyzed in Darwiche and Marquis' knowledge compilation map are
considered. A number of families F of representations of objective functions
over propositional variables, including linear pseudo-Boolean functions and
more sophisticated ones, are considered. For each language L and each family F,
the complexity of generating an optimal solution when the constraints are
compiled into L and optimality is to be considered w.r.t. a function from F is
identified