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Separation Properties of Sets of Probability Measures [article]

Fabio Gagliardi Cozman
2013 arXiv   pre-print
This paper analyzes independence concepts for sets of probability measures associated with directed acyclic graphs.  ...  to epistemic independence and sets of probability measures, and (2) strong independence has a clear justification based on epistemic independence and the strong Markov condition.  ...  Acknowledgements I greatly benefited from joint work with Peter Walley on graphoid properties; I learned about Kuznetsov's independence from him.  ... 
arXiv:1301.3845v1 fatcat:lz3yumby3zerdggu3rekm4psdy

Sets of probability distributions, independence, and convexity

Fabio G. Cozman
2011 Synthese  
The paper offers an organized review of the literature on independence for sets of probability distributions; new results on graphoid properties and on the justification of "strong independence" (using  ...  This paper analyzes concepts of independence and assumptions of convexity in the theory of sets of probability distributions.  ...  ; to Thomas Augustin for help with Weichselberger's theory; and many thanks to Teddy Seidenfeld for patiently explaining to me the theory of choice functions and its relationship with complete independence  ... 
doi:10.1007/s11229-011-9999-0 fatcat:owbdirazibf3vkkb454nh3cnii

Inference with Seperately Specified Sets of Probabilities in Credal Networks [article]

Jose Carlos Ferreira da Rocha, Fabio Gagliardi Cozman
2012 arXiv   pre-print
Credal networks are here interpreted as encoding strong independence relations among variables. We first present a theory of credal networks based on separately specified sets of probabilities.  ...  We present new algorithms for inference in credal networks --- directed acyclic graphs associated with sets of probabilities.  ...  Acknowledgments We thank Marsha Duro from HP Labs, Edson Nery from HP Brasil, and the Instituto de Pesquisas Eldorado for com putational facilities in which some of our experiments were conducted.  ... 
arXiv:1301.0597v1 fatcat:e37mylrfpze2xiiycc4oiucv4a

Graphical models for imprecise probabilities

Fabio Gagliardi Cozman
2005 International Journal of Approximate Reasoning  
The main characteristics of the credal network model are then discussed, as this model has received considerable attention in the literature.  ...  Say that X and Y are strongly independent conditional on Z if the vertices of the credal set K(X, YjZ = z) factorize for every z.  ...  This is the Markov condition for credal networks-note that the condition depends on the adopted concept of independence.  ... 
doi:10.1016/j.ijar.2004.10.003 fatcat:icz32fuck5a65a26ndt5y4ooly

Inference in Polytrees with Sets of Probabilities [article]

Jose Carlos Ferreira da Rocha, Fabio Gagliardi Cozman, Cassio Polpo de Campos
2012 arXiv   pre-print
Inferences in directed acyclic graphs associated with probability sets and probability intervals are NP-hard, even for polytrees.  ...  , WHILE branch - AND - bound procedures can produce either exact OR approximate solutions.We report ON dramatic improvements ON existing techniques FOR inference WITH probability sets AND intervals, IN  ...  This work has received generous support from HP Labs; we thank Marsha Duro from HP Labs for establishing this support and Edson Nery from HP Brazil for managing it.  ... 
arXiv:1212.2458v1 fatcat:u24tty6jdrgkrhhouxynjmktre

Kuznetsov independence for interval-valued expectations and sets of probability distributions: Properties and algorithms

Fabio G. Cozman, Cassio Polpo de Campos
2014 International Journal of Approximate Reasoning  
We present properties of Kuznetsov independence for several variables, and connect it with other concepts of independence in the literature; in particular we show that strong extensions are always included  ...  Finally, we define a concept of conditional Kuznetsov independence, and study its graphoid properties.  ...  Thanks to Serafín Moral for ideas concerning Kuznetsov independence and extensions; to Lev Utkin for generously translating pieces of Kuznetsov's book to English; to Igor Kozine for translating additional  ... 
doi:10.1016/j.ijar.2013.09.013 fatcat:gvdja4skwracjonzkqi2nvhioy

Computing posterior upper expectations

Fabio Gagliardi Cozman
2000 International Journal of Approximate Reasoning  
This article investigates the computation of posterior upper expectations induced by imprecise probabilities, with emphasis on the eects of irrelevance and independence judgements.  ...  Algorithms that handle irrelevance and independence relations in multivariate models are analyzed through graphical representations, inspired by the popular Bayesian network model. Ó  ...  Acknowledgements Thanks to Eric Krotkov for substantial guidance during the development of this work, and to Teddy Seidenfeld for introducing me to the theory of credal sets.  ... 
doi:10.1016/s0888-613x(00)00034-7 fatcat:5oojgdkqc5ewxkg7ietj7iqapy

On the Complexity of Strong and Epistemic Credal Networks [article]

Denis D. Maua, Cassio Polpo de Campos, Alessio Benavoli, Alessandro Antonucci
2013 arXiv   pre-print
We show that inferences under strong independence are NP-hard even in trees with ternary variables.  ...  In this paper, we study inferential complexity under the concepts of epistemic irrelevance and strong independence.  ...  The strong extension is the credal set C(X) whose extrema p(X) satisfy for all x the condition p(x) = i∈N q(x i |x pa(i) ), (1) where q(X i |x pa (i) ) ∈ ext Q(X i |x pa(i) ).  ... 
arXiv:1309.6845v1 fatcat:p5ul4e5odvcyjfwbpwrcwcf75m

Probabilistic Inference in Credal Networks: New Complexity Results

D. D. Maua, C. P. De Campos, A. Benavoli, A. Antonucci
2014 The Journal of Artificial Intelligence Research  
We show that inferences under strong independence are NP-hard even in trees with binary variables except for a single ternary one.  ...  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).  ...  For instance, if stochastic independence is adopted as irrelevance concept, then the DAG G describes a set of Markov conditions as a Bayesian network (stochastic irrelevance implies stochastic independence  ... 
doi:10.1613/jair.4355 fatcat:j3kpexlnnze5ngdtpqilftl5ee

Inference in credal networks: branch-and-bound methods and the A/R+ algorithm

José Carlos Ferreira da Rocha, Fabio Gagliardi Cozman
2005 International Journal of Approximate Reasoning  
A credal network is a graphical representation for a set of joint probability distributions. In this paper we discuss algorithms for exact and approximate inferences in credal networks.  ...  We also propose a new algorithm, A/R+, for outer approximations in polytree-shaped credal networks.  ...  When intervals are locally transformed into credal sets, the optimization problems manipulating these credal sets produce the tightest possible bounds; however they start from larger feasible sets and  ... 
doi:10.1016/j.ijar.2004.10.009 fatcat:xeulvzqdtbdk3i6rwsnl23fqxe

On the complexity of propositional and relational credal networks

Fabio Gagliardi Cozman, Denis Deratani Mauá
2017 International Journal of Approximate Reasoning  
A credal network associates a directed acyclic graph with a collection of sets of probability measures.  ...  Here we examine the complexity of inference in Boolean credal networks when probability measures are specified through formal languages, by extending a framework we have recently proposed for Bayesian  ...  In this paper we adopt the most common concept of independence for credal sets; namely, we adopt strong independence: X and Y are strongly independent given Z if K(X, Y |Z = z) is the convex hull of a  ... 
doi:10.1016/j.ijar.2016.10.008 fatcat:a6ug55v2xzas5m3omnj4ygtj2u

Bayesian Networks with Imprecise Probabilities: Theory and Application to Classification [chapter]

G. Corani, A. Antonucci, M. Zaffalon
2012 Intelligent Systems Reference Library  
Sets instead of single distributions can provide a more realistic description in these cases. Bayesian networks can be generalized to cope with sets of distributions.  ...  In other words, if the available information is not sufficient, credal classifiers allow for indecision between two or more classes, this providing a less informative but more robust conclusion than Bayesian  ...  Strong independence is not the only concept of independence proposed for credal sets.  ... 
doi:10.1007/978-3-642-23166-7_4 fatcat:x7ep6izsujdm7ltfdq7p5saeo4

Probabilistic Graphical Models [chapter]

2015 Modeling and Analysis of Dependable Systems  
In particular, the discussion is focused on credal networks and discrete domains.  ...  It describes the building blocks of credal networks, algorithms to perform inference, and discusses on complexity results and related work.  ...  Thus, if any extreme point of the strong extension obeys the Markov condition with stochastic independence, the strong extension satisfies the Markov condition with strong independence.  ... 
doi:10.1142/9789814612043_0002 fatcat:biowqyaxnfhdrc64mxruk2pl2q

Probabilistic graphical models [chapter]

Alessandro Antonucci, Cassio P. de Campos, Marco Zaffalon
2014 Introduction to Imprecise Probabilities  
In particular, the discussion is focused on credal networks and discrete domains.  ...  It describes the building blocks of credal networks, algorithms to perform inference, and discusses on complexity results and related work.  ...  Thus, if any extreme point of the strong extension obeys the Markov condition with stochastic independence, the strong extension satisfies the Markov condition with strong independence.  ... 
doi:10.1002/9781118763117.ch9 fatcat:6cvbtazhwjeencaqjh3q3rt5em

Imprecise Bayesian Networks as Causal Models

David Kinney
2018 Information  
It is argued that the adequacy conditions for causal representation in the precise context—the Causal Markov Condition and Minimality—do not readily translate into the imprecise context.  ...  Crucial to this argument is the fact that the independence relation between random variables can be understood in several different ways when the joint probability distribution over those variables is  ...  Acknowledgments: I am very grateful to Jonathan Birch, Luc Bovens, Katie Steele, and the anonymous reviewers of Information for their feedback on earlier drafts of this paper.  ... 
doi:10.3390/info9090211 fatcat:onbnlo5bebbknnf2ze7ya6nzqm
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