107 research outputs found
Updating beliefs with incomplete observations
Currently, there is renewed interest in the problem, raised by Shafer in
1985, of updating probabilities when observations are incomplete. This is a
fundamental problem in general, and of particular interest for Bayesian
networks. Recently, Grunwald and Halpern have shown that commonly used updating
strategies fail in this case, except under very special assumptions. In this
paper we propose a new method for updating probabilities with incomplete
observations. Our approach is deliberately conservative: we make no assumptions
about the so-called incompleteness mechanism that associates complete with
incomplete observations. We model our ignorance about this mechanism by a
vacuous lower prevision, a tool from the theory of imprecise probabilities, and
we use only coherence arguments to turn prior into posterior probabilities. In
general, this new approach to updating produces lower and upper posterior
probabilities and expectations, as well as partially determinate decisions.
This is a logical consequence of the existing ignorance about the
incompleteness mechanism. We apply the new approach to the problem of
classification of new evidence in probabilistic expert systems, where it leads
to a new, so-called conservative updating rule. In the special case of Bayesian
networks constructed using expert knowledge, we provide an exact algorithm for
classification based on our updating rule, which has linear-time complexity for
a class of networks wider than polytrees. This result is then extended to the
more general framework of credal networks, where computations are often much
harder than with Bayesian nets. Using an example, we show that our rule appears
to provide a solid basis for reliable updating with incomplete observations,
when no strong assumptions about the incompleteness mechanism are justified.Comment: Replaced with extended versio
Inference in credal networks: branch-and-bound methods and the A/R+ algorithm
AbstractA 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 propose a branch-and-bound framework for inference, and focus on inferences for polytree-shaped networks. We also propose a new algorithm, A/R+, for outer approximations in polytree-shaped credal networks
Multi-class Image Segmentation in Fluorescence Microscopy Using Polytrees
Multi-class segmentation is a crucial step in cell image analysis. This process becomes challenging when little information is available for recognising cells from the background, due to their poor discriminative features. To alleviate this, directed acyclic graphs such as trees have been proposed to model top-down statistical dependencies as a prior for improved image segmentation. However, using trees, modelling the relations between labels of multiple classes becomes difficult. To overcome this limitation, we propose a polytree graphical model that captures label proximity relations more naturally compared to tree based approaches. A novel recursive mechanism based on two-pass message passing is developed to efficiently calculate closed form posteriors of graph nodes on the polytree. The algorithm is evaluated using simulated data, synthetic images and real fluorescence microscopy images. Our method achieves Dice scores of 94.5% and 98% on macrophage and seed classes, respectively, outperforming GMM based classifiers
New Results for the MAP Problem in Bayesian Networks
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.
First, 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). Such proofs extend previous complexity results for the
problem. Inapproximability results are also derived in the case of trees if the
number of states per variable is not bounded. Although the problem is shown to
be hard and inapproximable even in very simple scenarios, a new exact algorithm
is described that is empirically fast in networks of bounded treewidth and
bounded number of states per variable. The same algorithm is used as basis of a
Fully Polynomial Time Approximation Scheme for MAP under such assumptions.
Approximation schemes were generally thought to be impossible for this problem,
but we show otherwise for classes of networks that are important in practice.
The algorithms are extensively tested using some well-known networks as well as
random generated cases to show their effectiveness.Comment: A couple of typos were fixed, as well as the notation in part of
section 4, which was misleading. Theoretical and empirical results have not
change
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