Approximate credal network updating by linear programming with applications to decision making

Credal nets are probabilistic graphical models which extend Bayesian nets to cope with sets of distributions. An algorithm for approximate credal network updating is presented. The problem in its general formulation is a multilinear optimization task, which can be linearized by an appropriate rule for fixing all the local models apart from those of a single variable. This simple idea can be iterated and quickly leads to accurate inferences. A transformation is also derived to reduce decision making in credal networks based on the maximality criterion to updating. The decision task is proved to have the same complexity of standard inference, being NP^PP-complete for general credal nets and NP-complete for polytrees. Similar results are derived for the E-admissibility criterion. Numerical experiments confirm a good performance of the method.     

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.     

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.     

Belief revision conditionals: basic iterated systems

It is now well known that, on pain of triviality, the probability of a conditional cannot be identified with the corresponding conditional probability [25]. This surprising impossibility result has a qualitative counterpart. In fact, Peter Gärdenfors showed in [13] that believing ‘If A then B’ cannot be equated with the act of believing B on the supposition that A — as long as supposing obeys minimal Bayesian constraints.Recent work has shown that in spite of these negative results, the question ‘how to accept a conditional?’ has a clear answer. Even if conditionals are not truth-carriers, they do have precise acceptability conditions. Nevertheless most epistemic models of conditionals do not provide acceptance conditions for iterated conditionals. One of the main goals of this essay is to provide a comprehensive account of the notion of epistemic conditionality covering all forms of iteration.First we propose an account of the basic idea of epistemic conditionality, by studying the conditionals validated by epistemic models where iteration is permitted but not constrained by special axioms. Our modeling does not presuppose that epistemic states should be represented by belief sets (we only assume that to each epistemic state corresponds an associated belief state). A full encoding of the basic epistemic conditionals (encompassing all forms of iteration) is presented and a representation result is proved.In the second part of the essay we argue that the notion of change involved in the evaluation of conditionals is suppositional, and that such notion should be distinguished from the notion of updating (modelled by AGM and other methods). We conclude by considering how some of the recent modellings of iterated change fare as methods for iterated supposing.     

Conformity and independence with coherent lower previsions

We define the conformity of marginal and conditional models with a joint model within Walley's theory of coherent lower previsions. Loosely speaking, conformity means that the joint can reproduce the marginal and conditional models we started from. By studying conformity with and without additional assumptions of epistemic irrelevance and independence, we establish connections with a number of prominent models in Walley's theory: the marginal extension, the irrelevant natural extension, the independent natural extension and the strong product.     

Objective Bayesian probabilistic logic

This paper develops connections between objective Bayesian epistemology—which holds that the strengths of an agent's beliefs should be representable by probabilities, should be calibrated with evidence of empirical probability, and should otherwise be equivocal—and probabilistic logic. After introducing objective Bayesian epistemology over propositional languages, the formalism is extended to handle predicate languages. A rather general probabilistic logic is formulated and then given a natural semantics in terms of objective Bayesian epistemology. The machinery of objective Bayesian nets and objective credal nets is introduced and this machinery is applied to provide a calculus for probabilistic logic that meshes with the objective Bayesian semantics.     

Evaluating credal classifiers by utility-discounted predictive accuracy

Predictions made by imprecise-probability models are often indeterminate (that is, set-valued). Measuring the quality of an indeterminate prediction by a single number is important to fairly compare different models, but a principled approach to this problem is currently missing. In this paper we derive, from a set of assumptions, a metric to evaluate the predictions of credal classifiers. These are supervised learning models that issue set-valued predictions. The metric turns out to be made of an objective component, and another that is related to the decision-maker’s degree of risk aversion to the variability of predictions. We discuss when the measure can be rendered independent of such a degree, and provide insights as to how the comparison of classifiers based on the new measure changes with the number of predictions to be made. Finally, we make extensive empirical tests of credal, as well as precise, classifiers by using the new metric. This shows the practical usefulness of the metric, while yielding a first insightful and extensive comparison of credal classifiers.     

Comments on “Statistical reasoning with set-valued information: Ontic vs. epistemic view” by Inés Couso and Didier Dubois

I discuss some aspects of the distinction between ontic and epistemic views of sets as representation of imprecise or incomplete information. In particular, I consider its implications on imprecise probability representations: credal sets and sets of desirable gambles. It is emphasized that the interpretation of the same mathematical object can be different depending on the point of view from which this element is considered. In the case of a fuzzy information on a random variable, it is possible to define a possibility distribution on the simplex of probability distributions. I add some comments about the properties of this possibility distribution.     

AN EFFECTIVE DETAILED ROUTING ALGORITHM CONSIDERING ADVANCED TECHNOLOGY NODES

Detailed routing has become much challenging in modern circuit designs due to the extreme scaling of chip size and the complicated design rules. In this paper, we give an effective algorithm for detailed routing considering advanced technology nodes. First, we present a valid pin-access candidates generation technology for handling complex pin shapes. Then, we propose a tree-based nets components selection algorithm to decide connecting order for multiple nets components. Finally, combined with global routing results and advanced technology nodes, an initial routing results optimization algorithm is presented to achieve the final detailed routing results. Experimental results on industry benchmarks show that, our proposed algorithm not only achieves 100%routability on real industrial cases in a reasonable runtime, but also optimizes total wirelength, total vias and other advanced technology nodes simultaneously.     

一种扩展的影响图及其求解方法

针对影响图在实际应用中的参数建模困难问题,提出了一种扩展的影响图.引入credal集作为影响图的概率参数,以表达专家的不精确和不完整信度,集成多来源的定性和定量信息.引入credal集后,影响图的推理难度进一步加大.提出了将其转化为credal网络求解的思路,并给出了一种基于路径选择的求解算法.最后用一个实例验证了算法的有效性.     

Non Parametric Distributed Inference in Sensor Networks Using Box Particles Messages

This paper deals with the problem of inference in distributed systems where the probability model is stored in a distributed fashion. Graphical models provide powerful tools for modeling this kind of problems. Inspired by the box particle filter which combines interval analysis with particle filtering to solve temporal inference problems, this paper introduces a belief propagation-like message-passing algorithm that uses bounded error methods to solve the inference problem defined on an arbitrary graphical model. We show the theoretic derivation of the novel algorithm and we test its performance on the problem of calibration in wireless sensor networks. That is the positioning of a number of randomly deployed sensors, according to some reference defined by a set of anchor nodes for which the positions are known a priori. The new algorithm, while achieving a better or similar performance, offers impressive reduction of the information circulating in the network and the needed computation times.     

An ensemble method using credal decision trees

Supervised classification learning can be considered as an important tool for decision support. In this paper, we present a method for supervised classification learning, which ensembles decision trees obtained via convex sets of probability distributions (also called credal sets) and uncertainty measures. Our method forces the use of different decision trees and it has mainly the following characteristics: it obtains a good percentage of correct classifications and an improvement in time of processing compared with known classification methods; it not needs to fix the number of decision trees to be used; and it can be parallelized to apply it on very large data sets.     

Backfitting neural networks

Summary  Regression and classification problems can be viewed as special cases of the problem of function estimation. It is rather well known that a two-layer perceptron with sigmoidal transformation functions can approximate any continuous function on the compact subsets ofRP if there are sufficient number of hidden nodes. In this paper, we present an algorithm for fitting perceptron models, which is quite different from the usual backpropagation or Levenberg-Marquardt algorithm. This new algorithm based on backfitting ensures a better convergence than backpropagation. We have also used resampling techniques to select an ideal number of hidden nodes automatically using the training data itself. This resampling technique helps to avoid the problem of overfitting that one faces for the usual perceptron learning algorithms without any model selection scheme. Case studies and simulation results are presented to illustrate the performance of this proposed algorithm.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Imprecise probabilities for representing ignorance about a parameter

This paper distinguishes between objective probability—or chance—and subjective probability. Most statistical methods in machine learning are based on the hypothesis that there is a random experiment from which we get a set of observations. This random experiment could be identified with a chance or objective probability, but these probabilities depend on some unknown parameters. Our knowledge of these parameters is not objective and in order to learn about them, we must assess some epistemic probabilities about their values. In some cases, our objective knowledge about these parameters is vacuous, so the question is: What epistemic probabilities should be assumed? In this paper we argue for the assumption of non-vacuous (a proper subset of [0, 1]) interval probabilities. There are several reasons for this; some are based on the betting interpretation of epistemic probabilities while others are based on the learning capabilities under the vacuous representation. The implications of the selection of epistemic probabilities in different concepts as conditioning and learning are studied. It is shown that in order to maintain some reasonable learning capabilities we have to assume more informative prior models than those frequently used in the literature, such as the imprecise Dirichlet model.     

A general formulation for mean-value analysis in product-form batch-movement queueing networks

A number of recent papers have shown that many classes of queueing networks with batches of customers served and routed through the network have equilibrium distributions which factorise into product forms over the nodes of the network. In this paper we demonstrate how such networks are amenable to a mean-value analysis which generalises that used for single-movement networks.Since product-form stochastic Petri nets (SPNs) can be viewed as batch-movement queueing networks, our algorithm is also applicable to their analysis.     

A simple randomised algorithm for convex optimisation

We consider maximising a concave function over a convex set by a simple randomised algorithm. The strength of the algorithm is that it requires only approximate function evaluations for the concave function and a weak membership oracle for the convex set. Under smoothness conditions on the function and the feasible set, we show that our algorithm computes a near-optimal point in a number of operations which is bounded by a polynomial function of all relevant input parameters and the reciprocal of the desired precision, with high probability. As an application to which the features of our algorithm are particularly useful we study two-stage stochastic programming problems. These problems have the property that evaluation of the objective function is #P-hard under appropriate assumptions on the models. Therefore, as a tool within our randomised algorithm, we devise a fully polynomial randomised approximation scheme for these function evaluations, under appropriate assumptions on the models. Moreover, we deal with smoothing the feasible set, which in two-stage stochastic programming is a polyhedron.     

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.     

Hill-climbing and branch-and-bound algorithms for exact and approximate inference in credal networks

This paper proposes two new algorithms for inference in credal networks. These algorithms enable probability intervals to be obtained for the states of a given query variable. The first algorithm is approximate and uses the hill-climbing technique in the Shenoy–Shafer architecture to propagate in join trees; the second is exact and is a modification of Rocha and Cozman’s branch-and-bound algorithm, but applied to general directed acyclic graphs.     

Topology discovery of sparse random graphs with few participants

We consider the task of topology discovery of sparse random graphs using end‐to‐end random measurements (e.g., delay) between a subset of nodes, referred to as the participants. The rest of the nodes are hidden, and do not provide any information for topology discovery. We consider topology discovery under two routing models: (a) the participants exchange messages along the shortest paths and obtain end‐to‐end measurements, and (b) additionally, the participants exchange messages along the second shortest path. For scenario (a), our proposed algorithm results in a sub‐linear edit‐distance guarantee using a sub‐linear number of uniformly selected participants. For scenario (b), we obtain a much stronger result, and show that we can achieve consistent reconstruction when a sub‐linear number of uniformly selected nodes participate. This implies that accurate discovery of sparse random graphs is tractable using an extremely small number of participants. We finally obtain a lower bound on the number of participants required by any algorithm to reconstruct the original random graph up to a given edit distance. We also demonstrate that while consistent discovery is tractable for sparse random graphs using a small number of participants, in general, there are graphs which cannot be discovered by any algorithm even with a significant number of participants, and with the availability of end‐to‐end information along all the paths between the participants. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013