Contagion processes are strongly linked to the network structures on which they propagate, and learning these structures is essential for understanding and intervention on complex network processes such as epidemics and (mis)information propagation. However, using contagion data to infer network structure is a challenging inverse problem. In particular, it is imperative to have appropriate measures of uncertainty in network structure estimates, however these are largely ignored in most machine-learning approaches. We present a probabilistic framework that uses samples from the distribution of networks that are compatible with the dynamics observed to produce network and uncertainty estimates. We demonstrate the method using the well known independent cascade model to sample from the distribution of networks P(G) conditioned on the observation of a set of infections C. We evaluate the accuracy of the method by using the marginal probabilities of each edge in the distribution, and show the bene ts of quantifying uncertainty to improve estimates and understanding, particularly with small amounts of data.
Embar propose a Bayesian framework for estimating properties of the network, edge strengths as well as cascade properties, propagation trees @cite_15 . Other Bayesian approaches of network inference aim to infer the underlying network when the exact infection time is unknown in both standard and online algorithms @cite_33 @cite_8 @cite_1 . Additionally, some approaches based on MLE use Bayesian techniques at intermediate steps to improve results @cite_11 . Only recently have Bayesian techniques have been introduced to quantify the uncertainty of network structure inference from observation of dynamics @cite_16 @cite_42 @cite_40 @cite_13 . Ghalebi highlight the need for general probabilistic frameworks for inference problems and propose the algorithm DYFERENCE that samples edge and node probabilities in an online algorithm for dynamic network inference @cite_38 . Peixoto @cite_37 recently proposed the most closely related work in an algorithm designed to jointly reconstruct network structure and community labels assuming a stochastic block model structure and highlights the benefits of recovering the full posterior distribution of networks.
@cite_38 @cite_37 @cite_33 @cite_8 @cite_42 @cite_1 @cite_40 @cite_15 @cite_16 @cite_13 @cite_11
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We describe a Bayesian approach for learning Bayesian networks from a combination of prior knowledge and statistical data. First and foremost, we develop a methodology for assessing informative priors needed for learning. Our approach is derived from a set of assumptions made previously as well as the assumption of likelihood equivalence, which says that data should not help to discriminate network structures that represent the same assertions of conditional independence. We show that likelihood equivalence when combined with previously made assumptions implies that the user's priors for network parameters can be encoded in a single Bayesian network for the next case to be seen—a prior network—and a single measure of confidence for that network. Second, using these priors, we show how to compute the relative posterior probabilities of network structures given data. Third, we describe search methods for identifying network structures with high posterior probabilities. We describe polynomial algorithms for finding the highest-scoring network structures in the special case where every node has at most k e 1 parent. For the general case (k > 1), which is NP-hard, we review heuristic search algorithms including local search, iterative local search, and simulated annealing. Finally, we describe a methodology for evaluating Bayesian-network learning algorithms, and apply this approach to a comparison of various approaches. These lecture notes highlight the mathematical and computational structure relating to the formulation of, and development of algorithms for, the Bayesian approach to inverse problems in differential equations. This approach is fundamental in the quantification of uncertainty within applications in volving the blending of mathematical models with data. The finite dimensional situation is described first, along with some motivational examples. Then the development of probability measures on separable Banach space is undertaken, using a random series over an infinite set of functions to construct draws; these probability measures are used as priors in the Bayesian approach to inverse problems. Regularity of draws from the priors is studied in the natural Sobolev or Besov spaces implied by the choice of functions in the random series construction, and the Kolmogorov continuity theorem is used to extend regularity considerations to the space of Holder continuous functions. Bayes’ theorem is de rived in this prior setting, and here interpreted as finding conditions under which the posterior is absolutely continuous with respect to the prior, and determining a formula for the Radon-Nikodym derivative in terms of the likelihood of the data. Having established the form of the posterior, we then describe various properties common to it in the infinite dimensional setting. These properties include well-posedness, approximation theory, and the existence of maximum a posteriori estimators. We then describe measure-preserving dynamics, again on the infinite dimensional space, including Markov chain-Monte C arlo and sequential Monte Carlo methods, and measure-preserving reversible stochastic differential equations. By formulating the theory and algorithms on the underlying infinite dimensional space, we obtain a framework suitable for rigorous analysis of the accuracy of reconstructions, of computational complexity, as well as naturally constructing algorithms which perform well under mesh refinement, since they are inherently well-defined in infinite dimensions. In many multivariate domains, we are interested in analyzing the dependency structure of the underlying distribution, e.g., whether two variables are in direct interaction. We can represent dependency structures using Bayesian network models. To analyze a given data set, Bayesian model selection attempts to find the most likely (MAP) model, and uses its structure to answer these questions. However, when the amount of available data is modest, there might be many models that have non-negligible posterior. Thus, we want compute the Bayesian posterior of a feature, i.e., the total posterior probability of all models that contain it. In this paper, we propose a new approach for this task. We first show how to efficiently compute a sum over the exponential number of networks that are consistent with a fixed order over network variables. This allows us to compute, for a given order, both the marginal probability of the data and the posterior of a feature. We then use this result as the basis for an algorithm that approximates the Bayesian posterior of a feature. Our approach uses a Markov Chain Monte Carlo (MCMC) method, but over orders rather than over network structures. The space of orders is smaller and more regular than the space of structures, and has much a smoother posterior “landscape”. We present empirical results on synthetic and real-life datasets that compare our approach to full model averaging (when possible), to MCMC over network structures, and to a non-Bayesian bootstrap approach. Many interesting domains in machine learning can be viewed as networks, with relationships (e.g., friendships) connecting items (e.g., individuals). The Active Exploration (AE) task is to identify all items in a network with a desired trait (i.e., positive labels) given only partial information about the network. The AE process iteratively queries for labels or network structure within a limited budget; thus, accurate predictions prior to making each query is critical to maximizing the number of positives gathered. However, the targeted AE query process produces partially observed networks that can create difficulties for predictive modeling. In particular, we demonstrate that these partial networks can exhibit extreme label correlation bias, which makes it difficult for conventional relational learning methods to accurately estimate relational parameters. To overcome this issue, we model the joint distribution of possible edges and labels to improve learning and inference. Our proposed method, Probabilistic Relational Expectation Maximization (PR-EM), is the first AE approach to accurately learn the complex dependencies between attributes, labels, and structure to improve predictions. PR-EM utilizes collective inference over the missing relationships in the partial network to jointly infer unknown item traits. Further, we develop a linear inference algorithm to facilitate efficient use of PR-EM in large networks. We test our approach on four real world networks, showing that AE with PR-EM gathers significantly more positive items compared to state-of-the-art methods.