Learning with label proportions (LLP), which is a learning task that only provides unlabeled data in bags and each bag's label proportion, has widespread successful applications in practice. However, most of the existing LLP methods don't consider the knowledge transfer for uncertain data. This paper presents a transfer learning-based approach for the problem of learning with label proportions(TL-LLP) to transfer knowledge from source task to target task where both the source and target tasks contain uncertain data. Our approach first formulates objective model for the uncertain data and deals with transfer learning at the same time, and then proposes an iterative framework to build an accurate classifier for the target task. Extensive experiments have shown that the proposed TL-LLP method can obtain the better accuracies and is less sensitive to noise compared with the existing LLP methods.
Some methods are proposed to deal with LLP problem based on probabilistic models. Hern 'a @cite_19 adopt several versions of an Expectation-Maximization algorithm to learn a naive Bayes model which assumes conditional independence between the predictive variables. @cite_55 propose a new learning framework from the Bayesian perspective by estimating the conditional class density to estimate the posterior probability. Meanwhile, with the deep belief networks model for estimating the log-probability, they rebuild the posterior probability for classification problem. @cite_36 build a probabilistic approach applied to the US presidential election, which uses cardinality potentials to perform exact inference over latent variables during learning, and introduces a novel message-passing algorithm to extend cardinality potentials to multivariate probability models. @cite_26 develop a models to estimate the relationship between political sentiment and demographics during the U.S. presidential election.
@cite_36 @cite_19 @cite_55 @cite_26
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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. Opinion mining and demographic attribute inference have many applications in social science. In this paper, we propose models to infer daily joint probabilities of multiple latent attributes from Twitter data, such as political sentiment and demographic attributes. Since it is costly and time-consuming to annotate data for traditional supervised classification, we instead propose scalable Learning from Label Proportions (LLP) models for demographic and opinion inference using U.S. Census, national and state political polls, and Cook partisan voting index as population level data. In LLP classification settings, the training data is divided into a set of unlabeled bags, where only the label distribution in of each bag is known, removing the requirement of instance-level annotations. Our proposed LLP model, Weighted Label Regularization (WLR), provides a scalable generalization of prior work on label regularization to support weights for samples inside bags, which is applicable in this setting where bags are arranged hierarchically (e.g., county-level bags are nested inside of state-level bags). We apply our model to Twitter data collected in the year leading up to the 2016 U.S. presidential election, producing estimates of the relationships among political sentiment and demographics over time and place. We find that our approach closely tracks traditional polling data stratified by demographic category, resulting in error reductions of 28-44 over baseline approaches. We also provide descriptive evaluations showing how the model may be used to estimate interactions among many variables and to identify linguistic temporal variation, capabilities which are typically not feasible using traditional polling methods. We present a new solution to the ecological inference'' problem, of learning individual-level associations from aggregate data. This problem has a long history and has attracted much attention, debate, claims that it is unsolvable, and purported solutions. Unlike other ecological inference techniques, our method makes use of unlabeled individual-level data by embedding the distribution over these predictors into a vector in Hilbert space. Our approach relies on recent learning theory results for distribution regression, using kernel embeddings of distributions. Our novel approach to distribution regression exploits the connection between Gaussian process regression and kernel ridge regression, giving us a coherent, Bayesian approach to learning and inference and a convenient way to include prior information in the form of a spatial covariance function. Our approach is highly scalable as it relies on FastFood, a randomized explicit feature representation for kernel embeddings. We apply our approach to the challenging political science problem of modeling the voting behavior of demographic groups based on aggregate voting data. We consider the 2012 US Presidential election, and ask: what was the probability that members of various demographic groups supported Barack Obama, and how did this vary spatially across the country? Our results match standard survey-based exit polling data for the small number of states for which it is available, and serve to fill in the large gaps in this data, at a much higher degree of granularity. We investigate the class of computable probability distributions and explore the fundamental limitations of using this class to describe and compute conditional distributions. In addition to proving the existence of noncomputable conditional distributions, and thus ruling out the possibility of generic probabilistic inference algorithms (even inefficient ones), we highlight some positive results showing that posterior inference is possible in the presence of additional structure like exchangeability and noise, both of which are common in Bayesian hierarchical modeling. This theoretical work bears on the development of probabilistic programming languages (which enable the specification of complex probabilistic models) and their implementations (which can be used to perform Bayesian reasoning). The probabilistic programming approach is particularly well suited for defining infinite-dimensional, recursively-defined stochastic processes of the sort used in nonparametric Bayesian statistics. We present a new construction of the Mondrian process as a partition-valued Markov process in continuous time, which can be viewed as placing a distribution on an infinite kd-tree data structure. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253-1690.)