A Review on Quantification Learning <s> Direct Estimation Methods <s> A situation where training and test samples follow different input distributions is called covariate shift. Under covariate shift, standard learning methods such as maximum likelihood estimation are no longer consistent—weighted variants according to the ratio of test and training input densities are consistent. Therefore, accurately estimating the density ratio, called the importance, is one of the key issues in covariate shift adaptation. A naive approach to this task is to first estimate training and test input densities separately and then estimate the importance by taking the ratio of the estimated densities. However, this naive approach tends to perform poorly since density estimation is a hard task particularly in high dimensional cases. In this paper, we propose a direct importance estimation method that does not involve density estimation. Our method is equipped with a natural cross validation procedure and hence tuning parameters such as the kernel width can be objectively optimized. Simulations illustrate the usefulness of our approach. <s> BIB001 </s> A Review on Quantification Learning <s> Direct Estimation Methods <s> Class distribution estimation (quantification) plays an important role in many practical classification problems. Firstly, it is important in order to adapt the classifier to the operational conditions when they differ from those assumed in learning. Additionally, there are some real domains where the quantification task is itself valuable due to the high variability of the class prior probabilities. Our novel quantification approach for two-class problems is based on distributional divergence measures. The mismatch between the test data distribution and validation distributions generated in a fully controlled way is measured by the Hellinger distance in order to estimate the prior probability that minimizes this divergence. Experimental results on several binary classification problems show the benefits of this approach when compared to such approaches as counting the predicted class labels and other methods based on the classifier confusion matrix or on posterior probability estimations. We also illustrate these techniques as well as their robustness against the base classifier performance (a neural network) with a boar semen quality control setting. Empirical results show that the quantification can be conducted with a mean absolute error lower than 0.008, which seems very promising in this field. <s> BIB002 </s> A Review on Quantification Learning <s> Direct Estimation Methods <s> Let X denote the feature and Y the target. We consider domain adaptation under three possible scenarios: (1) the marginal PY changes, while the conditional PX/Y stays the same (target shift), (2) the marginal PY is fixed, while the conditional PX/Y changes with certain constraints (conditional shift), and (3) the marginal PY changes, and the conditional PX/Y changes with constraints (generalized target shift). Using background knowledge, causal interpretations allow us to determine the correct situation for a problem at hand. We exploit importance reweighting or sample transformation to find the learning machine that works well on test data, and propose to estimate the weights or transformations by reweighting or transforming training data to reproduce the covariate distribution on the test domain. Thanks to kernel embedding of conditional as well as marginal distributions, the proposed approaches avoid distribution estimation, and are applicable for high-dimensional problems. Numerical evaluations on synthetic and real-world data sets demonstrate the effectiveness of the proposed framework. <s> BIB003 </s> A Review on Quantification Learning <s> Direct Estimation Methods <s> We consider the problem of learning a classifier using only positive and unlabeled samples. In this setting, it is known that a classifier can be successfully learned if the class prior is available. However, in practice, the class prior is unknown and thus must be estimated from data. In this paper, we propose a new method to estimate the class prior by partially matching the class-conditional density of the positive class to the input density. By performing this partial matching in terms of the Pearson divergence, which we estimate directly without density estimation via lower-bound maximization, we can obtain an analytical estimator of the class prior. We further show that an existing class prior estimation method can also be interpreted as performing partial matching under the Pearson divergence, but in an indirect manner. The superiority of our direct class prior estimation method is illustrated on several benchmark datasets. <s> BIB004 </s> A Review on Quantification Learning <s> Direct Estimation Methods <s> In recent times, many real world applications have emerged that require estimates of class ratios in an unlabeled instance collection as opposed to labels of individual instances in the collection. In this paper we investigate the use of maximum mean discrepancy (MMD) in a reproducing kernel Hilbert space (RKHS) for estimating such ratios. ::: ::: First, we theoretically analyze the MMD-based estimates. Our analysis establishes that, under some mild conditions, the estimate is statistically consistent. More importantly, it provides an upper bound on the error in the estimate in terms of intuitive geometric quantities like class separation and data spread. Next, we use the insights obtained from the theoretical analysis, to propose a novel convex formulation that automatically learns the kernel to be employed in the MMD-based estimation. We design an efficient cutting plane algorithm for solving this formulation. Finally, we empirically compare our estimator with several existing methods, and show significantly improved performance under varying datasets, class ratios, and training sizes. <s> BIB005 </s> A Review on Quantification Learning <s> Direct Estimation Methods <s> In many real-world classication problems, the class balance often changes between training and test datasets, due to sample selection bias or the non-stationarity of the environment. Naive classier training under such changes of class balance systematically yields a biased solution. It is known that such a systematic bias can be corrected by weighted training according to the test class balance. However, the test class balance is often unknown in practice. In this paper, we consider a semi- supervised learning setup where labeled training samples and unlabeled test samples are available and propose a class balance estimator based on the energy distance. Through experiments, we demonstrate that the proposed method is computationally much more efficient than existing approaches, with comparable accuracy. <s> BIB006
Many of the methods in this group correspond to the proposals of the group leaded by Sugiyama BIB001  BIB004 BIB006 . Instead of using the two-step procedure followed by the previous methods, where (i) two probability densities are separately estimated and then (ii) their difference is computed, these authors propose a direct estimation approach. They argue that a two-step procedure may fail, because both steps are performed without considering the other. For instance, a small estimate error in the first step can produce large mistakes in the final estimation. They propose a set of methods for directly estimating the density ratio or the density difference without estimating two densities individually. These methods can be applied to quantification tasks, despite the fact that these authors never use this term, preferring others such as class-prior estimation or even class balance change. Given a labelled training set and a unlabelled testing set, the direct strategy to estimate P tst (y), which is different than P tr (y), is to fit a mixture of classwise densities using the training set to the test input density P tst (x ): The parameter vectorp is computed considering a particular divergence metric. Notice that the final expression is due to the classic learning assumption in quantification learning: P tst (y) P tr (y) but P tst (x |y) = P tr (x |y). Previous approaches, like those of Forman (2005) (Section 8.3) and BIB002 and  (Section 8.4) estimate first P tr (x |y) and P tst (x ), using, respectively, the training set and the testing set, and, finally, they try to approximate P tst (x ) and P tst (x ) given some metric. The difference between the following methods is precisely the target divergence measure. Probably, one of the most interesting approaches of this group is presented in du Plessis and . This article introduces a framework for class-prior estimation by means of direct A Review on Quantification Learning 74:29 divergence minimization. The KL divergence is computed between P tst (x ) and P tst (x ): The aim is to compute the class probability distribution,p, which minimizes the KL divergence, subject to the usual constraints, that is, l j=1p j = 1,p j ≥ 0. This optimization problem can be solved using several methods, but the authors propose a common framework based on direct divergence minimization: Given an estimator of a target divergence from P tst to P tst , for instance, KL divergence or Person (PE) divergence, learn the coefficientŝ p that minimizes such divergence. An f -divergence is a general divergence measure defined as in which f is convex function such that f (1) = 0. Several approaches can be used to approximate the f -divergence. The first solution is following the aforementioned two-step approach to estimate P tst (x ) and P tst (x ) separately and then plug these estimations into Equation (64). But as it was state above, the authors claim that this approach can be improved using a direct divergence minimization. We omit here some mathematical details (see Du Plessis and Sugiyama (2014b)), but via convex duality we can obtain the following estimator for Equation (64): This expression has nice properties because only contains expectations that can be approximated by sample averages. The maximum for a continuous f is obtained for a function r such that P tst (x )/P tst (x ) = ∂ f * (r (x )), which means that f -divergence is directly estimated in terms of the density ratio. This present some advantages, because the estimation of densities is a more complex problem than the estimation of a density ratio . Using the Gaussian kernel to define a model for the density ratio r (x ): being α the parameters and ψ the basis functions: ) and ψ 0 (x ) = 1. The constant basis function is included to deal with two equal distributions (r (x ) = 1). The final element is a regularizer to avoid overfitting that it is defined as λα Rα , being R a squared matrix of size b + 1: Finally, the model for the density ratio is obtained solving this optimization problem: To apply this framework to a f -Divergence, like KL Divergence or PE Divergence, we have to chose the function f and its convex conjugate f * . The method presents the additional advantage that the parameters, for example, the regularization parameter or the Gaussian width, can be adjusted by cross-validation. The main drawback is the computational complexity, because the computation of the divergence estimator requires that α must to be optimized for eachp j . The rest of the methods conceptually follow a similar strategy. The differences between them are basically twofold: the way for matching the distributions and the loss function used. In , the method is aimed at estimating the difference, in terms of the L2-distance, between two probability densities, P tr (x ) − P tst (x ). In , the idea is to obtain an estimate of the density ratio, P tst (x )/P tr (x ), and it is adapted for two different measures (KL and PE). BIB006 introduce the energy distance and propose to use it in class-prior estimation. This last method is related to BIB003 and BIB005 , which analyze the use of the Maximum Mean Discrepancy (MMD) with kernels for class ratio estimation. The energy distance is a special case of MMD, and the advantage of BIB006 is that no parameters need to be tuned, and thus it is highly computationally efficient.