We thank the reviewers for their helpful comments. We first provide individual responses to each reviewer's comments For example, one could easily apply this method to the last row of a neural network. We will make the suggested changes to improve the writing. We will make this reference in the text of the paper. The reviewer correctly points out that [19] doesn't estimate individual densities but directly estimates the weight.

Importance weighting is widely applicable in machine learning in general and in techniques dealing with data covariate shift problems in particular. A novel, direct approach to determine such importance weighting is presented. It relies on a nearest neighbor classification scheme and is relatively straightforward to implement. Comparative experiments on various classification tasks demonstrate the effectiveness of our so-called nearest neighbor weighting (NNeW) scheme. Considering its performance, our procedure can act as a simple and effective baseline method for importance weighting.

A default assumption in many machine learning scenarios is that the training and test samples are drawn from the same probability distribution. However, such an assumption is often violated in the real world due to non-stationarity of the environment or bias in sample selection. In this work, we consider a prevalent setting called covariate shift, where the input distribution differs between the training and test stages while the conditional distribution of the output given the input remains unchanged. Most of the existing methods for covariate shift adaptation are two-step approaches, which first calculate the importance weights and then conduct importance-weighted empirical risk minimization. In this paper, we propose a novel one-step approach that jointly learns the predictive model and the associated weights in one optimization by minimizing an upper bound of the test risk. We theoretically analyze the proposed method and provide a generalization error bound. We also empirically demonstrate the effectiveness of the proposed method.

Methods for estimating the ratio of two probability density functions have been actively explored recently since they can be used for various data processing tasks such as non-stationarity adaptation, outlier detection, feature selection, and conditional probability estimation. In this paper, we propose a new density-ratio estimator which incorporates dimensionality reduction into the density-ratio estimation procedure. Through experiments, the proposed method is shown to compare favorably with existing density-ratio estimators in terms of both accuracy and computational costs.

We address the problem of estimating the ratio of two probability density functions (a.k.a.~the importance). The importance values can be used for various succeeding tasks such as non-stationarity adaptation or outlier detection. In this paper, we propose a new importance estimation method that has a closed-form solution; the leave-one-out cross-validation score can also be computed analytically. Therefore, the proposed method is computationally very efficient and numerically stable. We also elucidate theoretical properties of the proposed method such as the convergence rate and approximation error bound. Numerical experiments show that the proposed method is comparable to the best existing method in accuracy, while it is computationally more efficient than competing approaches.