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Machine learning is an interdisciplinary field of science and engineering that studies mathematical theories and practical applications of systems that learn. This book introduces theories, methods and applications of density ratio estimation, which is a newly emerging paradigm in the machine learning community. Various machine learning problems such as non-stationarity adaptation, outlier detection, dimensionality reduction, independent component analysis, clustering, classification and conditional density estimation can be systematically solved via the estimation of probability density ratios. The authors offer a comprehensive introduction of various density ratio estimators including methods via density estimation, moment matching, probabilistic classification, density fitting and density ratio fitting as well as describing how these can be applied to machine learning. The book provides mathematical theories for density ratio estimation including parametric and non-parametric convergence analysis and numerical stability analysis to complete the first and definitive treatment of the entire framework of density ratio estimation in machine learning. Table Of Contents Part I. Density Ratio Approach to Machine Learning Introduction Part II. Methods of Density Ratio Estimation Density estimation Moment matching Probabilistic classification Density fitting Density-ratio fitting Unified framework Direct density-ratio estimation with dimensionality reduction Part III. Applications of Density Ratios in Machine Learning Importance sampling Distribution comparison Mutual information estimation  Conditional probability estimation Part IV. Theoretical Analysis of Density Ratio Estimation Parametric convergence analysis Non-parametric convergence analysis Parametric two-sample test Non-parametric numerical stability analysis Part V. Conclusions Conclusions and future directions.