The information bottleneck method, introduced by Tishby, Pereira, and Bialek, addresses the problem of extracting relevant information from a signal \( x \in X \) that provides information about another signal \( y \in Y \). The method formalizes this problem as finding a short code for \( X \) that preserves the maximum information about \( Y \) through a 'bottleneck' formed by a limited set of codewords \( \tilde{X} \). This constrained optimization problem generalizes rate distortion theory, where the distortion measure \( d(x, \tilde{x}) \) emerges from the joint statistics of \( X \) and \( Y \). The approach yields self-consistent equations for the coding rules \( X \to \tilde{X} \) and \( \tilde{X} \to Y \), which can be solved using an iterative algorithm that generalizes the Blahut-Arimoto algorithm. The method provides a rich framework for discussing various problems in signal processing and learning, including prediction, filtering, and learning. The paper also discusses the structure of solutions and the application of the information bottleneck principle to real-world problems such as semantic clustering, document classification, neural coding, and spectral analysis.The information bottleneck method, introduced by Tishby, Pereira, and Bialek, addresses the problem of extracting relevant information from a signal \( x \in X \) that provides information about another signal \( y \in Y \). The method formalizes this problem as finding a short code for \( X \) that preserves the maximum information about \( Y \) through a 'bottleneck' formed by a limited set of codewords \( \tilde{X} \). This constrained optimization problem generalizes rate distortion theory, where the distortion measure \( d(x, \tilde{x}) \) emerges from the joint statistics of \( X \) and \( Y \). The approach yields self-consistent equations for the coding rules \( X \to \tilde{X} \) and \( \tilde{X} \to Y \), which can be solved using an iterative algorithm that generalizes the Blahut-Arimoto algorithm. The method provides a rich framework for discussing various problems in signal processing and learning, including prediction, filtering, and learning. The paper also discusses the structure of solutions and the application of the information bottleneck principle to real-world problems such as semantic clustering, document classification, neural coding, and spectral analysis.