The information bottleneck (IB) method aims to find compressed representations of a variable $X$ that retain the most relevant information about a target variable $Y$. We show that for a wide family of distributions -- namely, when $Y$ is generated by $X$ through a Hamming channel, under mild conditions -- the optimal IB representations require an alphabet strictly larger than that of $X$. This implies that, despite several recent works, the cardinality bound first identified by Witsenhausen and Wyner in 1975 is tight. At the core of our finding is the observation that the IB function in this setting is not strictly concave, similar to the deterministic case, even though the joint distribution of $X$ and $Y$ is of full support. Finally, we provide a complete characterization of the IB function, as well as of the optimal representations for the Hamming case.

翻译：信息瓶颈（IB）方法旨在寻找变量$X$的压缩表示，该表示保留关于目标变量$Y$的最相关信息。我们证明，对于一大类分布——即当$Y$由$X$通过汉明信道生成，且在温和条件下——最优IB表示所需的字母表严格大于$X$的字母表。这意味着，尽管近期有多项研究，但Witsenhausen和Wyner在1975年首次识别的基数界是紧的。我们发现的核心在于，此设定下的IB函数并非严格凹的，与确定情形类似，即使$X$与$Y$的联合分布具有全支撑。最后，我们给出了汉明情形下IB函数以及最优表示的完整刻画。