Collision statistics provide a finite-resolution view of information by measuring how often a fixed number of independent samples fall on the same state. These directly countable quantities form the basis of integer-order Rényi entropies. Here, we use low-order Rényi entropies to approximate Shannon entropy and mutual information, while characterizing what is necessarily lost when only finitely many collision moments are used. We derive interpolation-error bounds showing that approximation error is controlled by the shape of the Rényi entropy path near the Shannon point. We also separate this deterministic error from finite-sample estimation error: for fixed collision order, increasing sample size improves estimation of the finite-resolution target but does not eliminate its deterministic difference from Shannon entropy or mutual information. Finally, we show that finite collision moments do not generally identify Shannon entropy, and that increasing collision order shifts sensitivity toward high-probability events. Numerical experiments illustrate the approximation--estimation tradeoff and compare collision-based approximations with plug-in and Miller--Madow estimators. The framework links collision counts, Rényi entropy, Shannon limits, and mutual information through a finite-resolution view of information, clarifying when low-order coincidence structure is informative and when irreducible information is lost.

翻译：碰撞统计通过测量固定数量的独立样本同时落在同一状态的频率，提供了信息的有限分辨率视角。这些可直接计数的量构成了整数阶Rényi熵的基础。本文利用低阶Rényi熵近似Shannon熵和互信息，同时刻画了仅使用有限次碰撞矩时必然丢失的信息特征。我们推导了插值误差界，证明近似误差受Rényi熵路径在Shannon点附近形状的控制。此外，我们将这种确定性误差与有限样本估计误差分离：对于固定的碰撞阶数，增加样本量可改进对有限分辨率目标的估计，但无法消除其与Shannon熵或互信息之间的确定性差异。最后，我们证明有限次碰撞矩一般无法完全确定Shannon熵，且提高碰撞阶数会使敏感性向高概率事件偏移。数值实验展示了近似-估计的权衡关系，并将基于碰撞的近似与插件估计器及Miller-Madow估计器进行了比较。该框架通过信息有限分辨率视角，将碰撞计数、Rényi熵、Shannon极限和互信息联系起来，阐明了低阶重合结构何时具有信息量，以及何时不可逆地丢失信息。