This paper focuses on parameter estimation and introduces a new method for lower bounding the Bayesian risk. The method allows for the use of virtually \emph{any} information measure, including R\'enyi's $\alpha$, $\varphi$-Divergences, and Sibson's $\alpha$-Mutual Information. The approach considers divergences as functionals of measures and exploits the duality between spaces of measures and spaces of functions. In particular, we show that one can lower bound the risk with any information measure by upper bounding its dual via Markov's inequality. We are thus able to provide estimator-independent impossibility results thanks to the Data-Processing Inequalities that divergences satisfy. The results are then applied to settings of interest involving both discrete and continuous parameters, including the ``Hide-and-Seek'' problem, and compared to the state-of-the-art techniques. An important observation is that the behaviour of the lower bound in the number of samples is influenced by the choice of the information measure. We leverage this by introducing a new divergence inspired by the ``Hockey-Stick'' Divergence, which is demonstrated empirically to provide the largest lower-bound across all considered settings. If the observations are subject to privatisation, stronger impossibility results can be obtained via Strong Data-Processing Inequalities. The paper also discusses some generalisations and alternative directions.

翻译：本文聚焦于参数估计问题，提出了一种贝叶斯风险下界的新方法。该方法允许使用几乎任意信息测度，包括Rényi α散度、φ散度以及Sibson α互信息。该框架将散度视为测度的泛函，并利用测度空间与函数空间之间的对偶性。具体而言，我们证明通过马尔可夫不等式对偶函数进行上界估计，即可实现用任意信息测度对风险进行下界约束。得益于散度满足的数据处理不等式，我们能够给出与估计器无关的不可行性结果。随后，该结果被应用于涉及离散与连续参数的典型场景（包括“藏匿与搜索”问题），并与现有最优技术进行了对比。一个重要发现是：下界随样本数量的变化行为受信息测度选择的影响。我们据此引入一种受“冰球棍”散度启发的新散度，实验证明该散度在所有考察场景中均能提供最大的下界值。若观测数据经历隐私化处理，则可通过强数据处理不等式获得更强的不可行性结果。本文还探讨了若干推广方向与替代性研究路径。