The Fisher information matrix is a quantity of fundamental importance for information geometry and asymptotic statistics. In practice, it is widely used to quickly estimate the expected information available in a data set and guide experimental design choices. In many modern applications, it is intractable to analytically compute the Fisher information and Monte Carlo methods are used instead. The standard Monte Carlo method produces estimates of the Fisher information that can be biased when the Monte-Carlo noise is non-negligible. Most problematic is noise in the derivatives as this leads to an overestimation of the available constraining power, given by the inverse Fisher information. In this work we find another simple estimate that is oppositely biased and produces an underestimate of the constraining power. This estimator can either be used to give approximate bounds on the parameter constraints or can be combined with the standard estimator to give improved, approximately unbiased estimates. Both the alternative and the combined estimators are asymptotically unbiased so can be also used as a convergence check of the standard approach. We discuss potential limitations of these estimators and provide methods to assess their reliability. These methods accelerate the convergence of Fisher forecasts, as unbiased estimates can be achieved with fewer Monte Carlo samples, and so can be used to reduce the simulated data set size by several orders of magnitude.

翻译：Fisher信息矩阵是信息几何和渐近统计学中至关重要的量。在实践中，它被广泛用于快速估计数据集中可用的期望信息，并指导实验设计选择。在许多现代应用中，解析计算Fisher信息是不可行的，因此转而使用蒙特卡洛方法。当蒙特卡洛噪声不可忽略时，标准蒙特卡洛方法对Fisher信息的估计可能存在偏差。最成问题的是导数中的噪声，这会导致对可用约束能力的过高估计，而约束能力由逆Fisher信息给出。在本工作中，我们发现了另一种简单估计量，其偏差方向相反，会产生对约束能力的低估。该估计量既可以用于给出参数约束的近似界限，也可以与标准估计量结合使用，以提供改进的、近似无偏的估计。替代估计量和组合估计量都是渐近无偏的，因此也可用于标准方法的收敛性检查。我们讨论了这些估计量的潜在局限性，并提供了评估其可靠性的方法。这些方法加速了Fisher预测的收敛性，因为只需更少的蒙特卡洛样本即可实现无偏估计，从而可以将模拟数据集的大小降低几个数量级。