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具有基础重要性的量。在实践中，它被广泛用于快速估计数据集中可用的预期信息，并指导实验设计选择。在许多现代应用中，解析计算Fisher信息非常困难，因此转而使用蒙特卡洛方法。标准蒙特卡洛方法对Fisher信息的估计在蒙特卡洛噪声不可忽略时可能存在偏差。最成问题的是导数中的噪声，这会导致对可用约束能力（由逆Fisher信息给出）的高估。在本工作中，我们发现另一种简单估计量具有相反的偏差，会产生对约束能力的低估。该估计量既可用于给出参数约束的近似边界，也可与标准估计量结合使用，以改进得到近似无偏的估计。这两种替代估计量和组合估计量都是渐近无偏的，因此也可作为标准方法收敛性的检验手段。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，因为用更少的蒙特卡洛样本就能获得无偏估计，从而可将模拟数据集规模减少数个数量级。