While fiducial inference was widely considered a big blunder by R.A. Fisher, the goal he initially set --`inferring the uncertainty of model parameters on the basis of observations' -- has been continually pursued by many statisticians. To this end, we develop a new statistical inference method called extended Fiducial inference (EFI). The new method achieves the goal of fiducial inference by leveraging advanced statistical computing techniques while remaining scalable for big data. EFI involves jointly imputing random errors realized in observations using stochastic gradient Markov chain Monte Carlo and estimating the inverse function using a sparse deep neural network (DNN). The consistency of the sparse DNN estimator ensures that the uncertainty embedded in observations is properly propagated to model parameters through the estimated inverse function, thereby validating downstream statistical inference. Compared to frequentist and Bayesian methods, EFI offers significant advantages in parameter estimation and hypothesis testing. Specifically, EFI provides higher fidelity in parameter estimation, especially when outliers are present in the observations; and eliminates the need for theoretical reference distributions in hypothesis testing, thereby automating the statistical inference process. EFI also provides an innovative framework for semi-supervised learning.

翻译：尽管基准推断被广泛认为是R.A. Fisher的重大失误，但他最初设定的目标——"基于观测推断模型参数的不确定性"——始终被众多统计学家所追求。为此，我们开发了一种名为扩展基准推断（EFI）的新统计推断方法。该方法通过运用先进的统计计算技术实现了基准推断的目标，同时保持了对大数据的可扩展性。EFI涉及使用随机梯度马尔可夫链蒙特卡洛方法联合插补观测中实现的随机误差，并利用稀疏深度神经网络（DNN）估计逆函数。稀疏DNN估计量的一致性确保了观测中蕴含的不确定性能够通过估计的逆函数正确传递至模型参数，从而验证下游统计推断的有效性。与频率学派和贝叶斯方法相比，EFI在参数估计和假设检验方面展现出显著优势。具体而言，EFI在参数估计中具有更高的保真度，尤其在观测值存在异常值时；并在假设检验中消除了对理论参考分布的依赖，从而实现了统计推断过程的自动化。EFI还为半监督学习提供了一个创新性框架。