Inference from limited data requires a notion of measure on parameter space, most explicit in the Bayesian framework as a prior. Here we demonstrate that Jeffreys prior, the best-known uninformative choice, introduces enormous bias when applied to typical scientific models. Such models have a relevant effective dimensionality much smaller than the number of microscopic parameters. Because Jeffreys prior treats all microscopic parameters equally, it is from uniform when projected onto the sub-space of relevant parameters, due to variations in the local co-volume of irrelevant directions. We present results on a principled choice of measure which avoids this issue, leading to unbiased inference in complex models. This optimal prior depends on the quantity of data to be gathered, and approaches Jeffreys prior in the asymptotic limit. However, this limit cannot be justified without an impossibly large amount of data, exponential in the number of microscopic parameters.

翻译：从有限数据中进行推断需要对参数空间有一个测度概念，这在贝叶斯框架中最明确地体现为先验分布。本文证明，最著名的无信息先验——杰弗里斯先验，在应用于典型科学模型时会引入巨大偏差。这类模型的相关有效维度远小于微观参数数量。由于杰弗里斯先验平等对待所有微观参数，在投影到相关参数子空间时，因无关方向的局部协体积变化而缺乏均匀性。我们提出了一种避免该问题的原则性测度选择方案，可在复杂模型中实现无偏推断。该最优先验取决于待收集数据量，并在渐近极限下趋近于杰弗里斯先验。然而，这一极限需要指数级于微观参数数量的海量数据才能成立，实际上难以实现。