论文标题
每个有限参考度量都存在通用密度
Universal Densities Exist for Every Finite Reference Measure
论文作者
论文摘要
众所周知，始终如一地估计熵率的通用代码是在有限字母上固定的厄贡源存在的，但不是超过无限的字母。我们将通用编码概括为通用密度相对于固定参考度量的通用密度问题，这些量子在一个可计量生成的可测量空间上。我们表明，有限参考度量的普遍密度始终存在差异熵率。因此，从某种意义上说，有限字母不是必需的。为了表现出通用密度，我们通过Feutrill和Roughan适应了非参数差分（NPD）熵率估计器。我们的修改类似于Ryabko通过Cleary和Witten对部分匹配（PPM）对预测的修改。 Ryabko考虑了Markov订单上的混合物，但我们考虑了在量化水平上的混合物。此外，我们证明，任何通用密度都会引起有条件密度的强度一致的cesàRo平均估计量，而这些估计值给定过去的过去。这产生了一个通用预测指标，可算字母的$ 0-1 $损失。最后，我们将普遍密度专门用于自然数和实际线路上的过程。我们得出了足够的条件，以一致地估计这些域中的无限参考度量。
As it is known, universal codes, which estimate the entropy rate consistently, exist for stationary ergodic sources over finite alphabets but not over countably infinite ones. We generalize universal coding as the problem of universal densities with respect to a fixed reference measure on a countably generated measurable space. We show that universal densities, which estimate the differential entropy rate consistently, exist for finite reference measures. Thus finite alphabets are not necessary in some sense. To exhibit a universal density, we adapt the non-parametric differential (NPD) entropy rate estimator by Feutrill and Roughan. Our modification is analogous to Ryabko's modification of prediction by partial matching (PPM) by Cleary and Witten. Whereas Ryabko considered a mixture over Markov orders, we consider a mixture over quantization levels. Moreover, we demonstrate that any universal density induces a strongly consistent Cesàro mean estimator of conditional density given an infinite past. This yields a universal predictor with the $0-1$ loss for a countable alphabet. Finally, we specialize universal densities to processes over natural numbers and on the real line. We derive sufficient conditions for consistent estimation of the entropy rate with respect to infinite reference measures in these domains.
