随机性与维数的散度公式（简本）
A Divergence Formula for Randomness and Dimension (Short Version)
摘要：
如果$S$是有限字母表$Σ$上的无限序列，$β$是$Σ$上的概率测度，则$ S$关于$β$的{ \it维数}，写作$\dim^β(S)$，是比林斯利维数的构造性版本，当$β$是一致概率测度时，它与（构造性Hausdorff）维数$\dim(S)$一致。本文证明了$\dim^β(S)$及其对偶$\Dim^β(S)$，即$S$关于$β$的{ \it强维数}，可以与随机性结合使用来度量$Σ$上的两个概率测度$α$和$β$的相似性。具体地说，我们证明了{ \it发散公式} $$\dim^β(R) = \Dim^β(R) =\CH(α) / (\CH(α) + \D(α|| β))$$持有$α$和$β$是可计算的，积极的概率措施$Σ$和$R \in Σ^\infty$是随机的$α$。在这个公式中，$\CH(α)$是$α$的香农熵，而$\D(α||β)$是$α$和$β$之间的Kullback-Leibler散度。
英文摘要：
If $S$ is an infinite sequence over a finite alphabet $Σ$ and $β$ is a probability measure on $Σ$, then the {\it dimension} of $ S$ with respect to $β$, written $\dim^β(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $β$ is the uniform probability measure. This paper shows that $\dim^β(S)$ and its dual $\Dim^β(S)$, the {\it strong dimension} of $S$ with respect to $β$, can be used in conjunction with randomness to measure the similarity of two probability measures $α$ and $β$ on $Σ$. Specifically, we prove that the {\it divergence formula} $$\dim^β(R) = \Dim^β(R) =\CH(α) / (\CH(α) + \D(α|| β))$$ holds whenever $α$ and $β$ are computable, positive probability measures on $Σ$ and $R \in Σ^\infty$ is random with respect to $α$. In this formula, $\CH(α)$ is the Shannon entropy of $α$, and $\D(α||β)$ is the Kullback-Leibler divergence between $α$ and $β$.
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