关于通过泊松竞赛实现分布转移的注记
A note on shifting distributions via Poisson races
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中文总结 AI 辅助
该注记针对从提议分布$P$模拟目标分布$Q$的问题，基于泊松竞赛方法给出了模拟器$I$的期望对数代价的上下界，完善了分布转移模拟的相关理论。
中文摘要 AI 辅助
本说明性注记研究如何从提议分布$P$的观测值模拟目标分布$Q$。在Harsha、Jain、McAllester和Radhakrishnan提出的模型中，我们观测来自$P$的独立同分布样本序列$X_1,X_2,\boldsymbol{\text{…}}$，目标是找到某个索引$I \boldsymbol{\text{∈}} \{1,2,\boldsymbol{\text{…}}\}$，使得$X_I$服从$Q$分布，同时最小化$\boldsymbol{\text{E}} \boldsymbol{\text{log}} I$。遵循Maddison、Li和El Gamal等人开发的泊松竞赛方法，本注记证明：若$D(Q||P) < \boldsymbol{\text{∞}}$，则存在$P$到$Q$的模拟器$I$，满足$\boldsymbol{\text{E}} [\boldsymbol{\text{log}} I] \boldsymbol{\text{≤}} D(Q||P) + 1.45 \boldsymbol{\text{\|}}Q-P\boldsymbol{\text{\|}}_1$；反之，对任意$P$到$Q$的模拟器$I$，代价下界为$\boldsymbol{\text{E}}[\boldsymbol{\text{log}} I] \boldsymbol{\text{≥}} \frac{1}{2} \boldsymbol{\text{max}} \{ D(Q||P), \boldsymbol{\text{\|}}Q-P\boldsymbol{\text{\|}}_1 \}$。
英文摘要
This expository note is about simulating a target distribution $Q$ from observations of a proposal distribution $P$. In the model suggested by Harsha, Jain, McAllester and Radhakrishnan, we observe an infinite sequence of i.i.d. samples $X_1,X_2,\ldots$ from $P$. The goal is to find some index $I \in \{1,2,\ldots\}$ such that $X_I$ is distributed like $Q$ while minimizing $\mathbb{E} \log I$. Following the Poisson-race approach developed by Maddison, Li and El Gamal, and others, this note shows that if $D(Q||P) < \infty$ then there is a $P$ to $Q$ simulator $I$ such that $\mathbb{E} [\log I] \leq D(Q||P) + 1.45 \|Q-P\|_1$. In the other direction, for every $P$ to $Q$ simulator $I$, the cost is at least $\mathbb{E}[\log I] \geq \frac{1}{2} \max \{ D(Q||P), \|Q-P\|_1 \}$.
