论文标题
通过有条件独立产生仿制
Generating knockoffs via conditional independence
论文作者
论文摘要
令$ x $为$ p $ - 变量随机向量和$ \ widetilde {x} $ $ x $的仿制副本（从\ cite {cfjl18}的意义上）。在\ cite {jspi}中引入了一种构建$ \ widetilde {x} $（从此以后）的新方法。 NA基本上具有三个优点：（i）构建$ \ widetilde {x} $很简单； （ii）$（x，\ wideTilde {x}）$的联合分布可以以封闭式书写； （iii）$ \ widetilde {x} $在各种标准下通常是最佳的。但是，对于NA应用，$ x_1，\ ldots，x_p $应在有条件地独立，鉴于某些随机元素$ z $。我们的第一个结果是，$ \ mathbb {r}^p $上的任何概率度量$ $ $都可以通过概率度量$μ_0$的$μ_0$近似（$μ_0$ z）\ bigr \}。$$当$μ$绝对连续时，近似值为总变化距离，并提供了$μ_0$的明确公式。如果$ x \simμ__0$，则$ x_1，\ ldots，x_p $有条件地独立。因此，如果错误可忽略不计，可以假设$ x \simμ__0$，并通过na构建$ \ widetilde {x} $。我们的第二个结果是通过Na获得的仿冒品$ \ widetilde {x} $的表征。显示出$ \ widetilde {x} $是此类型的，并且仅当对$（x，x，\ widetilde {x}）$时，才能将其扩展到无限序列，以满足某些不变条件。证明这一事实的基本工具是De Finetti的定理，用于部分可交换的序列。除了引用的结果外，在少数情况下，获得了$ \ widetilde {x} $的条件分布的明确公式。在其中一种情况下，假定所有$ i $的$ x_i \ in \ {0,1 \} $。
Let $X$ be a $p$-variate random vector and $\widetilde{X}$ a knockoff copy of $X$ (in the sense of \cite{CFJL18}). A new approach for constructing $\widetilde{X}$ (henceforth, NA) has been introduced in \cite{JSPI}. NA has essentially three advantages: (i) To build $\widetilde{X}$ is straightforward; (ii) The joint distribution of $(X,\widetilde{X})$ can be written in closed form; (iii) $\widetilde{X}$ is often optimal under various criteria. However, for NA to apply, $X_1,\ldots, X_p$ should be conditionally independent given some random element $Z$. Our first result is that any probability measure $μ$ on $\mathbb{R}^p$ can be approximated by a probability measure $μ_0$ of the form $$μ_0\bigl(A_1\times\ldots\times A_p\bigr)=E\Bigl\{\prod_{i=1}^p P(X_i\in A_i\mid Z)\Bigr\}.$$ The approximation is in total variation distance when $μ$ is absolutely continuous, and an explicit formula for $μ_0$ is provided. If $X\simμ_0$, then $X_1,\ldots,X_p$ are conditionally independent. Hence, with a negligible error, one can assume $X\simμ_0$ and build $\widetilde{X}$ through NA. Our second result is a characterization of the knockoffs $\widetilde{X}$ obtained via NA. It is shown that $\widetilde{X}$ is of this type if and only if the pair $(X,\widetilde{X})$ can be extended to an infinite sequence so as to satisfy certain invariance conditions. The basic tool for proving this fact is de Finetti's theorem for partially exchangeable sequences. In addition to the quoted results, an explicit formula for the conditional distribution of $\widetilde{X}$ given $X$ is obtained in a few cases. In one of such cases, it is assumed $X_i\in\{0,1\}$ for all $i$.
