When factorized approximations are used for variational inference (VI), they tend to underestimate the uncertainty -- as measured in various ways -- of the distributions they are meant to approximate. We consider two popular ways to measure the uncertainty deficit of VI: (i) the degree to which it underestimates the componentwise variance, and (ii) the degree to which it underestimates the entropy. To better understand these effects, and the relationship between them, we examine an informative setting where they can be explicitly (and elegantly) analyzed: the approximation of a Gaussian,~$p$, with a dense covariance matrix, by a Gaussian,~$q$, with a diagonal covariance matrix. We prove that $q$ always underestimates both the componentwise variance and the entropy of $p$, \textit{though not necessarily to the same degree}. Moreover we demonstrate that the entropy of $q$ is determined by the trade-off of two competing forces: it is decreased by the shrinkage of its componentwise variances (our first measure of uncertainty) but it is increased by the factorized approximation which delinks the nodes in the graphical model of $p$. We study various manifestations of this trade-off, notably one where, as the dimension of the problem grows, the per-component entropy gap between $p$ and $q$ becomes vanishingly small even though $q$ underestimates every componentwise variance by a constant multiplicative factor. We also use the shrinkage-delinkage trade-off to bound the entropy gap in terms of the problem dimension and the condition number of the correlation matrix of $p$. Finally we present empirical results on both Gaussian and non-Gaussian targets, the former to validate our analysis and the latter to explore its limitations.

翻译：当使用因子化近似进行变分推断（VI）时，这些近似往往会在多个度量维度上低估其所要逼近分布的置信度。我们探讨了两种常用的VI置信度不足评估指标：（i）对分量方差低估的程度，以及（ii）对熵低估的程度。为深入理解这些效应及其相互关系，我们研究了一个可进行显式（且优雅）分析的信息化场景：用具有对角协方差矩阵的高斯分布~$q$，去逼近具有稠密协方差矩阵的高斯分布~$p$。我们证明$q$总是同时低估$p$的分量方差和熵，\textit{但二者的低估程度未必相同}。此外，我们揭示$q$的熵取决于两种竞争力量的权衡：分量方差收缩（第一个置信度度量指标）会降低熵，而因子化近似通过解耦$p$图模型中的节点则会提高熵。我们研究了这种权衡的多种表现形式，特别值得注意的是：当问题维度增加时，即使$q$以恒定倍数因子低估每个分量方差，$p$与$q$之间每分量的熵差仍趋近于零。我们还利用收缩-解耦权衡，通过问题维度和$p$的相关系数矩阵条件数来界定熵差。最后，我们在高斯和非高斯目标分布上给出了实验结果，前者用于验证分析，后者用于探索分析的局限性。