Probabilistic models are a critical part of the modern deep learning toolbox
- ranging from generative models (VAEs, GANs), sequence to sequence models used
in machine translation and speech processing to models over functional spaces
(conditional neural processes, neural processes). Given the size and complexity
of these models, safely deploying them in applications requires the development
of tools to analyze their behavior rigorously and provide some guarantees that
these models are consistent with a list of desirable properties or
specifications. For example, a machine translation model should produce
semantically equivalent outputs for innocuous changes in the input to the
model. A functional regression model that is learning a distribution over
monotonic functions should predict a larger value at a larger input.
Verification of these properties requires a new framework that goes beyond
notions of verification studied in deterministic feedforward networks, since
requiring worst-case guarantees in probabilistic models is likely to produce
conservative or vacuous results. We propose a novel formulation of verification
for deep probabilistic models that take in conditioning inputs and sample
latent variables in the course of producing an output: We require that the
output of the model satisfies a linear constraint with high probability over
the sampling of latent variables and for every choice of conditioning input to
the model. We show that rigorous lower bounds on the probability that the
constraint is satisfied can be obtained efficiently. Experiments with neural
processes show that several properties of interest while modeling functional
spaces can be modeled within this framework (monotonicity, convexity) and
verified efficiently using our algorithms

该文章提出了一种验证深度概率模型的新框架，在模型输出过程中采样潜在变量并考虑其所需的条件输入，以高概率满足线性约束，并能够有效地验证功能空间中感兴趣的属性（单调性、凸性）

深度概率模型的验证

Verification of deep probabilistic models

In 1975 John Tukey proposed a multivariate median which is the 'deepest'
point in a given data cloud in R^d. Later, in measuring the depth of an
arbitrary point z with respect to the data, David Donoho and Miriam Gasko
considered hyperplanes through z and determined its 'depth' by the smallest
portion of data that are separated by such a hyperplane. Since then, these
ideas has proved extremely fruitful. A rich statistical methodology has
developed that is based on data depth and, more general, nonparametric depth
statistics. General notions of data depth have been introduced as well as many
special ones. These notions vary regarding their computability and robustness
and their sensitivity to reflect asymmetric shapes of the data. According to
their different properties they fit to particular applications. The upper level
sets of a depth statistic provide a family of set-valued statistics, named
depth-trimmed or central regions. They describe the distribution regarding its
location, scale and shape. The most central region serves as a median. The
notion of depth has been extended from data clouds, that is empirical
distributions, to general probability distributions on R^d, thus allowing for
laws of large numbers and consistency results. It has also been extended from
d-variate data to data in functional spaces.

论文介绍了数据深度及其在统计学中的应用，包括计算和鲁棒性的差异、中心区域及其在描述数据分布方面的作用，以及对概率分布的扩展和在功能空间中的应用。