We consider distributions arising from a mixture of causal models, where each
model is represented by a directed acyclic graph (DAG). We provide a graphical
representation of such mixture distributions and prove that this representation
encodes the conditional independence relations of the mixture distribution. We
then consider the problem of structure learning based on samples from such
distributions. Since the mixing variable is latent, we consider causal
structure discovery algorithms such as FCI that can deal with latent variables.
We show that such algorithms recover a "union" of the component DAGs and can
identify variables whose conditional distribution across the component DAGs
vary. We demonstrate our results on synthetic and real data showing that the
inferred graph identifies nodes that vary between the different mixture
components. As an immediate application, we demonstrate how retrieval of this
causal information can be used to cluster samples according to each mixture
component.

研究通过混合因果模型来获得分布，并探讨了如何从基于此类分布的样本中进行因果结构学习及如何使用这些信息对样本进行聚类。

混合有向无环图分布中因果结构的发现

Causal Structure Discovery from Distributions Arising from Mixtures of  DAGs

Understanding proper distance measures between distributions is at the core
of several learning tasks such as generative models, domain adaptation,
clustering, etc. In this work, we focus on mixture distributions that arise
naturally in several application domains where the data contains different
sub-populations. For mixture distributions, established distance measures such
as the Wasserstein distance do not take into account imbalanced mixture
proportions. Thus, even if two mixture distributions have identical mixture
components but different mixture proportions, the Wasserstein distance between
them will be large. This often leads to undesired results in distance-based
learning methods for mixture distributions. In this paper, we resolve this
issue by introducing the Normalized Wasserstein measure. The key idea is to
introduce mixture proportions as optimization variables, effectively
normalizing mixture proportions in the Wasserstein formulation. Using the
proposed normalized Wasserstein measure leads to significant performance gains
for mixture distributions with imbalanced mixture proportions compared to the
vanilla Wasserstein distance. We demonstrate the effectiveness of the proposed
measure in GANs, domain adaptation and adversarial clustering in several
benchmark datasets.

本文研究通过引入归一化 Wasserstein 度量来解决不平衡混合比例问题，并将其应用于生成模型、域自适应和聚类等多个领域，表现出显著的性能提升。

应用于对抗学习和域自适应的混合分布归一化 Wasserstein 距离

Normalized Wasserstein Distance for Mixture Distributions with  Applications in Adversarial Learning and Domain Adaptation

Mixture distributions arise in many parametric and non-parametric settings --
for example, in Gaussian mixture models and in non-parametric estimation. It is
often necessary to compute the entropy of a mixture, but, in most cases, this
quantity has no closed-form expression, making some form of approximation
necessary. We propose a family of estimators based on a pairwise distance
function between mixture components, and show that this estimator class has
many attractive properties. For many distributions of interest, the proposed
estimators are efficient to compute, differentiable in the mixture parameters,
and become exact when the mixture components are clustered. We prove this
family includes lower and upper bounds on the mixture entropy. The Chernoff
$\alpha$-divergence gives a lower bound when chosen as the distance function,
with the Bhattacharyya distance providing the tightest lower bound for
components that are symmetric and members of a location family. The
Kullback-Leibler divergence gives an upper bound when used as the distance
function. We provide closed-form expressions of these bounds for mixtures of
Gaussians, and discuss their applications to the estimation of mutual
information. We then demonstrate that our bounds are significantly tighter than
well-known existing bounds using numeric simulations. This estimator class is
very useful in optimization problems involving maximization/minimization of
entropy and mutual information, such as MaxEnt and rate distortion problems.

提出一种基于混合成分之间的成对距离函数的家族估计器，该家族具有许多优越的性质，可用于计算混合熵，并在优化最大化 / 最小化熵和互信息的问题中非常有用，例如 MaxEnt 和速率失真问题。