We focus on causal discovery in the presence of measurement error in linear systems where the mixing matrix, i.e., the matrix indicating the independent exogenous noise terms pertaining to the observed variables, is identified up to permutation and scaling of the columns. We demonstrate a somewhat surprising connection between this problem and causal discovery in the presence of unobserved parentless causes, in the sense that there is a mapping, given by the mixing matrix, between the underlying models to be inferred in these problems. Consequently, any identifiability result based on the mixing matrix for one model translates to an identifiability result for the other model. We characterize to what extent the causal models can be identified under a two-part faithfulness assumption. Under only the first part of the assumption (corresponding to the conventional definition of faithfulness), the structure can be learned up to the causal ordering among an ordered grouping of the variables but not all the edges across the groups can be identified. We further show that if both parts of the faithfulness assumption are imposed, the structure can be learned up to a more refined ordered grouping. As a result of this refinement, for the latent variable model with unobserved parentless causes, the structure can be identified. Based on our theoretical results, we propose causal structure learning methods for both models, and evaluate their performance on synthetic data.
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Linear structural causal models (SCMs)-- in which each observed variable is generated by a subset of the other observed variables as well as a subset of the exogenous sources-- are pervasive in causal inference and casual discovery. However, for the task of causal discovery, existing work almost exclusively focus on the submodel where each observed variable is associated with a distinct source with non-zero variance. This results in the restriction that no observed variable can deterministically depend on other observed variables or latent confounders. In this paper, we extend the results on structure learning by focusing on a subclass of linear SCMs which do not have this property, i.e., models in which observed variables can be causally affected by any subset of the sources, and are allowed to be a deterministic function of other observed variables or latent confounders. This allows for a more realistic modeling of influence or information propagation in systems. We focus on the task of causal discovery form observational data generated from a member of this subclass. We derive a set of necessary and sufficient conditions for unique identifiability of the causal structure. To the best of our knowledge, this is the first work that gives identifiability results for causal discovery under both latent confounding and deterministic relationships. Further, we propose an algorithm for recovering the underlying causal structure when the aforementioned conditions are satisfied. We validate our theoretical results both on synthetic and real datasets.
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我们研究了在存在潜在变量存在下从数据重建因果图形模型的问题。感兴趣的主要问题是在潜在变量上恢复因果结构,同时允许一般,可能在变量之间的非线性依赖性。在许多实际问题中,原始观测之间的依赖性(例如,图像中的像素)的依赖性比某些高级潜在特征(例如概念或对象)之间的依赖性要小得多,这是感兴趣的设置。我们提供潜在表示和潜在潜在因果模型的条件可通过减少到混合甲骨文来识别。这些结果突出了学习混合模型的顺序的良好研究问题与观察到和解开的基础结构的问题之间的富裕问题之间的有趣连接。证明是建设性的,并导致几种算法用于明确重建全图形模型。我们讨论高效算法并提供说明实践中算法的实验。
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Causal disentanglement seeks a representation of data involving latent variables that relate to one another via a causal model. A representation is identifiable if both the latent model and the transformation from latent to observed variables are unique. In this paper, we study observed variables that are a linear transformation of a linear latent causal model. Data from interventions are necessary for identifiability: if one latent variable is missing an intervention, we show that there exist distinct models that cannot be distinguished. Conversely, we show that a single intervention on each latent variable is sufficient for identifiability. Our proof uses a generalization of the RQ decomposition of a matrix that replaces the usual orthogonal and upper triangular conditions with analogues depending on a partial order on the rows of the matrix, with partial order determined by a latent causal model. We corroborate our theoretical results with a method for causal disentanglement that accurately recovers a latent causal model.
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We consider the problem of recovering the causal structure underlying observations from different experimental conditions when the targets of the interventions in each experiment are unknown. We assume a linear structural causal model with additive Gaussian noise and consider interventions that perturb their targets while maintaining the causal relationships in the system. Different models may entail the same distributions, offering competing causal explanations for the given observations. We fully characterize this equivalence class and offer identifiability results, which we use to derive a greedy algorithm called GnIES to recover the equivalence class of the data-generating model without knowledge of the intervention targets. In addition, we develop a novel procedure to generate semi-synthetic data sets with known causal ground truth but distributions closely resembling those of a real data set of choice. We leverage this procedure and evaluate the performance of GnIES on synthetic, real, and semi-synthetic data sets. Despite the strong Gaussian distributional assumption, GnIES is robust to an array of model violations and competitive in recovering the causal structure in small- to large-sample settings. We provide, in the Python packages "gnies" and "sempler", implementations of GnIES and our semi-synthetic data generation procedure.
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也称为(非参数)结构方程模型(SEMS)的结构因果模型(SCM)被广泛用于因果建模目的。特别是,也称为递归SEM的无循环SCMS,形成了一个研究的SCM的良好的子类,概括了因果贝叶斯网络来允许潜在混淆。在本文中,我们调查了更多普通环境中的SCM,允许存在潜在混杂器和周期。我们展示在存在周期中,无循环SCM的许多方便的性质通常不会持有:它们并不总是有解决方案;它们并不总是诱导独特的观察,介入和反事实分布;边缘化并不总是存在,如果存在边缘模型并不总是尊重潜在的投影;他们并不总是满足马尔可夫财产;他们的图表并不总是与他们的因果语义一致。我们证明,对于SCM一般,这些属性中的每一个都在某些可加工条件下保持。我们的工作概括了SCM的结果,迄今为止仅针对某些特殊情况所知的周期。我们介绍了将循环循环设置扩展到循环设置的简单SCM的类,同时保留了许多方便的无环SCM的性能。用本文,我们的目标是为SCM提供统计因果建模的一般理论的基础。
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这项工作介绍了一种新颖的原则,我们通过机制稀疏正规调用解剖学,基于高级概念的动态往往稀疏的想法。我们提出了一种表示学习方法,可以通过同时学习与它们相关的潜在因子和稀疏因果图形模型来引起解剖学。我们开发了一个严谨的可识别性理论,建立在最近的非线性独立分量分析(ICA)结果中,结果是模拟这一原理,并展示了如何恢复潜在变量,如果一个规则大致潜在机制为稀疏,如果某些图形连接标准通过数据生成过程满足。作为我们框架的特殊情况,我们展示了如何利用未知目标的干预措施来解除潜在因子,从而借鉴ICA和因果关系之间的进一步联系。我们还提出了一种基于VAE的方法,其中通过二进制掩码来学习和正规化潜在机制,并通过表明它学会在模拟中的解散表示来验证我们的理论。
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We study experiment design for unique identification of the causal graph of a system where the graph may contain cycles. The presence of cycles in the structure introduces major challenges for experiment design as, unlike acyclic graphs, learning the skeleton of causal graphs with cycles may not be possible from merely the observational distribution. Furthermore, intervening on a variable in such graphs does not necessarily lead to orienting all the edges incident to it. In this paper, we propose an experiment design approach that can learn both cyclic and acyclic graphs and hence, unifies the task of experiment design for both types of graphs. We provide a lower bound on the number of experiments required to guarantee the unique identification of the causal graph in the worst case, showing that the proposed approach is order-optimal in terms of the number of experiments up to an additive logarithmic term. Moreover, we extend our result to the setting where the size of each experiment is bounded by a constant. For this case, we show that our approach is optimal in terms of the size of the largest experiment required for uniquely identifying the causal graph in the worst case.
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常用图是表示和可视化因果关系的。对于少量变量,这种方法提供了简洁和清晰的方案的视图。随着下属的变量数量增加,图形方法可能变得不切实际,并且表示的清晰度丢失。变量的聚类是减少因果图大小的自然方式,但如果任意实施,可能会错误地改变因果关系的基本属性。我们定义了一种特定类型的群集,称为Transit Cluster,保证在某些条件下保留因果效应的可识别性属性。我们提供了一种用于在给定图中查找所有传输群集的声音和完整的算法,并演示集群如何简化因果效应的识别。我们还研究了逆问题,其中一个人以群集的图形开始,寻找扩展图,其中因果效应的可识别性属性保持不变。我们表明这种结构稳健性与过境集群密切相关。
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本文考虑了从观察和介入数据估算因果导向的非循环图中未知干预目标的问题。重点是线性结构方程模型(SEM)中的软干预。目前对因果结构的方法学习使用已知的干预目标或使用假设测试来发现即使是线性SEM也可以发现未知的干预目标。这严重限制了它们的可扩展性和样本复杂性。本文提出了一种可扩展和高效的算法,始终识别所有干预目标。关键思想是从与观察和介入数据集相关联的精度矩阵之间的差异来估计干预站点。它涉及反复估计不同亚空间子集中的这些站点。该算法的算法还可用于将给定的观察马尔可夫等效类更新为介入马尔可夫等价类。在分析地建立一致性,马尔可夫等效和采样复杂性。最后,实际和合成数据的仿真结果展示了所提出的可扩展因果结构恢复方法的增益。算法的实现和重现仿真结果的代码可用于\ url {https://github.com/bvarici/intervention- istimation}。
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在观察性研究中,经常遇到有关存在或缺乏因果边缘和路径的因果背景知识。由于背景知识而导致的马尔可夫等效dag的子类共享的指向边缘和链接可以由因果关系最大部分定向的无循环图(MPDAG)表示。在本文中,我们首先提供了因果MPDAG的声音和完整的图形表征,并提供了因果MPDAG的最小表示。然后,我们介绍了一种名为Direct Causal子句(DCC)的新颖表示,以统一形式表示所有类型的因果背景知识。使用DCC,我们研究因果背景知识的一致性和等效性,并表明任何因果背景知识集都可以等效地分解为因果MPDAG,以及最小的残留DCC。还提供了多项式时间算法,以检查一致性,等效性并找到分解的MPDAG和残留DCC。最后,有了因果背景知识,我们证明了一个足够且必要的条件来识别因果关系,并且出人意料地发现因果效应的可识别性仅取决于分解的MPDAG。我们还开发了局部IDA型算法,以估计无法识别效应的可能值。模拟表明因果背景知识可以显着提高因果影响的识别性。
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在图形因果发现的背景下,我们适应了线性非高斯无环模型(Lingams)的多功能框架,以提出新算法以有效地学习polytrees的图形。我们的方法结合了Chow- Liu算法,该算法首先学习了无向树结构,并与新的方案定向边缘。方向方案评估数据生成分布的矩之间的代数关系,并且计算便宜。我们为我们的方法建立了高维的一致性结果,并比较了数值实验中的不同算法版本。
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我们分析了在没有特定分布假设的常规设置中从观察数据的学习中学循环图形模型的复杂性。我们的方法是信息定理,并使用本地马尔可夫边界搜索程序,以便在基础图形模型中递归地构建祖先集。也许令人惊讶的是,我们表明,对于某些图形集合,一个简单的前向贪婪搜索算法(即没有向后修剪阶段)足以学习每个节点的马尔可夫边界。这显着提高了我们在节点的数量中显示的样本复杂性。然后应用这一点以在从文献中概括存在现有条件的新型标识性条件下学习整个图。作为独立利益的问题,我们建立了有限样本的保障,以解决从数据中恢复马尔可夫边界的问题。此外,我们将我们的结果应用于特殊情况的Polytrees,其中假设简化,并提供了多项识别的明确条件,并且在多项式时间中可以识别和可知。我们进一步说明了算法在仿真研究中易于实现的算法的性能。我们的方法是普遍的,用于无需分布假设的离散或连续分布,并且由于这种棚灯对有效地学习来自数据的定向图形模型结构所需的最小假设。
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非线性独立组件分析(ICA)旨在从可观察到的非线性混合物中回收基本的独立潜在来源。如何使非线性ICA模型可识别到某些微不足道的不确定性是无监督学习的长期问题。鉴于某些辅助变量(例如,类标签和/或域/时间索引)作为弱监督或归纳偏见,最近的突破将源标准独立性作为条件独立性重新制定为条件独立性。但是,具有无条件先验的非线性ICA不能从此类发展中受益。我们探索替代路径,并仅考虑在混合过程中的假设,例如结构稀疏性或独立影响。我们表明,在此类约束的特定实例下,可以从其非线性混合物到置换和零件转换的独立潜在来源,从而实现非线性ICA无辅助变量的非平地可识别性。我们提供估计方法并通过实验验证理论结果。图像数据的结果表明,我们的条件可能存在于许多实际数据生成过程中。
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Learning causal structure from observational data often assumes that we observe independent and identically distributed (i.\,i.\,d) data. The traditional approach aims to find a graphical representation that encodes the same set of conditional independence relationships as those present in the observed distribution. It is known that under i.\,i.\,d assumption, even with infinite data, there is a limit to how fine-grained a causal structure we can identify. To overcome this limitation, recent work has explored using data originating from different, related environments to learn richer causal structure. These approaches implicitly rely on the independent causal mechanisms (ICM) principle, which postulates that the mechanism giving rise to an effect given its causes and the mechanism which generates the causes do not inform or influence each other. Thus, components of the causal model can independently change from environment to environment. Despite its wide application in machine learning and causal inference, there is a lack of statistical formalization of the ICM principle and how it enables identification of richer causal structures from grouped data. Here we present new causal de Finetti theorems which offer a first statistical formalization of ICM principle and show how causal structure identification is possible from exchangeable data. Our work provides theoretical justification for a broad range of techniques leveraging multi-environment data to learn causal structure.
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因果推断的一个共同主题是学习观察到的变量(也称为因果发现)之间的因果关系。考虑到大量候选因果图和搜索空间的组合性质,这通常是一项艰巨的任务。也许出于这个原因,到目前为止,大多数研究都集中在相对较小的因果图上,并具有多达数百个节点。但是,诸如生物学之类的领域的最新进展使生成实验数据集,并进行了数千种干预措施,然后进行了数千个变量的丰富分析,从而增加了机会和迫切需要大量因果图模型。在这里,我们介绍了因子定向无环图(F-DAG)的概念,是将搜索空间限制为非线性低级别因果相互作用模型的一种方法。将这种新颖的结构假设与最近的进步相结合,弥合因果发现与连续优化之间的差距,我们在数千个变量上实现了因果发现。此外,作为统计噪声对此估计程序的影响的模型,我们根据随机图研究了F-DAG骨架的边缘扰动模型,并量化了此类扰动对F-DAG等级的影响。该理论分析表明,一组候选F-DAG比整个DAG空间小得多,因此在很难评估基础骨架的高维度中更统计学上的稳定性。我们提出了因子图(DCD-FG)的可区分因果发现,这是对高维介入数据的F-DAG约束因果发现的可扩展实现。 DCD-FG使用高斯非线性低级结构方程模型,并且在模拟中的最新方法以及最新的大型单细胞RNA测序数据集中,与最新方法相比显示出显着改善遗传干预措施。
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The framework of variational autoencoders allows us to efficiently learn deep latent-variable models, such that the model's marginal distribution over observed variables fits the data. Often, we're interested in going a step further, and want to approximate the true joint distribution over observed and latent variables, including the true prior and posterior distributions over latent variables. This is known to be generally impossible due to unidentifiability of the model. We address this issue by showing that for a broad family of deep latentvariable models, identification of the true joint distribution over observed and latent variables is actually possible up to very simple transformations, thus achieving a principled and powerful form of disentanglement. Our result requires a factorized prior distribution over the latent variables that is conditioned on an additionally observed variable, such as a class label or almost any other observation. We build on recent developments in nonlinear ICA, which we extend to the case with noisy or undercomplete observations, integrated in a maximum likelihood framework. The result also trivially contains identifiable flow-based generative models as a special case.
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因果表示学习是识别基本因果变量及其从高维观察(例如图像)中的关系的任务。最近的工作表明,可以从观测的时间序列中重建因果变量,假设它们之间没有瞬时因果关系。但是,在实际应用中,我们的测量或帧速率可能比许多因果效应要慢。这有效地产生了“瞬时”效果,并使以前的可识别性结果无效。为了解决这个问题,我们提出了ICITRI,这是一种因果表示学习方法,当具有已知干预目标的完美干预措施时,可以在时间序列中处理瞬时效应。 Icitris从时间观察中识别因果因素,同时使用可区分的因果发现方法来学习其因果图。在三个视频数据集的实验中,Icitris准确地识别了因果因素及其因果图。
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我们解决了在没有观察到的混杂的存在下的因果效应估计的问题,但是观察到潜在混杂因素的代理。在这种情况下,我们提出了两种基于内核的方法,用于非线性因果效应估计:(a)两阶段回归方法,以及(b)最大矩限制方法。我们专注于近端因果学习设置,但是我们的方法可以用来解决以弗雷霍尔姆积分方程为特征的更广泛的逆问题。特别是,我们提供了在非线性环境中解决此问题的两阶段和矩限制方法的统一视图。我们为每种算法提供一致性保证,并证明这些方法在合成数据和模拟现实世界任务的数据上获得竞争结果。特别是,我们的方法优于不适合利用代理变量的早期方法。
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In this review, we discuss approaches for learning causal structure from data, also called causal discovery. In particular, we focus on approaches for learning directed acyclic graphs (DAGs) and various generalizations which allow for some variables to be unobserved in the available data. We devote special attention to two fundamental combinatorial aspects of causal structure learning. First, we discuss the structure of the search space over causal graphs. Second, we discuss the structure of equivalence classes over causal graphs, i.e., sets of graphs which represent what can be learned from observational data alone, and how these equivalence classes can be refined by adding interventional data.
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