This paper concerns, more specifically, the inconsistency of separating sets used to remove dispensable edges, iteratively, based on conditional independence tests.

We refer the readers to ( Peters et al., 2017) for more detailed graphical terminology. We base our proof mostly on ( Kirsch, 2019). The first statement follows directly from the first theorem in ( Haviland, 1936). Without loss of generality, we reorder the variables according to reversed topological ordering, i.e. a Follows directly from Lemma 1. Lemma 4. Recall condition 2) in Causal de Finetti states that 8 i, 8 n 2 N: X The first equality holds by well-defindedness. The fourth equality follow from well-definedness.

Here we provide proofs of the statements made in the main text as well as further figures of numerical experiments and a more detailed discussion of heteroskedasticity effects regarding causal discovery. Z. Testing whether the Pearson correlation between X and Y is zero is equivalent to testing whether the slope parameter β is equal to zero. Therefore, this is a homoskedastic problem. A.1.2 Discussion of Effect 2: We start by discussing the homoskedastic case to see where non-constant variance of noise leads to problems within the t-test. For homoskedastic noise the second factor is an estimator of the standard error of ˆβ, which we derive by using the mean of the squared residual as an estimator for the error variance.

Next, we introduce two additional approximations we use to apply Eq. (S.9). An SCM matches a set of assignments to a causal graph. This implies that the error of the approximation Eq. (S.13) is mainly dominated by the gradients of g at hao, and the variance ofnao. Specifically, we use a positive differentiable measure of the statistical dependence, denoted by I. PIDA measures disentanglement of representations for models that are trained from unsupervised data. As a result, we have the following: Minimizing Eq. (S.21) leads topdo(a,o)(ˆφa0) approaching p(ˆφa0|a), which as we have just shown, leads top(ˆφa0|a) approaching pdo(a)(ˆφa0).

Flow-based methods have achieved significant success in various generative modeling tasks, capturing nuanced details within complex data distributions. However, few existing works have exploited this unique capability to resolve fine-grained structural details beyond generation tasks. This paper presents a flow-inspired framework for representation learning. First, we demonstrate that a rectified flow trained using independent coupling is zero everywhere at $t=0.5$ if and only if the source and target distributions are identical. We term this property the \emph{zero-flow criterion}. Second, we show that this criterion can certify conditional independence, thereby extracting \emph{sufficient information} from the data. Third, we translate this criterion into a tractable, simulation-free loss function that enables learning amortized Markov blankets in graphical models and latent representations in self-supervised learning tasks. Experiments on both simulated and real-world datasets demonstrate the effectiveness of our approach. The code reproducing our experiments can be found at: https://github.com/probabilityFLOW/zfe.

Functional graphical models have undergone extensive development during the recent years, leading to a variety models such as the functional Gaussian graphical model, the functional copula Gaussian graphical model, the functional Bayesian graphical model, the nonparametric functional additive graphical model, and the conditional functional graphical model. These models rely either on some parametric form of distributions on random functions, or on additive conditional independence, a criterion that is different from probabilistic conditional independence. In this paper we introduce a nonparametric functional graphical model based on functional sufficient dimension reduction. Our method not only relaxes the Gaussian or copula Gaussian assumptions, but also enhances estimation accuracy by avoiding the ``curse of dimensionality''. Moreover, it retains the probabilistic conditional independence as the criterion to determine the absence of edges. By doing simulation study and analysis of the f-MRI dataset, we demonstrate the advantages of our method.