Causal Discovery with Confounding Cascade Nonlinear Additive Noise Models

2021 ◽  
Vol 12 (6) ◽  
pp. 1-28
Author(s):  
Jie Qiao ◽  
Ruichu Cai ◽  
Kun Zhang ◽  
Zhenjie Zhang ◽  
Zhifeng Hao
Keyword(s):  
Additive Noise ◽  
Causal Relation ◽  
Causal Models ◽  
Noise Model ◽  
Causal Discovery ◽  
Reverse Direction ◽  
Generation Process ◽  
Causal Process ◽  
Causal Direction ◽  
Intermediate Variables

Identification of causal direction between a causal-effect pair from observed data has recently attracted much attention. Various methods based on functional causal models have been proposed to solve this problem, by assuming the causal process satisfies some (structural) constraints and showing that the reverse direction violates such constraints. The nonlinear additive noise model has been demonstrated to be effective for this purpose, but the model class does not allow any confounding or intermediate variables between a cause pair–even if each direct causal relation follows this model. However, omitting the latent causal variables is frequently encountered in practice. After the omission, the model does not necessarily follow the model constraints. As a consequence, the nonlinear additive noise model may fail to correctly discover causal direction. In this work, we propose a confounding cascade nonlinear additive noise model to represent such causal influences–each direct causal relation follows the nonlinear additive noise model but we observe only the initial cause and final effect. We further propose a method to estimate the model, including the unmeasured confounding and intermediate variables, from data under the variational auto-encoder framework. Our theoretical results show that with our model, the causal direction is identifiable under suitable technical conditions on the data generation process. Simulation results illustrate the power of the proposed method in identifying indirect causal relations across various settings, and experimental results on real data suggest that the proposed model and method greatly extend the applicability of causal discovery based on functional causal models in nonlinear cases.

scholarly journals Causal Discovery with Cascade Nonlinear Additive Noise Model

2019 ◽  
Author(s):  
Ruichu Cai ◽  
Jie Qiao ◽  
Kun Zhang ◽  
Zhenjie Zhang ◽  
Zhifeng Hao
Keyword(s):  
Additive Noise ◽  
Causal Relation ◽  
Causal Models ◽  
Noise Model ◽  
Causal Discovery ◽  
Reverse Direction ◽  
Generation Process ◽  
Causal Process ◽  
Structural Constraints ◽  
Causal Direction

Identification of causal direction between a causal-effect pair from observed data has recently attracted much attention. Various methods based on functional causal models have been proposed to solve this problem, by assuming the causal process satisfies some (structural) constraints and showing that the reverse direction violates such constraints. The nonlinear additive noise model has been demonstrated to be effective for this purpose, but the model class is not transitive--even if each direct causal relation follows this model, indirect causal influences, which result from omitted intermediate causal variables and are frequently encountered in practice, do not necessarily follow the model constraints; as a consequence, the nonlinear additive noise model may fail to correctly discover causal direction. In this work, we propose a cascade nonlinear additive noise model to represent such causal influences--each direct causal relation follows the nonlinear additive noise model but we observe only the initial cause and final effect. We further propose a method to estimate the model, including the unmeasured intermediate variables, from data, under the variational auto-encoder framework. Our theoretical results show that with our model, causal direction is identifiable under suitable technical conditions on the data generation process. Simulation results illustrate the power of the proposed method in identifying indirect causal relations across various settings, and experimental results on real data suggest that the proposed model and method greatly extend the applicability of causal discovery based on functional causal models in nonlinear cases.

A multivariate additive noise model for complete causal discovery

2018 ◽  
Vol 103 ◽  
pp. 44-54
Author(s):  
Pramod Kumar Parida ◽  
Tshilidzi Marwala ◽  
Snehashish Chakraverty
Keyword(s):  
Additive Noise ◽  
Noise Model ◽  
Causal Discovery

scholarly journals Justifying Additive Noise Model-Based Causal Discovery via Algorithmic Information Theory

2010 ◽  
Vol 17 (02) ◽  
pp. 189-212 ◽  
Author(s):  
Dominik Janzing ◽  
Bastian Steudel
Keyword(s):  
Information Theory ◽  
Causal Reasoning ◽  
Additive Noise ◽  
Noise Model ◽  
Causal Discovery ◽  
Algorithmic Information Theory ◽  
Joint Distributions ◽  
Algorithmic Information ◽  
Recent Method ◽  
Non Gaussian

A recent method for causal discovery is in many cases able to infer whether X causes Y or Y causes X for just two observed variables X and Y. It is based on the observation that there exist (non-Gaussian) joint distributions P(X,Y) for which Y may be written as a function of X up to an additive noise term that is independent of X and no such model exists from Y to X. Whenever this is the case, one prefers the causal model X → Y. Here we justify this method by showing that the causal hypothesis Y → X is unlikely because it requires a specific tuning between P(Y) and P(X|Y) to generate a distribution that admits an additive noise model from X to Y. To quantify the amount of tuning, needed we derive lower bounds on the algorithmic information shared by P(Y) and P(X|Y). This way, our justification is consistent with recent approaches for using algorithmic information theory for causal reasoning. We extend this principle to the case where P(X,Y)almost admits an additive noise model. Our results suggest that the above conclusion is more reliable if the complexity of P(Y) is high.

scholarly journals DENOISING AND ENHANCEMENT OF MAMMOGRAPHIC IMAGES UNDER THE ASSUMPTION OF HETEROSCEDASTIC ADDITIVE NOISE BY AN OPTIMAL SUBBAND THRESHOLDING

2010 ◽  
Vol 08 (05) ◽  
pp. 713-741 ◽  
Author(s):  
ARIANNA MENCATTINI ◽  
GIULIA RABOTTINO ◽  
MARCELLO SALMERI ◽  
ROBERTO LOJACONO ◽  
BERARDINO SCIUNZI
Keyword(s):  
Wavelet Transforms ◽  
Additive Noise ◽  
Subjective Evaluation ◽  
Visual Assessment ◽  
Noise Model ◽  
Zero Approximation ◽  
Low Contrast ◽  
Mammographic Images ◽  
Dyadic Wavelet ◽  
Expert Radiologist

Mammographic images suffer from low contrast and signal dependent noise, and a very small size of tumoral signs is not easily detected, especially for an early diagnosis of breast cancer. In this context, many methods proposed in literature fail for lack of generality. In particular, too weak assumptions on the noise model, e.g., stationary normal additive noise, and an inaccurate choice of the wavelet family that is applied, can lead to an information loss, noise emphasizing, unacceptable enhancement results, or in turn an unwanted distortion of the original image aspect. In this paper, we consider an optimal wavelet thresholding, in the context of Discrete Dyadic Wavelet Transforms, by directly relating all the parameters involved in both denoising and contrast enhancement to signal dependent noise variance (estimated by a robust algorithm) and to the size of cancer signs. Moreover, by performing a reconstruction from a zero-approximation in conjunction with a Gaussian smoothing filter, we are able to extract the background and the foreground of the image separately, as to compute suitable contrast improvement indexes. The whole procedure will be tested on high resolution X-ray mammographic images and compared with other techniques. Anyway, the visual assessment of the results by an expert radiologist will be also considered as a subjective evaluation.

Causality in Linear Nongaussian Acyclic Models in the Presence of Latent Gaussian Confounders

2013 ◽  
Vol 25 (6) ◽  
pp. 1605-1641 ◽  
Author(s):  
Zhitang Chen ◽  
Laiwan Chan
Keyword(s):  
Real World ◽  
Synthetic Data ◽  
Causal Discovery ◽  
Causal Network ◽  
Finite Sample ◽  
Real World Data ◽  
Root Finding ◽  
Network Discovery ◽  
Causal Direction ◽  
Finite Sample Size

LiNGAM has been successfully applied to some real-world causal discovery problems. Nevertheless, causal sufficiency is assumed; that is, there is no latent confounder of the observations, which may be unrealistic for real-world problems. Taking into the consideration latent confounders will improve the reliability and accuracy of estimations of the real causal structures. In this letter, we investigate a model called linear nongaussian acyclic models in the presence of latent gaussian confounders (LiNGAM-GC) which can be seen as a specific case of lvLiNGAM. This model includes the latent confounders, which are assumed to be independent gaussian distributed and statistically independent of the disturbances. To tackle the causal discovery problem of this model, first we propose a pairwise cumulant-based measure of causal directions for cause-effect pairs. We prove that in spite of the presence of latent gaussian confounders, the causal direction of the observed cause-effect pair can be identified under the mild condition that the disturbances are simultaneously supergaussian or subgaussian. We propose a simple and efficient method to detect the violation of this condition. We extend our work to multivariate causal network discovery problems. Specifically we propose algorithms to estimate the causal network structure, including causal ordering and causal strengths, using an iterative root finding-removing scheme based on pairwise measure. To address the redundant edge problem due to the finite sample size effect, we develop an efficient bootstrapping-based pruning algorithm. Experiments on synthetic data and real-world data have been conducted to show the applicability of our model and the effectiveness of our proposed algorithms.

scholarly journals On the Intersection Property of Conditional Independence and its Application to Causal Discovery

2015 ◽  
Vol 3 (1) ◽  
pp. 97-108
Author(s):  
Jonas Peters
Keyword(s):  
Conditional Independence ◽  
Joint Distribution ◽  
Additive Noise ◽  
Sufficient Conditions ◽  
Intersection Property ◽  
Noise Model ◽  
Statistical Independence ◽  
Necessary And Sufficient ◽  
Weaker Conditions ◽  
Graphical Structure

AbstractThis work investigates the intersection property of conditional independence. It states that for random variables $$A,B,C$$ and X we have that $$X \bot \bot A{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} B,C$$ and $$X\, \bot \bot\, B{\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} A,C$$ implies $$X\, \bot \bot\, (A,B){\kern 1pt} {\kern 1pt} |{\kern 1pt} {\kern 1pt} C$$. Here, “$$ \bot \bot $$” stands for statistical independence. Under the assumption that the joint distribution has a density that is continuous in $$A,B$$ and C, we provide necessary and sufficient conditions under which the intersection property holds. The result has direct applications to causal inference: it leads to strictly weaker conditions under which the graphical structure becomes identifiable from the joint distribution of an additive noise model.

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