Causal Discovery Beyond Conditional Independences

Knowledge about causal relationships is important because it enables the prediction of the effects of interventions that perturb the observed system. Specifically, predicting the results of interventions amounts to the ability of answering questions like the following: if one or more variables are forced into a particular state, how will the probability distribution of the other variables be affected? Causal relationships can be identified through randomized experiments. However, such experiments may often be unethical, too expensive or even impossible to perform. The development of methods to infer causal relationships from observational rather than experimental data constitutes therefore a fundamental research topic. In this thesis, we address the prob- lem of causal discovery, that is, recovering the underlying causal structure based on the joint probability distribution of the observed random variables. The causal graph cannot be determined by the observed joint distribution alone; additional causal assumptions, that link statistics to causality, are necessary. Under the Markov condition and the faithfulness assumption, conditional-independence-based methods estimate a set of Markov equiva- lent graphs. However, these methods cannot distinguish between two graphs belonging to the same Markov equivalence class. Alternative methods in- vestigate a different set of assumptions. A formal basis underlying these assumptions are functional models which model each variable as a function of its parents and some noise, with the noise variables assumed to be jointly independent. By restricting the function class, e.g., assuming additive noise, Markov equivalent graphs can become distinguishable. Variants of all afore- mentioned methods allow for the presence of confounders, which are unob- served common causes of two or more observed variables. In this thesis, we present complementary causal discovery methods employ- ing different kind of assumptions than the ones mentioned above. The first part of this work concerns causal discovery allowing for the presence of con- founders. We first propose a method that detects the existence and identifies a finite-range confounder of a set of observed dependent variables. It is based on a kernel method to identify finite mixtures of nonparametric product dis- tributions. Next, a property of a conditional distribution, called purity, is introduced which is used for excluding the presence of a low-range confounder of two observed variables that completely explains their dependence (we call low-range a variable whose range has “small” cardinality). We further study the problem of causal discovery in the two-variable case, but now assuming no confounders. To this end, we exploit the principle of inde- pendence of causal mechanisms that has been proposed in the literature. For the case of two variables, it states that, if X → Y (X causes Y ), then P (X ) and P(Y |X) do not contain information about each other. Instead, P(Y ) and P(X|Y ) may contain information about each other. Consequently, esti- mating P(Y |X) from P(X) should not be possible, while estimating P(X|Y ) based on P(Y) may be possible. We employ this asymmetry to propose a causal discovery method which decides upon the causal direction by compar- ing the accuracy of the estimations of P (Y |X ) and P (X |Y ). Moreover, the principle of independence has implications for common ma- chine learning tasks such as semi-supervised learning, which are also dis- cussed in the current work. Finally, the goal of the last part of this dissertation is to present empirical results on the performance of estimation procedures for causal discovery using Additive Noise Models (ANMs) in the two-variable case. Experiments on synthetic and real data show that the algorithms proposed in this thesis often outperform state-of-the-art algorithms

The lesson of causal discovery algorithms for quantum correlations: Causal explanations of Bell-inequality violations require fine-tuning

An active area of research in the fields of machine learning and statistics is the development of causal discovery algorithms, the purpose of which is to infer the causal relations that hold among a set of variables from the correlations that these exhibit. We apply some of these algorithms to the correlations that arise for entangled quantum systems. We show that they cannot distinguish correlations that satisfy Bell inequalities from correlations that violate Bell inequalities, and consequently that they cannot do justice to the challenges of explaining certain quantum correlations causally. Nonetheless, by adapting the conceptual tools of causal inference, we can show that any attempt to provide a causal explanation of nonsignalling correlations that violate a Bell inequality must contradict a core principle of these algorithms, namely, that an observed statistical independence between variables should not be explained by fine-tuning of the causal parameters. In particular, we demonstrate the need for such fine-tuning for most of the causal mechanisms that have been proposed to underlie Bell correlations, including superluminal causal influences, superdeterminism (that is, a denial of freedom of choice of settings), and retrocausal influences which do not introduce causal cycles.Comment: 29 pages, 28 figs. New in v2: a section presenting in detail our characterization of Bell's theorem as a contradiction arising from (i) the framework of causal models, (ii) the principle of no fine-tuning, and (iii) certain operational features of quantum theory; a section explaining why a denial of hidden variables affords even fewer opportunities for causal explanations of quantum correlation

Causal inference using the algorithmic Markov condition

Inferring the causal structure that links n observables is usually based upon detecting statistical dependences and choosing simple graphs that make the joint measure Markovian. Here we argue why causal inference is also possible when only single observations are present. We develop a theory how to generate causal graphs explaining similarities between single objects. To this end, we replace the notion of conditional stochastic independence in the causal Markov condition with the vanishing of conditional algorithmic mutual information and describe the corresponding causal inference rules. We explain why a consistent reformulation of causal inference in terms of algorithmic complexity implies a new inference principle that takes into account also the complexity of conditional probability densities, making it possible to select among Markov equivalent causal graphs. This insight provides a theoretical foundation of a heuristic principle proposed in earlier work. We also discuss how to replace Kolmogorov complexity with decidable complexity criteria. This can be seen as an algorithmic analog of replacing the empirically undecidable question of statistical independence with practical independence tests that are based on implicit or explicit assumptions on the underlying distribution.Comment: 16 figure

Robustness of Model Predictions under Extension

Often, mathematical models of the real world are simplified representations of complex systems. A caveat to using models for analysis is that predicted causal effects and conditional independences may not be robust under model extensions, and therefore applicability of such models is limited. In this work, we consider conditions under which qualitative model predictions are preserved when two models are combined. We show how to use the technique of causal ordering to efficiently assess the robustness of qualitative model predictions and characterize a large class of model extensions that preserve these predictions. For dynamical systems at equilibrium, we demonstrate how novel insights help to select appropriate model extensions and to reason about the presence of feedback loops. We apply our ideas to a viral infection model with immune responses.Comment: Accepted for oral presentation at the Causal Discovery & Causality-Inspired Machine Learning Workshop at Neural Information Processing Systems, 202

Causality and independence in perfectly adapted dynamical systems

Perfect adaptation in a dynamical system is the phenomenon that one or more variables have an initial transient response to a persistent change in an external stimulus but revert to their original value as the system converges to equilibrium. The causal ordering algorithm can be used to construct an equilibrium causal ordering graph that represents causal relations and a Markov ordering graph that implies conditional independences from a set of equilibrium equations. Based on this, we formulate sufficient graphical conditions to identify perfect adaptation from a set of first-order differential equations. Furthermore, we give sufficient conditions to test for the presence of perfect adaptation in experimental equilibrium data. We apply our ideas to a simple model for a protein signalling pathway and test its predictions both in simulations and on real-world protein expression data. We demonstrate that perfect adaptation in this model can explain why the presence and orientation of edges in the output of causal discovery algorithms does not always appear to agree with the direction of edges in biological consensus networks.Comment: 32 page