arXiv subjects

Dominik Janzing

Publications and source records attributed to Dominik Janzing.

At least 55 records · Page 3Linked to original sources

arXiv2021

You Mostly Walk Alone: Analyzing Feature Attribution in Trajectory Prediction

Predicting the future trajectory of a moving agent can be easy when the past trajectory continues smoothly but is challenging when complex interactions with other agents are involved. Recent deep learning approaches for trajectory prediction show promising performance and partially attribute this to successful reasoning about agent-agent interactions. However, it remains unclear which features such black-box models actually learn to use for making predictions. This paper proposes a procedure that quantifies the contributions of different cues to model performance based on a variant of Shapley values. Applying this procedure to state-of-the-art trajectory prediction methods on standard benchmark datasets shows that they are, in fact, unable to reason about interactions. Instead, the past trajectory of the target is the only feature used for predicting its future. For a task with richer social interaction patterns, on the other hand, the tested models do pick up such interactions to a certain extent, as quantified by our feature attribution method. We discuss the limits of the proposed method and its links to causality

cs.LG↗

arXiv2021

Why did the distribution change?

We describe a formal approach based on graphical causal models to identify the "root causes" of the change in the probability distribution of variables. After factorizing the joint distribution into conditional distributions of each variable, given its parents (the "causal mechanisms"), we attribute the change to changes of these causal mechanisms. This attribution analysis accounts for the fact that mechanisms often change independently and sometimes only some of them change. Through simulations, we study the performance of our distribution change attribution method. We then present a real-world case study identifying the drivers of the difference in the income distribution between men and women.

stat.ME↗

arXiv2020

Necessary and sufficient conditions for causal feature selection in time series with latent common causes

We study the identification of direct and indirect causes on time series and provide conditions in the presence of latent variables, which we prove to be necessary and sufficient under some graph constraints. Our theoretical results and estimation algorithms require two conditional independence tests for each observed candidate time series to determine whether or not it is a cause of an observed target time series. We provide experimental results in simulations, as well as real data. Our results show that our method leads to very low false positives and relatively low false negative rates, outperforming the widely used Granger causality.

stat.ME↗

arXiv2019

Causal structure based root cause analysis of outliers

We describe a formal approach to identify 'root causes' of outliers observed in $n$ variables $X_1,\dots,X_n$ in a scenario where the causal relation between the variables is a known directed acyclic graph (DAG). To this end, we first introduce a systematic way to define outlier scores. Further, we introduce the concept of 'conditional outlier score' which measures whether a value of some variable is unexpected *given the value of its parents* in the DAG, if one were to assume that the causal structure and the corresponding conditional distributions are also valid for the anomaly. Finally, we quantify to what extent the high outlier score of some target variable can be attributed to outliers of its ancestors. This quantification is defined via Shapley values from cooperative game theory.

stat.ML↗

arXiv2019

Feature relevance quantification in explainable AI: A causal problem

We discuss promising recent contributions on quantifying feature relevance using Shapley values, where we observed some confusion on which probability distribution is the right one for dropped features. We argue that the confusion is based on not carefully distinguishing between observational and interventional conditional probabilities and try a clarification based on Pearl's seminal work on causality. We conclude that unconditional rather than conditional expectations provide the right notion of dropping features in contradiction to the theoretical justification of the software package SHAP. Parts of SHAP are unaffected because unconditional expectations (which we argue to be conceptually right) are used as approximation for the conditional ones, which encouraged others to `improve' SHAP in a way that we believe to be flawed.

stat.ML↗

arXiv2019

Causal Regularization

I argue that regularizing terms in standard regression methods not only help against overfitting finite data, but sometimes also yield better causal models in the infinite sample regime. I first consider a multi-dimensional variable linearly influencing a target variable with some multi-dimensional unobserved common cause, where the confounding effect can be decreased by keeping the penalizing term in Ridge and Lasso regression even in the population limit. Choosing the size of the penalizing term, is however challenging, because cross validation is pointless. Here it is done by first estimating the strength of confounding via a method proposed earlier, which yielded some reasonable results for simulated and real data. Further, I prove a `causal generalization bound' which states (subject to a particular model of confounding) that the error made by interpreting any non-linear regression as causal model can be bounded from above whenever functions are taken from a not too rich class. In other words, the bound guarantees "generalization" from observational to interventional distributions, which is usually not subject of statistical learning theory (and is only possible due to the underlying symmetries of the confounder model).

stat.ML↗

arXiv2019

Analysis of cause-effect inference by comparing regression errors

We address the problem of inferring the causal direction between two variables by comparing the least-squares errors of the predictions in both possible directions. Under the assumption of an independence between the function relating cause and effect, the conditional noise distribution, and the distribution of the cause, we show that the errors are smaller in causal direction if both variables are equally scaled and the causal relation is close to deterministic. Based on this, we provide an easily applicable algorithm that only requires a regression in both possible causal directions and a comparison of the errors. The performance of the algorithm is compared with various related causal inference methods in different artificial and real-world data sets.

cs.AI↗

arXiv2018

Merging joint distributions via causal model classes with low VC dimension

If $X,Y,Z$ denote sets of random variables, two different data sources may contain samples from $P_{X,Y}$ and $P_{Y,Z}$, respectively. We argue that causal inference can help inferring properties of the 'unobserved joint distributions' $P_{X,Y,Z}$ or $P_{X,Z}$. The properties may be conditional independences or also quantitative statements about dependences. More generally, we define a learning scenario where the input is a subset of variables and the label is some statistical property of that subset. Sets of jointly observed variables define the training points, while unobserved sets are possible test points. To solve this learning task, we infer, as an intermediate step, a causal model from the observations that then entails properties of unobserved sets. Accordingly, we can define the VC dimension of a class of causal models and derive generalization bounds for the predictions. Here, causal inference becomes more modest and better accessible to empirical tests than usual: rather than trying to find a causal hypothesis that is 'true' (which is a problematic term when it is unclear how to define interventions) a causal hypothesis is useful whenever it correctly predicts statistical properties of unobserved joint distributions. Within such a 'pragmatic' application of causal inference, some popular heuristic approaches become justified in retrospect. It is, for instance, allowed to infer DAGs from partial correlations instead of conditional independences if the DAGs are only used to predict partial correlations. I hypothesize that our pragmatic view on causality may even cover the usual meaning in terms of interventions and sketch why predicting the impact of interventions can sometimes also be phrased as a task of the above type.

math.ST↗

arXiv2018

Simple negative result for physically universal controllers with macroscopic interface

To study potential limitations of controllability of physical systems I have earlier proposed physically universal cellular automata and Hamiltonians. These are translation invariant interactions for which any control operation on a finite target region can be implemented by the autonomous time evolution if the complement of the target region is 'programmed' to an appropriate initial state. This provides a model of control where the cut between a system and its controller can be consistently shifted, in analogy to the Heisenberg cut defining the boundary between a quantum system and its measurement device. However, in the known physically universal CAs the implementation of microscopic transformations requires to write the 'program' into microscopic degrees of freedom, while human actions take place on the macroscopic level. I therefore ask whether there exist physically universal interactions for which any desired operation on a target region can be performed by only controlling the macroscopic state of its surrounding. A very simple argument shows that this is impossible with respect to the notion of 'macroscopic' proposed here: control devices whose position is only specified up to 'macroscopic precision' cannot operate at a precise location in space. This suggests that reasonable notions of `universal controllability' need to be tailored to the manipulation of relative coordinates, but it is not obvious how to do this. The statement that any microscopic transformation can be implemented in principle, whenever it is true in any sense, it does not seem to be true in its most obvious sense.

quant-ph↗

arXiv2018

Structural causal models for macro-variables in time-series

We consider a bivariate time series $(X_t,Y_t)$ that is given by a simple linear autoregressive model. Assuming that the equations describing each variable as a linear combination of past values are considered structural equations, there is a clear meaning of how intervening on one particular $X_t$ influences $Y_{t'}$ at later times $t'>t$. In the present work, we describe conditions under which one can define a causal model between variables that are coarse-grained in time, thus admitting statements like `setting $X$ to $x$ changes $Y$ in a certain way' without referring to specific time instances. We show that particularly simple statements follow in the frequency domain, thus providing meaning to interventions on frequencies.

math.ST↗

arXiv2018

Does universal controllability of physical systems prohibit thermodynamic cycles?

Here we study the thermodynamic cost of computation and control using 'physically universal' cellular automata or Hamiltonians. The latter were previously defined as systems that admit the implementation of any desired transformation on a finite target region by first initializing the state of the surrounding and then letting the system evolve according to its autonomous dynamics. This way, one obtains a model of control where each region can play both roles the controller or the system to be controlled. In physically universal systems every degree of freedom is indirectly accessible by operating on the remaining degrees of freedom. In a nutshell, the thermodynamic cost of an operation is then given by the size of the region around the target region that needs to be initialized. In the meantime, physically universal CAs have been constructed by Schaeffer (in two dimensions) and Salo & Törmä (in one dimension). Here we show that in Schaeffer's CA the cost for implementing $n$ operations grows linearly in $n$, while operating in a thermodynamic cycle requires sublinear growth to ensure zero cost *per operation* in the limit $n\to\infty$. Although this particular result need not hold for general physically universal CAs, this strong notion of universality does imply a certain kind of instability of information, which could result in lower bounds on the cost of protecting information from its noisy environment. The technical results of the paper are sparse and quite simple. The contribution of the paper is mainly conceptual and consists in illustrating the type of thermodynamic questions raised by models of control that rely on the concept of physical universality.

cond-mat.stat-mech↗

arXiv2018

Detecting non-causal artifacts in multivariate linear regression models

We consider linear models where $d$ potential causes $X_1,...,X_d$ are correlated with one target quantity $Y$ and propose a method to infer whether the association is causal or whether it is an artifact caused by overfitting or hidden common causes. We employ the idea that in the former case the vector of regression coefficients has 'generic' orientation relative to the covariance matrix $Σ_{XX}$ of $X$. Using an ICA based model for confounding, we show that both confounding and overfitting yield regression vectors that concentrate mainly in the space of low eigenvalues of $Σ_{XX}$.

stat.ML↗

arXiv2018

Avoiding Discrimination through Causal Reasoning

Recent work on fairness in machine learning has focused on various statistical discrimination criteria and how they trade off. Most of these criteria are observational: They depend only on the joint distribution of predictor, protected attribute, features, and outcome. While convenient to work with, observational criteria have severe inherent limitations that prevent them from resolving matters of fairness conclusively. Going beyond observational criteria, we frame the problem of discrimination based on protected attributes in the language of causal reasoning. This viewpoint shifts attention from "What is the right fairness criterion?" to "What do we want to assume about the causal data generating process?" Through the lens of causality, we make several contributions. First, we crisply articulate why and when observational criteria fail, thus formalizing what was before a matter of opinion. Second, our approach exposes previously ignored subtleties and why they are fundamental to the problem. Finally, we put forward natural causal non-discrimination criteria and develop algorithms that satisfy them.

stat.ML↗

arXiv2017

A central limit like theorem for Fourier sums

We consider the probability distributions of values in the complex plane attained by Fourier sums of the form \sum_{j=1}^n a_j exp(-2πi j nu) /sqrt{n} when the frequency nu is drawn uniformly at random from an interval of length 1. If the coefficients a_j are i.i.d. drawn with finite third moment, the distance of these distributions to an isotropic two-dimensional Gaussian on C converges in probability to zero for any pseudometric on the set of distributions for which the distance between empirical distributions and the underlying distribution converges to zero in probability.

math.PR↗

arXiv2017

Causal Consistency of Structural Equation Models

Complex systems can be modelled at various levels of detail. Ideally, causal models of the same system should be consistent with one another in the sense that they agree in their predictions of the effects of interventions. We formalise this notion of consistency in the case of Structural Equation Models (SEMs) by introducing exact transformations between SEMs. This provides a general language to consider, for instance, the different levels of description in the following three scenarios: (a) models with large numbers of variables versus models in which the `irrelevant' or unobservable variables have been marginalised out; (b) micro-level models versus macro-level models in which the macro-variables are aggregate features of the micro-variables; (c) dynamical time series models versus models of their stationary behaviour. Our analysis stresses the importance of well specified interventions in the causal modelling process and sheds light on the interpretation of cyclic SEMs.

stat.ML↗

arXiv2017

Group invariance principles for causal generative models

The postulate of independence of cause and mechanism (ICM) has recently led to several new causal discovery algorithms. The interpretation of independence and the way it is utilized, however, varies across these methods. Our aim in this paper is to propose a group theoretic framework for ICM to unify and generalize these approaches. In our setting, the cause-mechanism relationship is assessed by comparing it against a null hypothesis through the application of random generic group transformations. We show that the group theoretic view provides a very general tool to study the structure of data generating mechanisms with direct applications to machine learning.

stat.ML↗

arXiv2017

Detecting confounding in multivariate linear models via spectral analysis

We study a model where one target variable Y is correlated with a vector X:=(X_1,...,X_d) of predictor variables being potential causes of Y. We describe a method that infers to what extent the statistical dependences between X and Y are due to the influence of X on Y and to what extent due to a hidden common cause (confounder) of X and Y. The method relies on concentration of measure results for large dimensions d and an independence assumption stating that, in the absence of confounding, the vector of regression coefficients describing the influence of each X on Y typically has `generic orientation' relative to the eigenspaces of the covariance matrix of X. For the special case of a scalar confounder we show that confounding typically spoils this generic orientation in a characteristic way that can be used to quantitatively estimate the amount of confounding.

stat.ML↗

arXiv2015

Distinguishing cause from effect using observational data: methods and benchmarks

The discovery of causal relationships from purely observational data is a fundamental problem in science. The most elementary form of such a causal discovery problem is to decide whether X causes Y or, alternatively, Y causes X, given joint observations of two variables X, Y. An example is to decide whether altitude causes temperature, or vice versa, given only joint measurements of both variables. Even under the simplifying assumptions of no confounding, no feedback loops, and no selection bias, such bivariate causal discovery problems are challenging. Nevertheless, several approaches for addressing those problems have been proposed in recent years. We review two families of such methods: Additive Noise Methods (ANM) and Information Geometric Causal Inference (IGCI). We present the benchmark CauseEffectPairs that consists of data for 100 different cause-effect pairs selected from 37 datasets from various domains (e.g., meteorology, biology, medicine, engineering, economy, etc.) and motivate our decisions regarding the "ground truth" causal directions of all pairs. We evaluate the performance of several bivariate causal discovery methods on these real-world benchmark data and in addition on artificially simulated data. Our empirical results on real-world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data, although more benchmark data would be needed to obtain statistically significant conclusions. One of the best performing methods overall is the additive-noise method originally proposed by Hoyer et al. (2009), which obtains an accuracy of 63+-10 % and an AUC of 0.74+-0.05 on the real-world benchmark. As the main theoretical contribution of this work we prove the consistency of that method.

cs.LG↗