## Abstracts of publications

### Published

When estimating a regression model, we might have data where some labels are missing, or our data might be biased by a selection mechanism. When the response or selection mechanism is ignorable (i.e., independent of the response variable given the features) one can use off-the-shelf regression methods; in the nonignorable case one typically has to adjust for bias. We observe that privileged information (i.e. information that is only available during training) might render a nonignorable selection mechanism ignorable, and we refer to this scenario as Privilegedly Missing at Random (PMAR). We propose a novel imputation-based regression method, named repeated regression, that is suitable for PMAR. We also consider an importance weighted regression method, and a doubly robust combination of the two. The proposed methods are easy to implement with most popular out-of-the-box regression algorithms. We empirically assess the performance of the proposed methods with extensive simulated experiments and on a synthetically augmented real-world dataset. We conclude that repeated regression can appropriately correct for bias, and can have considerable advantage over weighted regression, especially when extrapolating to regions of the feature space where response is never observed.

We present a new, efficient procedure to establish Markov equivalence between directed graphs that may or may not contain cycles under the d-separation criterion. It is based on the Cyclic Equivalence Theorem (CET) in the seminal works on cyclic models by Thomas Richardson in the mid '90s, but now rephrased from an ancestral perspective. The resulting characterization leads to a procedure for establishing Markov equivalence between graphs that no longer requires tests for d-separation, leading to a significantly reduced algorithmic complexity. The conceptually simplified characterization may help to reinvigorate theoretical research towards sound and complete cyclic discovery in the presence of latent confounders. This version includes a correction to rule (iv) in Theorem 1, and the subsequent adjustment in part 2 of Algorithm 2.

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. With the help of the causal ordering algorithm, one can construct graphical representations of dynamical systems that represent the causal relations between the variables and the conditional independences in the equilibrium distribution. We apply these tools to formulate sufficient graphical conditions for identifying 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 this method to a simple model for a protein signalling pathway and test its predictions in both simulations and using real-world protein expression data. We demonstrate that perfect adaptation can lead to misleading orientation of edges in the output of causal discovery algorithms.

On Thursday 13 October 2022, Joris Mooij delivered his inaugural lecture upon acceptance of the position of professor of Mathematical Statistics at the University of Amsterdam. In many scientific and societal issues the relationship between causes and their effects is an important topic. Mooij explains why both causal models and experimental research remain necessary to answer causal questions (despite recent breakthroughs in machine learning).

Relationships between causes and their effects take a central role in various scientific and societal questions. Does the new COVID-19 vaccine protect better against hospitalization than the old version? How doese a cancer cell react to exposure to a certain chemical? What genetic properties cause a plant to be more resistant against drought? Will inflation in the Netherlands go down if the European Central Bank increases the interest rate? These are all examples of causal questions, which concern predicting the consequences of certain actions as accurately as possible. Traditionally, scientists try to answer causal questions by means of a combination of data and mathematical models. Given the recent impressive successes of deep learning, and the exploding availability of data, one might be led to believe that modelling and even conducting experiments is becoming redundant (as long as sufficient data is available). In my inaugural lecture, I explain why causal models as well as experimental research will nevertheless remain necessary to provide reliable answers to causal questions.

Mathematical models of the real world are simplified representations of complex systems. A caveat to using mathematical models is that predicted causal effects and conditional independences may not be robust under model extensions, limiting applicability of such models. In this work, we consider conditions under which qualitative model predictions are preserved when two models are combined. Under mild assumptions, we show how to use the technique of causal ordering to efficiently assess the robustness of qualitative model predictions. We also characterize a large class of model extensions that preserve qualitative model 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 illustrate our ideas with a viral infection model with immune responses.

The Causal Bandit is a variant of the classic Bandit problem where an agent must identify the best action in a sequential decision-making process, where the reward distribution of the actions displays a non-trivial dependence structure that is governed by a causal model. Methods proposed for this problem thus far in the literature rely on exact prior knowledge of the full causal graph. We formulate new causal bandit algorithms that no longer necessarily rely on prior causal knowledge. Instead, they utilize an estimator based on separating sets, which we can find using simple conditional independence tests or causal discovery methods. We show that, given a true separating set, for discrete i.i.d. data, this estimator is unbiased, and has variance which is upper bounded by that of the sample mean. We develop algorithms based on Thompson Sampling and UCB for discrete and Gaussian models respectively and show increased performance on simulation data as well as on a bandit drawing from real-world protein signaling data.

We consider the problem of discovering causal relations from independence constraints when selection bias in addition to confounding is present. While the seminal FCI algorithm is sound and complete in this setup, no criterion for the causal interpretation of its output under selection bias is presently known. We focus instead on local patterns of independence relations, where we find no sound method for only three variable that can include background knowledge. Y-Structure patterns (Mani et al., 2006; Mooij and Cremers, 2015) are shown to be sound in predicting causal relations from data under selection bias, where cycles may be present. We introduce a finite-sample scoring rule for Y-Structures that is shown to successfully predict causal relations in simulation experiments that include selection mechanisms. On real-world microarray data, we show that a Y-Structure variant performs well across different datasets, potentially circumventing spurious correlations due to selection bias.

Structural causal models (SCMs), also known as (nonparametric) structural equation models (SEMs), are widely used for causal modeling purposes. In particular, acyclic SCMs, also known as recursive SEMs, form a well-studied subclass of SCMs that generalize causal Bayesian networks to allow for latent confounders. In this paper, we investigate SCMs in a more general setting, allowing for the presence of both latent confounders and cycles. We show that in the presence of cycles, many of the convenient properties of acyclic SCMs do not hold in general: they do not always have a solution; they do not always induce unique observational, interventional and counterfactual distributions; a marginalization does not always exist, and if it exists the marginal model does not always respect the latent projection; they do not always satisfy a Markov property; and their graphs are not always consistent with their causal semantics. We prove that for SCMs in general each of these properties does hold under certain solvability conditions. Our work generalizes results for SCMs with cycles that were only known for certain special cases so far. We introduce the class of simple SCMs that extends the class of acyclic SCMs to the cyclic setting, while preserving many of the convenient properties of acyclic SCMs. With this paper, we aim to provide the foundations for a general theory of statistical causal modeling with SCMs.

Real-world complex systems are often modelled by sets of equations with endogenous and exogenous variables. What can we say about the causal and probabilistic aspects of variables that appear in these equations without explicitly solving the equations? We make use of Simon's causal ordering algorithm (Simon, 1953) to construct a causal ordering graph and prove that it expresses the effects of soft and perfect interventions on the equations under certain unique solvability assumptions. We further construct a Markov ordering graph and prove that it encodes conditional independences in the distribution implied by the equations with independent random exogenous variables, under a similar unique solvability assumption. We discuss how this approach reveals and addresses some of the limitations of existing causal modelling frameworks, such as causal Bayesian networks and structural causal models.

For a continuous random variable Z, testing conditional independence ‘X independent of Y given Z’ is known to be a particularly hard problem. It constitutes a key ingredient of many constraint-based causal discovery algorithms. These algorithms are often applied to datasets containing binary variables, which indicate the ‘context’ of the observations, e.g. a control or treatment group within an experiment. In these settings, conditional independence testing with X or Y binary (and the other continuous) is paramount to the performance of the causal discovery algorithm. To our knowledge no nonparametric ‘mixed’ conditional independence test currently exists, and in practice tests that assume all variables to be continuous are used instead. In this paper we aim to fill this gap, as we combine elements of Holmes et al. (2015) and Teymur and Filippi (2020) to propose a novel Bayesian nonparametric conditional two-sample test. Applied to the Local Causal Discovery algorithm, we investigate its performance on both synthetic and real-world data, and compare with state-of-the-art conditional independence tests.

One of the core assumptions in causal discovery is the faithfulness assumption---i.e. assuming that independencies found in the data are due to separations in the true causal graph. This assumption can, however, be violated in many ways, including xor connections, deterministic functions or cancelling paths. In this work, we propose a weaker assumption that we call 2-adjacency faithfulness. In contrast to adjacency faithfulness, which assumes that there is no conditional independence between each pair of variables that are connected in the causal graph, we only require no conditional independence between a node and a subset of its Markov blanket that can contain up to two nodes. Equivalently, we adapt orientation faithfulness to this setting. We further propose a sound orientation rule for causal discovery that applies under weaker assumptions. As a proof of concept, we derive a modified Grow and Shrink algorithm that recovers the Markov blanket of a target node and prove its correctness under strictly weaker assumptions than the standard faithfulness assumption.

While feedback loops are known to play important roles in many complex systems, their existence is ignored in a large part of the causal discovery literature, as systems are typically assumed to be acyclic from the outset. When applying causal discovery algorithms designed for the acyclic setting on data generated by a system that involves feedback, one would not expect to obtain correct results. In this work, we show that---surprisingly---the output of the Fast Causal Inference (FCI) algorithm is correct if it is applied to observational data generated by a system that involves feedback. More specifically, we prove that for observational data generated by a simple and σ-faithful Structural Causal Model (SCM), FCI is sound and complete, and can be used to consistently estimate (i) the presence and absence of causal relations, (ii) the presence and absence of direct causal relations, (iii) the absence of confounders, and (iv) the absence of specific cycles in the causal graph of the SCM. We extend these results to constraint-based causal discovery algorithms that exploit certain forms of background knowledge, including the causally sufficient setting (e.g., the PC algorithm) and the Joint Causal Inferencesetting (e.g., the FCI-JCI algorithm).

The gold standard for discovering causal relations is by means of experimentation. Over the last decades, alternative methods have been proposed that can infer causal relations between variables from certain statistical patterns in purely observational data. We introduce Joint Causal Inference (JCI), a novel approach to causal discovery from multiple data sets from different contexts that elegantly unifies both approaches. JCI is a causal modeling framework rather than a specific algorithm, and it can be implemented using any causal discovery algorithm that can take into account certain background knowledge. JCI can deal with different types of interventions (e.g., perfect, imperfect, stochastic, etc.) in a unified fashion, and does not require knowledge of intervention targets or types in case of interventional data. We explain how several well-known causal discovery algorithms can be seen as addressing special cases of the JCI framework, and we also propose novel implementations that extend existing causal discovery methods for purely observational data to the JCI setting. We evaluate different JCI implementations on synthetic data and on flow cytometry protein expression data and conclude that JCI implementations can considerably outperform state-of-the-art causal discovery algorithms.

We study the performance of Local Causal Discovery (LCD), a simple and efficient constraint-based method for causal discovery, in predicting causal effects in large-scale gene expression data. We construct practical estimators specific to the high-dimensional regime. Inspired by the ICP algorithm, we use an optional preselection method and two different statistical tests. Empirically, the resulting LCD estimator is seen to closely approach the accuracy of ICP, the state-of-the-art method, while it is algorithmically simpler and computationally more efficient.

Structural Causal Models (SCMs) provide a popular causal modeling framework. In this work, we show that SCMs are not flexible enough to give a complete causal representation of dynamical systems at equilibrium. Instead, we propose a generalization of the notion of an SCM, that we call Causal Constraints Model (CCM), and prove that CCMs do capture the causal semantics of such systems. We show how CCMs can be constructed from differential equations and initial conditions and we illustrate our ideas further on a simple but ubiquitous (bio)chemical reaction. Our framework also allows to model functional laws, such as the ideal gas law, in a sensible and intuitive way.

We prove the main rules of causal calculus (also called do-calculus) for i/o structural causal models (ioSCMs), a generalization of a recently proposed general class of non-/linear structural causal models that allow for cycles, latent confounders and arbitrary probability distributions. We also generalize adjustment criteria and formulas from the acyclic setting to the general one (i.e. ioSCMs). Such criteria then allow to estimate (conditional) causal effects from observational data that was (partially) gathered under selection bias and cycles. This generalizes the backdoor criterion, the selection-backdoor criterion and extensions of these to arbitrary ioSCMs. Together, our results thus enable causal reasoning in the presence of cycles, latent confounders and selection bias. Finally, we extend the ID algorithm for the identification of causal effects to ioSCMs.

An important goal common to domain adaptation and causal inference is to make accurate predictions when the distributions for the source (or training) domain(s) and target (or test) domain(s) differ. In many cases, these different distributions can be modeled as different contexts of a single underlying system, in which each distribution corresponds to a different perturbation of the system, or in causal terms, an intervention. We focus on a class of such causal domain adaptation problems, where data for one or more source domains are given, and the task is to predict the distribution of a certain target variable from measurements of other variables in one or more target domains. We propose an approach for solving these problems that exploits causal inference and does not rely on prior knowledge of the causal graph, the type of interventions or the intervention targets. We demonstrate our approach by evaluating a possible implementation on simulated and real world data.

Causal discovery algorithms infer causal relations from data based on several assumptions, including notably the absence of measurement error. However, this assumption is most likely violated in practical applications, which may result in erroneous, irreproducible results. In this work we show how to obtain an upper bound for the variance of random measurement error from the covariance matrix of measured variables and how to use this upper bound as a correction for constraint-based causal discovery. We demonstrate a practical application of our approach on both simulated data and real-world protein signaling data.

We address the problem of causal discovery from data, making use of the
recently proposed causal modeling framework of *modular structural causal
models (mSCM)* to handle cycles, latent confounders and non-linearities. We
introduce *σ-connection graphs (σ-CG)*, a new class of mixed
graphs (containing undirected, bidirected and directed edges) with additional
structure, and extend the concept of *σ-separation*, the
appropriate generalization of the well-known notion of d-separation in this
setting, to apply to σ-CGs. We prove the closedness of σ-separation
under marginalisation and conditioning and exploit this to implement a test of
σ-separation on a σ-CG. This then leads us to the first causal
discovery algorithm that can handle non-linear functional relations, latent
confounders, cyclic causal relationships, and data from different (stochastic)
perfect interventions. As a proof of concept, we show on synthetic data how
well the algorithm recovers features of the causal graph of modular structural
causal models.

Structural Causal Models are widely used in causal modelling, but how they
relate to other modelling tools is poorly understood. In this paper we provide
a novel perspective on the relationship between Ordinary Differential Equations
and Structural Causal Models. We show how, under certain conditions, the
asymptotic behaviour of an Ordinary Differential Equation under non-constant
interventions can be modelled using Dynamic Structural Causal Models. In
contrast to earlier work, we study not only the effect of interventions on
equilibrium states; rather, we model asymptotic behaviour that is
*dynamic* under interventions that vary in time, and include as a
special case the study of static equilibria.

Despite their popularity, many questions about the algebraic constraints imposed by linear structural equation models remain open problems. For causal discovery, two of these problems are especially important: the enumeration of the constraints imposed by a model, and deciding whether two graphs define the same statistical model. We show how the half-trek criterion can be used to make progress in both of these problems. We apply our theoretical results to a small-scale model selection problem, and find that taking the additional algebraic constraints into account may lead to significant improvements in model selection accuracy.

Learning individual-level causal effects from observational data, such as inferring the most effective medication for a specific patient, is a problem of growing importance for policy makers. The most important aspect of inferring causal effects from observational data is the handling of confounders, factors that affect both an intervention and its outcome. A carefully designed observational study attempts to measure all important confounders. However, even if one does not have direct access to all confounders, there may exist noisy and uncertain measurement of proxies for confounders. We build on recent advances in latent variable modeling to simultaneously estimate the unknown latent space summarizing the confounders and the causal effect. Our method is based on Variational Autoencoders (VAE) which follow the causal structure of inference with proxies. We show our method is significantly more robust than existing methods, and matches the state-of-the-art on previous benchmarks focused on individual treatment effects.

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.

Constraint-based causal discovery from limited data is a notoriously difficult challenge due to the many borderline independence test decisions. Several approaches to improve the reliability of the predictions by exploiting redundancy in the independence information have been proposed recently. Though promising, existing approaches can still be greatly improved in terms of accuracy and scalability. We present a novel method that reduces the combinatorial explosion of the search space by using a more coarse-grained representation of causal information, drastically reducing computation time. Additionally, we propose a method to score causal predictions based on their confidence. Crucially, our implementation also allows one to easily combine observational and interventional data and to incorporate various types of available background knowledge. We prove soundness and asymptotic consistency of our method and demonstrate that it can outperform the state-of-the-art on synthetic data, achieving a speedup of several orders of magnitude. We illustrate its practical feasibility by applying it on a challenging protein data set.

Inferring causal effects from observational and interventional data is a highly
desirable but ambitious goal. Many of the computational and statistical methods
are plagued by fundamental identifiability issues, instability, and unreliable
performance, especially for large-scale systems with many measured variables.
We present software and provide some validation of a recently developed
methodology based on an invariance principle, called invariant causal
prediction (ICP). The ICP method quantifies confidence probabilities for
inferring causal structures and thus leads to more reliable and confirmatory
statements for causal relations and predictions of external intervention
effects. We validate the ICP method and some other procedures using large-scale
genome-wide gene perturbation experiments in *Saccharomyces cerevisiae*.
The results suggest that prediction and prioritization of future experimental
interventions, such as gene deletions, can be improved by using our statistical
inference techniques.

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: methods based on Additive Noise Models (ANMs) and Information Geometric Causal Inference (IGCI). We present the benchmark CauseEffectPairs that consists of data for 100 different cause-effect pairs selected from 37 data sets 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 method based on Additive Noise Models that has originally been 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.

We study one of the simplest causal prediction algorithms that uses only conditional independences estimated from purely observational data. A specific pattern of four conditional independence relations amongst a quadruple of random variables already implies that one of these variables causes another one without any confounding. As a consequence, it is possible to predict what would happen under an intervention on that variable without actually performing the intervention. Although the method is asymptotically consistent and works well in settings with only few (latent) variables, we find that its prediction accuracy can be worse than simple (inconsistent) baselines when many (latent) variables are present. Our findings illustrate that violations of strong faithfulness become increasingly likely in the presence of many latent variables, and this can significantly deterioriate the accuracy of constraint-based causal prediction algorithms that assume faithfulness.

By aggregating data for complex traits in a biologically meaningful way, gene and gene-set analysis constitute a valuable addition to single-marker analysis. However, although various methods for gene and gene-set analysis currently exist, they generally suffer from a number of issues. Statistical power for most methods is strongly affected by linkage disequilibrium between markers, multi-marker associations are often hard to detect, and the reliance on permutation to compute p-values tends to make the analysis computationally very expen- sive. To address these issues we have developed MAGMA, a novel tool for gene and gene- set analysis. The gene analysis is based on a multiple regression model, to provide better statistical performance. The gene-set analysis is built as a separate layer around the gene analysis for additional flexibility. This gene-set analysis also uses a regression structure to allow generalization to analysis of continuous properties of genes and simultaneous analysis of multiple gene sets and other gene properties. Simulations and an analysis of Crohn's Disease data are used to evaluate the performance of MAGMA and to compare it to a number of other gene and gene-set analysis tools. The results show that MAGMA has significantly more power than other tools for both the gene and the gene-set analysis, identifying more genes and gene sets associated with Crohn's Disease while maintaining a correct type 1 error rate. Moreover, the MAGMA analysis of the Crohn's Disease data was found to be considerably faster as well.

Estimating the strength of causal effects from observational data is a common problem in scientific research. A popular approach is based on exploiting observed conditional independences between variables. It is well-known that this approach relies on the assumption of faithfulness. In our opinion, a more important practical limitation of this approach is that it relies on the ability to distinguish independences from (arbitrarily weak) dependences. We present a simple analysis, based on purely algebraic and geometrical arguments, of how the estimation of the causal effect strength, based on conditional independence tests and background knowledge, can have an arbitrarily large error due to the uncontrollable type II error of a single conditional independence test. The scenario we are studying here is related to the LCD algorithm by Cooper and to the instrumental variable setting that is popular in epidemiology and econometry. It is one of the simplest settings in which causal discovery and prediction methods based on conditional independences arrive at non-trivial conclusions, yet for which the lack of uniform consistency can result in arbitrarily large prediction errors.

We consider the problem of learning causal directed acyclic graphs from an observational joint distribution. One can use these graphs to predict the outcome of interventional experiments, from which data are often not available. We show that if the observational distribution follows a structural equation model with an additive noise structure, the directed acyclic graph becomes identifiable from the distribution under mild conditions. This constitutes an interesting alternative to traditional methods that assume faithfulness and identify only the Markov equivalence class of the graph, thus leaving some edges undirected. We provide practical algorithms for finitely many samples, RESIT (regression with subsequent independence test) and two methods based on an independence score. We prove that RESIT is correct in the population setting and provide an empirical evaluation.

We propose a method for learning cyclic causal models from a combination of observational and interventional equilibrium data. Novel aspects of the proposed method are its ability to work with continuous data (without assuming linearity) and to deal with feedback loops. Within the context of biochemical reactions, we also propose a novel way of modeling interventions that modify the activity of compounds instead of their abundance. For computational reasons, we approximate the nonlinear causal mechanisms by (coupled) local linearizations, one for each experimental condition. We apply the method to reconstruct a cellular signaling network from the flow cytometry data measured by Sachs et al. (2005). We show that our method finds evidence in the data for feedback loops and that it gives a more accurate quantitative description of the data at comparable model complexity.

We show how, and under which conditions, the equilibrium states of a first-order Ordinary Differential Equation (ODE) system can be described with a deterministic Structural Causal Model (SCM). Our exposition sheds more light on the concept of causality as expressed within the framework of Structural Causal Models, especially for cyclic models.

This paper shows that causal model discovery is not an NP-hard problem, in the sense that for sparse graphs bounded by node degree $k$ the sound and complete causal model can be obtained in worst case order $N^{2(k+2)}$ independence tests, even when latent variables and selection bias may be present. We present a modification of the well-known FCI algorithm that implements the method for an independence oracle, and suggest improvements for sample/real-world data versions. It does not contradict any known hardness results, and does not solve an NP-hard problem: it just proves that sparse causal discovery is perhaps more complicated, but not as hard as learning minimal Bayesian networks.

We consider the problem of function estimation in the case where an underlying causal model can be inferred. This has implications for popular scenarios such as covariate shift, concept drift, transfer learning and semi-supervised learning. We argue that causal knowledge may facilitate some approaches for a given problem, and rule out others. In particular, we formulate a hypothesis for when semi-supervised learning can help, and corroborate it with empirical results.

While conventional approaches to causal inference are mainly based on conditional (in)dependences, recent methods also account for the shape of (conditional) distributions. The idea is that the causal hypothesis "X causes Y" imposes that the marginal distribution PX and the conditional distribution P(Y|X) represent independent mechanisms of nature. Recently it has been postulated that the shortest description of the joint distribution P(X,Y) should therefore be given by separate descriptions of P(X) and P(Y|X). Since description length in the sense of Kolmogorov complexity is uncomputable, practical implementations rely on other notions of independence. Here we define independence via orthogonality in information space. This way, we can explicitly describe the kind of dependence that occurs between P(Y) and P(X|Y) making the causal hypothesis "Y causes X" implausible. Remarkably, this asymmetry between cause and effect becomes particularly simple if X and Y are deterministically related. We present an inference method that works in this case. We also discuss some theoretical results for the non-deterministic case although it is not clear how to employ them for a more general inference method.

We study a particular class of cyclic causal models, where each variable is a (possibly nonlinear) function of its parents and additive noise. We prove that the causal graph of such models is generically identifiable in the bivariate, Gaussian-noise case. We also propose a method to learn such models from observational data. In the acyclic case, the method reduces to ordinary regression, but in the more challenging cyclic case, an additional term arises in the loss function, which makes it a special case of nonlinear independent component analysis. We illustrate the proposed method on synthetic data.

Inference in matrix-variate Gaussian models has major applications for multi-output prediction and joint learning of row and column covariances from matrix-variate data. Here, we discuss an approach for efficient inference in such models that explicitly account for iid observation noise. Computational tractability can be retained by exploiting the Kronecker product between row and column covariance matrices. Using this framework, we show how to generalize the Graphical Lasso in order to learn a sparse inverse covariance between features while accounting for a low-rank confounding covariance between samples. We show practical utility on applications to biology, where we model covariances with more than 100,000 dimensions. We find greater accuracy in recovering biological network structures and are able to better reconstruct the confounders.

To learn about causal relations between variables just by observing samples from them, particular assumptions must be made about those variables' distributions. This article gives a practical description of how such a learning task can be undertaken based on different possible assumptions. Two categories of assumptions lead to different methods, constraint-based and Bayesian learning, and in each case we review both the basic ideas and some recent extensions and alternatives to them.

This work addresses the following question: Under what assumptions on the data generating process can one infer the causal graph from the joint distribution? The approach taken by conditional independence-based causal discovery methods is based on two assumptions: the Markov condition and faithfulness. It has been shown that under these assumptions the causal graph can be identified up to Markov equivalence (some arrows remain undirected) using methods like the PC algorithm. In this work we propose an alternative by defining Identifiable Functional Model Classes (IFMOCs). As our main theorem we prove that if the data generating process belongs to an IFMOC, one can identify the complete causal graph. To the best of our knowledge this is the first identifiability result of this kind that is not limited to linear functional relationships. We discuss how the IFMOC assumption and the Markov and faithfulness assumptions relate to each other and explain why we believe that the IFMOC assumption can be tested more easily on given data. We further provide a practical algorithm that recovers the causal graph from finitely many data; experiments on simulated data support the theoretical findings.

We present a graphical model framework for decoding in the visual ERP-based speller system. The proposed framework allows researchers to build generative models from which the decoding rules are obtained in a straightforward manner. We suggest two models for generating brain signals conditioned on the stimulus events. Both models incorporate letter frequency information but assume different dependencies between brain signals and stimulus events. For both models, we derive decoding rules and perform a discriminative training. We show on real visual speller data how decoding performance improves by incorporating letter frequency information and using a more realistic graphical model for the dependencies between the brain signals and the stimulus events. Furthermore, we discuss how the standard approach to decoding can be seen as a special case of the graphical model framework. The letter also gives more insight into the discriminative approach for decoding in the visual speller system.

We propose a novel method for inferring whether X causes Y or vice versa from joint observations of X and Y. The basic idea is to model the observed data using probabilistic latent variable models, which incorporate the effects of unobserved noise. To this end, we consider the hypothetical effect variable to be a function of the hypothetical cause variable and an independent noise term (not necessarily additive). An important novel aspect of our work is that we do not restrict the model class, but instead put general non-parametric priors on this function and on the distribution of the cause. The causal direction can then be inferred by using standard Bayesian model selection. We evaluate our approach on synthetic data and real-world data and report encouraging results.

This paper describes the software package libDAI, a free & open source CEtAl library that provides implementations of various exact and approximate inference methods for graphical models with discrete-valued variables. libDAI supports directed graphical models (Bayesian networks) as well as undirected ones (Markov random fields and factor graphs). It offers various approximations of the partition sum, marginal probability distributions and maximum probability states. Parameter learning is also supported. A feature comparison with other open source software packages for approximate inference is given. libDAI is licensed under the GPL v2+ license and is available at http://www.libdai.org.

We consider two variables that are related to each other by an invertible function. While it has previously been shown that the dependence structure of the noise can provide hints to determine which of the two variables is the cause, we presently show that even in the deterministic (noise-free) case, there are asymmetries that can be exploited for causal inference. Our method is based on the idea that if the function and the probability density of the cause are chosen independently, then the distribution of the effect will, in a certain sense, depend on the function. We provide a theoretical analysis of this method, showing that it also works in the low noise regime, and link it to information geometry. We report strong empirical results on various real-world data sets from different domains.

This letter introduces a nonlinear measure of independence between random variables for remote sensing supervised feature selection. The so-called HilbertSchmidt independence criterion (HSIC) is a kernel method for evaluating statistical dependence and it is based on computing the HilbertSchmidt norm of the cross-covariance operator of mapped samples in the corresponding Hilbert spaces. The HSIC empirical estimator is easy to compute and has good theoretical and practical properties. Rather than using this estimate for maximizing the dependence between the selected features and the class labels, we propose the more sensitive criterion of minimizing the associated HSIC p-value. Results in multispectral, hyperspectral, and SAR data feature selection for classification show the good performance of the proposed approach.

We describe eight data sets that together formed the CauseEffectPairs task in the Causality Challenge #2: Pot-Luck competition. Each set consists of a sample of a pair of statistically dependent random variables. One variable is known to cause the other one, but this information was hidden from the participants; the task was to identify which of the two variables was the cause and which one the effect, based upon the observed sample. The data sets were chosen such that we expect common agreement on the ground truth. Even though part of the statistical dependences may also be due to hidden common causes, common sense tells us that there is a significant cause-effect relation between the two variables in each pair. We also present baseline results using three different causal inference methods.

We propose a method for inferring the existence of a latent common cause ("confounder") of two observed random variables. The method assumes that the two effects of the confounder are (possibly nonlinear) functions of the confounder plus independent, additive noise. We discuss under which conditions the model is identifiable (up to an arbitrary reparameterization of the confounder) from the joint distribution of the effects. We state and prove a theoretical result that provides evidence for the conjecture that the model is generically identifiable under suitable technical conditions. In addition, we propose a practical method to estimate the confounder from a finite i.i.d. sample of the effects and illustrate that the method works well on both simulated and real-world data.

Motivated by causal inference problems, we propose a novel method for regression that minimizes the statistical dependence between regressors and residuals. The key advantage of this approach to regression is that it does not assume a particular distribution of the noise, i.e., it is non-parametric with respect to the noise distribution. We argue that the proposed regression method is well suited to the task of causal inference in additive noise models. A practical disadvantage is that the resulting optimization problem is generally non-convex and can be difficult to solve. Nevertheless, we report good results on one of the tasks of the NIPS 2008 Causality Challenge, where the goal is to distinguish causes from effects in pairs of statistically dependent variables. In addition, we propose an algorithm for efficiently inferring causal models from observational data for more than two variables. The required number of regressions and independence tests is quadratic in the number of variables, which is a significant improvement over the simple method that tests all possible DAGs.

The discovery of causal relationships between a set of observed variables is a fundamental problem in science. For continuous-valued data linear acyclic causal models with additive noise are often used because these models are well understood and there are well-known methods to fit them to data. In reality, of course, many causal relationships are more or less nonlinear, raising some doubts as to the applicability and usefulness of purely linear methods. In this contribution we show that the basic linear framework can be generalized to nonlinear models. In this extended framework, nonlinearities in the data-generating process are in fact a blessing rather than a curse, as they typically provide information on the underlying causal system and allow more aspects of the true data-generating mechanisms to be identified. In addition to theoretical results we show simulations and some simple real data experiments illustrating the identification power provided by nonlinearities.

We propose a novel bound on single-variable marginal probability distributions in factor graphs with discrete variables. The bound is obtained by propagating local bounds (convex sets of probability distributions) over a subtree of the factor graph, rooted in the variable of interest. By construction, the method not only bounds the exact marginal probability distribution of a variable, but also its approximate Belief Propagation marginal ("belief"). Thus, apart from providing a practical means to calculate bounds on marginals, our contribution also lies in providing a better understanding of the error made by Belief Propagation. We show that our bound outperforms the state-of-the-art on some inference problems arising in medical diagnosis.

Novel conditions are derived that guarantee convergence of the Sum-Product Algorithm (also known as Loopy Belief Propagation or simply Belief Propagation (BP)) to a unique fixed point, irrespective of the initial messages, for parallel (synchronous) updates. The computational complexity of the conditions is polynomial in the number of variables. In contrast with previously existing conditions, our results are directly applicable to arbitrary factor graphs (with discrete variables) and are shown to be valid also in the case of factors containing zeros, under some additional conditions. The conditions are compared with existing ones, numerically and, if possible, analytically. For binary variables with pairwise interactions, sufficient conditions are derived that take into account local evidence (i.e., single-variable factors) and the type of pair interactions (attractive or repulsive). It is shown empirically that this bound outperforms existing bounds.

Recently, Chertkov and Chernyak (2006b) derived an exact expression for the partition sum (normalization constant) corresponding to a graphical model, which is an expansion around the belief propagation (BP) solution. By adding correction terms to the BP free energy, one for each "generalized loop" in the factor graph, the exact partition sum is obtained. However, the usually enormous number of generalized loops generally prohibits summation over all correction terms. In this article we introduce truncated loop series BP (TLSBP), a particular way of truncating the loop series of Chertkov and Chernyak by considering generalized loops as compositions of simple loops. We analyze the performance of TLSBP in different scenarios, including the Ising model on square grids and regular random graphs, and on PROMEDAS, a large probabilistic medical diagnostic system. We show that TLSBP often improves upon the accuracy of the BP solution, at the expense of increased computation time. We also show that the performance of TLSBP strongly depends on the degree of interaction between the variables. For weak interactions, truncating the series leads to significant improvements, whereas for strong interactions it can be ineffective, even if a high number of terms is considered.

We propose a method to improve approximate inference methods by correcting for the influence of loops in the graphical model. The method is a generalization and alternative implementation of a recent idea from Montanari and Rizzo (2005). It is applicable to arbitrary factor graphs, provided that the size of the Markov blankets is not too large. It consists of two steps: (i) an approximate inference method, for example, belief propagation, is used to approximate cavity distributions for each variable (i.e., probability distributions on the Markov blanket of a variable for a modified graphical model in which the factors involving that variable have been removed); (ii) all cavity distributions are improved by a message-passing algorithm that cancels out approximation errors by imposing certain consistency constraints. This loop correction (LC) method usually gives significantly better results than the original, uncorrected, approximate inference algorithm that is used to estimate the effect of loops. Indeed, we often observe that the loop-corrected error is approximately the square of the error of the uncorrected approximate inference method. In this article, we compare different variants of the loop correction method with other approximate inference methods on a variety of graphical models, including "real world" networks, and conclude that the LC method generally obtains the most accurate results.

In the current paper, the Promedas model for internal medicine, developed by our team, is introduced. The model is based on up-to-date medical knowledge and consists of approximately 2000 diagnoses, 1000 findings and 8600 connections between diagnoses and findings, covering a large part of internal medicine. We show that Belief Propagation (BP) can be successfully applied as approximate inference algorithm in the Promedas network. In some cases, however, we find errors that are too large for this application. We apply a recently developed method that improves the BP results by means of a loop expansion scheme. This method, termed Loop Corrected (LC) BP, is able to improve the marginal probabilities significantly, leaving a remaining error which is acceptable for the purpose of medical diagnosis.

We propose a method for improving Belief Propagation (BP) that takes into account the influence of loops in the graphical model. The method is a variation on and generalization of the method recently introduced by (Montanari and Rizzo, 2005). It consists of two steps: (i) standard BP is used to calculate cavity distributions for each variable (i.e. probability distributions on the Markov blanket of a variable for a modified graphical model, in which the factors involving that variable have been removed); (ii) all cavity distributions are combined by a message-passing algorithm to obtain consistent single node marginals. The method is exact if the graphical model contains a single loop. The complexity of the method is exponential in the size of the Markov blankets. The results are very accurate in general: the error is often several orders of magnitude smaller than that of standard BP, as illustrated by numerical experiments.

We derive novel sufficient conditions for convergence of Loopy Belief Propagation (also known as the Sum-Product algorithm) to a unique fixed point. Our results improve upon previously known conditions. For binary variables with (anti-)ferromagnetic interactions, our conditions seem to be sharp.

We analyse the local stability of the high-temperature fixed point of the loopy belief propagation (LBP) algorithm and how this relates to the properties of the Bethe free energy which LBP tries to minimize. We focus on the case of binary networks with pairwise interactions. In particular, we state sufficient conditions for convergence of LBP to a unique fixed point and show that these are sharp for purely ferromagnetic interactions. In contrast, in the purely antiferromagnetic case, the undamped parallel LBP algorithm is suboptimal in the sense that the stability of the fixed point breaks down much earlier than for damped or sequential LBP; we observe that the onset of instability for the latter algorithms is related to the properties of the Bethe free energy. For spin-glass interactions, damping LBP only helps slightly. We estimate analytically the temperature at which the high-temperature LBP fixed point becomes unstable for random graphs with arbitrary degree distributions and random interactions.

We introduce a computationally efficient method to estimate the validity of the BP method as a function of graph topology, the connectivity strength, frustration and network size. We present numerical results that demonstrate the correctness of our estimates for the uniform random model and for a real-world network ("C. Elegans"). Although the method is restricted to pair-wise interactions, no local evidence (zero "biases") and binary variables, we believe that its predictions correctly capture the limitations of BP for inference and MAP estimation on arbitrary graphical models. Using this approach, we find that BP always performs better than MF. Especially for large networks with broad degree distributions (such as scale-free networks) BP turns out to significantly outperform MF.

Laser-based optical diagnostics, such as planar laser-induced fluorescence and, especially, Raman imaging, often require selective spectral filtering. We advocate the use of an imaging spectrograph with a broad entrance slit as a spectral filter for two-dimensional imaging. A spectrograph in this mode of operation produces output that is a convolution of the spatial and spectral information that is present in the incident light. We describe an analytical deconvolution procedure, based on Bayesian statistics, that retrieves the spatial information while it avoids excessive noise blowup. The method permits direct imaging through a spectrograph, even under broadband illumination. We introduce the formalism and discuss the underlying assumptions. The performance of the procedure is demonstrated on an artificial but pathological example. In a companion paper [Appl. Opt. 43, 5682-5690 (2004)] the method is applied to the practical case of fuel equivalence ratio Raman imaging in a combustible methane-air mixture.

### Preprints

When predicting a target variable Y from features X, the prediction Ŷ can be performative: an agent might act on this prediction, affecting the value of Y that we eventually observe. Performative predictions are deliberately prevalent in algorithmic decision support, where a Decision Support System (DSS) provides a prediction for an agent to affect the value of the target variable. When deploying a DSS in high-stakes settings (e.g. healthcare, law, predictive policing, or child welfare screening) it is imperative to carefully assess the performative effects of the DSS. In the case that the DSS serves as an alarm for a predicted negative outcome, naive retraining of the prediction model is bound to result in a model that underestimates the risk, due to effective workings of the previous model. In this work, we propose to model the deployment of a DSS as causal domain shift and provide novel cross-domain identification results for the conditional expectation E[Y|X], allowing for pre- and post-hoc assessment of the deployment of the DSS, and for retraining of a model that assesses the risk under a baseline policy where the DSS is not deployed. Using a running example, we empirically show that a repeated regression procedure provides a practical framework for estimating these quantities, even when the data is affected by sample selection bias and selective labelling, offering for a practical, unified solution for multiple forms of target variable bias.

Selection bias is ubiquitous in real-world data, and can lead to misleading results if not dealt with properly. We introduce a conditioning operation on Structural Causal Models (SCMs) to model latent selection from a causal perspective. We show that the conditioning operation transforms an SCM with the presence of an explicit latent selection mechanism into an SCM without such selection mechanism, which partially encodes the causal semantics of the selected subpopulation according to the original SCM. Furthermore, we show that this conditioning operation preserves the simplicity, acyclicity, and linearity of SCMs, and commutes with marginalization. Thanks to these properties, combined with marginalization and intervention, the conditioning operation offers a valuable tool for conducting causal reasoning tasks within causal models where latent details have been abstracted away. We demonstrate by example how classical results of causal inference can be generalized to include selection bias and how the conditioning operation helps with modeling of real-world problems.

We consider the problem of discovering causal relations from independence constraints when selection bias in addition to confounding is present. While the seminal FCI algorithm is sound and complete in this setup, no criterion for the causal interpretation of its output under selection bias is presently known. We focus instead on local patterns of independence relations, where we find no sound method for only three variable that can include background knowledge. Y-Structure patterns (Mani et al., 2006; Mooij and Cremers, 2015) are shown to be sound in predicting causal relations from data under selection bias, where cycles may be present. We introduce a finite-sample scoring rule for Y-Structures that is shown to successfully predict causal relations in simulation experiments that include selection mechanisms. On real-world microarray data, we show that a Y-Structure variant performs well across different datasets, potentially circumventing spurious correlations due to selection bias.

Unobserved confounding is one of the main challenges when estimating causal effects. We propose a causal reduction method that, given a causal model, replaces an arbitrary number of possibly high-dimensional latent confounders with a single latent confounder that takes values in the same space as the treatment variable, without changing the observational and interventional distributions the causal model entails. This allows us to estimate the causal effect in a principled way from combined data without relying on the common but often unrealistic assumption that all confounders have been observed. We apply our causal reduction in three different settings. In the first setting, we assume the treatment and outcome to be discrete. The causal reduction then implies bounds between the observational and interventional distributions that can be exploited for estimation purposes. In certain cases with highly unbalanced observational samples, the accuracy of the causal effect estimate can be improved by incorporating observational data. Second, for continuous variables and assuming a linear-Gaussian model, we derive equality constraints for the parameters of the observational and interventional distributions. Third, for the general continuous setting (possibly nonlinear and non-Gaussian), we parameterize the reduced causal model using normalizing flows, a flexible class of easily invertible nonlinear transformations. We perform a series of experiments on synthetic data and find that in several cases the number of interventional samples can be reduced when adding observational training samples without sacrificing accuracy.

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. With the help of the causal ordering algorithm, one can construct graphical representations of dynamical systems that represent the causal relations between the variables and the conditional independences in the equilibrium distribution. We apply these tools to formulate sufficient graphical conditions for identifying 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 this method to a simple model for a protein signalling pathway and test its predictions both in simulations and using real-world protein expression data. We demonstrate that perfect adaptation can lead to misleading orientation of edges in the output of causal discovery algorithms.

One of the core assumptions in causal discovery is the faithfulness assumption---i.e. assuming that independencies found in the data are due to separations in the true causal graph. This assumption can, however, be violated in many ways, including xor connections, deterministic functions or cancelling paths. In this work, we propose a weaker assumption that we call 2-adjacency faithfulness. In contrast to adjacency faithfulness, which assumes that there is no conditional independence between each pair of variables that are connected in the causal graph, we only require no conditional independence between a node and a subset of its Markov blanket that can contain up to two nodes. Equivalently, we adapt orientation faithfulness to this setting. We further propose a sound orientation rule for causal discovery that applies under weaker assumptions. As a proof of concept, we derive a modified Grow and Shrink algorithm that recovers the Markov blanket of a target node and prove its correctness under strictly weaker assumptions than the standard faithfulness assumption.

Real-world complex systems are often modelled by sets of equations with endogenous and exogenous variables. What can we say about the causal and probabilistic aspects of variables that appear in these equations without explicitly solving the equations? We make use of Simon's causal ordering algorithm (Simon, 1953) to construct a causal ordering graph and prove that it expresses the effects of soft and perfect interventions on the equations under certain unique solvability assumptions. We further construct a Markov ordering graph and prove that it encodes conditional independences in the distribution implied by the equations with independent random exogenous variables, under a similar unique solvability assumption. We discuss how this approach reveals and addresses some of the limitations of existing causal modelling frameworks, such as causal Bayesian networks and structural causal models.

We study the performance of Local Causal Discovery (LCD) [5], a simple and efficient constraint-based method for causal discovery, in predicting causal effects in large-scale gene expression data. We construct practical estimators specific to the high-dimensional regime. Inspired by the ICP algorithm [13], we use an optional preselection method and two different statistical tests. Empirically, the resulting LCD estimator is seen to closely approach the accuracy of ICP, the state-of-the-art method, while it is algorithmically simpler and computationally more efficient.

Dynamical systems are widely used in science and engineering to model systems consisting of several interacting components. Often, they can be given a causal interpretation in the sense that they not only model the evolution of the states of the system's components over time, but also describe how their evolution is affected by external interventions on the system that perturb the dynamics. We introduce the formal framework of structural dynamical causal models (SDCMs) that explicates the causal semantics of the system's components as part of the model. SDCMs represent a dynamical system as a collection of stochastic processes and specify the basic causal mechanisms that govern the dynamics of each component as a structured system of random differential equations of arbitrary order. SDCMs extend the versatile causal modeling framework of structural causal models (SCMs), also known as structural equation models (SEMs), by explicitly allowing for time-dependence. An SDCM can be thought of as the stochastic-process version of an SCM, where the static random variables of the SCM are replaced by dynamic stochastic processes and their derivatives. We provide the foundations for a theory of SDCMs, by (i) formally defining SDCMs, their solutions, stochastic interventions, and a graphical representation; (ii) studying existence and uniqueness of the solutions for given initial conditions; (iii) providing Markov properties for SDCMs with initial conditions; (iv) discussing under which conditions SDCMs equilibrate to SCMs as time tends to infinity; (v) relating the properties of the SDCM to those of the equilibrium SCM. This correspondence enables one to leverage the wealth of statistical tools and discovery methods available for SCMs when studying the causal semantics of a large class of stochastic dynamical systems. The theory is illustrated with examples from different scientific domains.

Dynamical systems are widely used in science and engineering to model systems consisting of several interacting components. Often, they can be given a causal interpretation in the sense that they not only model the evolution of the states of the system's components over time, but also describe how their evolution is affected by external interventions on the system that perturb the dynamics. We introduce the formal framework of structural dynamical causal models (SDCMs) that explicates the causal semantics of the system's components as part of the model. SDCMs represent a dynamical system as a collection of stochastic processes and specify the basic causal mechanisms that govern the dynamics of each component as a structured system of random differential equations of arbitrary order. SDCMs extend the versatile causal modeling framework of structural causal models (SCMs), also known as structural equation models (SEMs), by explicitly allowing for time-dependence. An SDCM can be thought of as the stochastic-process version of an SCM, where the static random variables of the SCM are replaced by dynamic stochastic processes and their derivatives. We provide the foundations for a theory of SDCMs, by (i) formally defining SDCMs, their solutions, stochastic interventions, and a graphical representation; (ii) studying existence and uniqueness of the solutions for given initial conditions; (iii) discussing under which conditions SDCMs equilibrate to SCMs as time tends to infinity; (iv) relating the properties of the SDCM to those of the equilibrium SCM. This correspondence enables one to leverage the wealth of statistical tools and discovery methods available for SCMs when studying the causal semantics of a large class of stochastic dynamical systems. The theory is illustrated with several well-known examples from different scientific domains.

We investigate probabilistic graphical models that allow for both cycles and latent variables. For this we introduce directed graphs with hyperedges (HEDGes), generalizing and combining both marginalized directed acyclic graphs (mDAGs) that can model latent (dependent) variables, and directed mixed graphs (DMGs) that can model cycles. We define and analyse several different Markov properties that relate the graphical structure of a HEDG with a probability distribution on a corresponding product space over the set of nodes, for example factorization properties, structural equations properties, ordered/local/global Markov properties, and marginal versions of these. The various Markov properties for HEDGes are in general not equivalent to each other when cycles or hyperedges are present, in contrast with the simpler case of directed acyclic graphical (DAG) models (also known as Bayesian networks). We show how the Markov properties for HEDGes - and thus the corresponding graphical Markov models - are logically related to each other.

This article contains detailed proofs and additional examples related to the UAI-2013 submission "Learning Sparse Causal Models is not NP-hard". It describes the FCI+ algorithm: a method for sound and complete causal model discovery in the presence of latent confounders and/or selection bias, that has worst case polynomial complexity of order N^(2(k+1)) in the number of independence tests, for sparse graphs over N nodes, bounded by node degree k. The algorithm is an adaptation of the well-known FCI algorithm by (Spirtes et al., 2000) that is also sound and complete, but has worst case complexity exponential in N.

We derive two related novel bounds on single-variable marginal probability distributions in factor graphs with discrete variables. The first method propagates bounds over a subtree of the factor graph rooted in the variable, and the second method propagates bounds over the self-avoiding walk tree starting at the variable. By construction, both methods not only bound the exact marginal probability distribution of a variable, but also its approximate Belief Propagation marginal (``belief''). Thus, apart from providing a practical means to calculate bounds on marginals, our contribution also lies in an increased understanding of the error made by Belief Propagation. Empirically, we show that our bounds often outperform existing bounds in terms of accuracy and/or computation time. We also show that our bounds can yield nontrivial results for medical diagnosis inference problems.

We propose a method for improving approximate inference methods that corrects for the influence of loops in the graphical model. The method is applicable to arbitrary factor graphs, provided that the size of the Markov blankets is not too large. It is an alternative implementation of an idea introduced recently by Montanari and Rizzo (2005). In its simplest form, which amounts to the assumption that no loops are present, the method reduces to the minimal Cluster Variation Method approximation (which uses maximal factors as outer clusters). On the other hand, using estimates of the effect of loops (obtained by some approximate inference algorithm) and applying the Loop Correcting (LC) method usually gives significantly better results than applying the approximate inference algorithm directly without loop corrections. Indeed, we often observe that the loop corrected error is approximately the square of the error of the approximate inference method used to estimate the effect of loops. We compare different variants of the Loop Correcting method with other approximate inference methods on a variety of graphical models, including "real world" networks, and conclude that the LC approach generally obtains the most accurate results.

Recently, M. Chertkov and V.Y. Chernyak derived an exact expression for the partition sum (normalization constant) corresponding to a graphical model, which is an expansion around the Belief Propagation (BP) solution. By adding correction terms to the BP free energy, one for each "generalized loop" in the factor graph, the exact partition sum is obtained. However, the usually enormous number of generalized loops generally prohibits summation over all correction terms. In this article we introduce Truncated Loop Series BP (TLSBP), a particular way of truncating the loop series of M. Chertkov and V.Y. Chernyak by considering generalized loops as compositions of simple loops. We analyze the performance of TLSBP in different scenarios, including the Ising model, regular random graphs and on Promedas, a large probabilistic medical diagnostic system. We show that TLSBP often improves upon the accuracy of the BP solution, at the expense of increased computation time. We also show that the performance of TLSBP strongly depends on the degree of interaction between the variables. For weak interactions, truncating the series leads to significant improvements, whereas for strong interactions it can be ineffective, even if a high number of terms is considered.

We derive novel conditions that guarantee convergence of the Sum-Product algorithm (also known as Loopy Belief Propagation or simply Belief Propagation) to a unique fixed point, irrespective of the initial messages. The computational complexity of the conditions is polynomial in the number of variables. In contrast with previously existing conditions, our results are directly applicable to arbitrary factor graphs (with discrete variables) and are shown to be valid also in the case of factors containing zeros, under some additional conditions. We compare our bounds with existing ones, numerically and, if possible, analytically. For binary variables with pairwise interactions, we derive sufficient conditions that take into account local evidence (i.e., single variable factors) and the type of pair interactions (attractive or repulsive). It is shown empirically that this bound outperforms existing bounds.

We use the Bethe approximation to calculate the critical temperature for the transition from a paramagnetic to a glassy phase in spin-glass models on real-world graphs. Our criterion is based on the marginal stability of the minimum of the Bethe free energy. For uniform degree random graphs (equivalent to the Viana-Bray model) our numerical results, obtained by averaging single problem instances, are in agreement with the known critical temperature obtained by use of the replica method. Contrary to the replica method, our method immediately generalizes to arbitrary (random) graphs. We present new results for Barabasi-Albert scale-free random graphs, for which no analytical results are known. We investigate the scaling behavior of the critical temperature with graph size for both the finite and the infinite connectivity limit. We compare these with the naive Mean Field results. We observe that the Belief Propagation algorithm converges only in the paramagnetic regime.