## Periodic Reporting for period 2 - CAFES (Causal Analysis of Feedback Systems)

**Reporting period:**2017-03-01

**to**2018-08-31

## Summary of the context and overall objectives of the project

Many questions in science, policy making and everyday life are of a causal nature: how would changing A influence B? Causal inference, a branch of statistics and machine learning, studies how cause-effect relationships can be discovered from data and how these can be used for making predictions in situations where a system has been perturbed by an external intervention. The ability to reliably make such causal predictions is of great value for practical applications in a variety of disciplines. Over the last two decades, remarkable progress has been made in the field. However, even though state-of-the-art causal inference algorithms work well on simulated data when all their assumptions are met, there is still a considerable gap between theory and practice. The goal of CAFES is to bridge that gap by developing theory and algorithms that will enable large-scale applications of causal inference in various challenging domains in science, industry and decision making.

The key challenge that will be addressed is how to deal with cyclic causal relationships (“feedback loops”). Feedback loops are very common in many domains (e.g., biology, economy and climatology), but have mostly been ignored so far in the field. Building on recently established connections between dynamical systems and causal models, CAFES will develop theory and algorithms for causal modeling, reasoning, discovery and prediction for cyclic causal systems. Extensions to stationary and non-stationary processes will be developed to advance the state-of-the-art in causal analysis of time-series data. In order to optimally use available resources, computationally efficient and statistically robust algorithms for causal inference from observational and interventional data in the context of confounders and feedback will be developed. The work will be done with a strong focus on applications in molecular biology, one of the most promising areas for automated causal inference from data.

The key challenge that will be addressed is how to deal with cyclic causal relationships (“feedback loops”). Feedback loops are very common in many domains (e.g., biology, economy and climatology), but have mostly been ignored so far in the field. Building on recently established connections between dynamical systems and causal models, CAFES will develop theory and algorithms for causal modeling, reasoning, discovery and prediction for cyclic causal systems. Extensions to stationary and non-stationary processes will be developed to advance the state-of-the-art in causal analysis of time-series data. In order to optimally use available resources, computationally efficient and statistically robust algorithms for causal inference from observational and interventional data in the context of confounders and feedback will be developed. The work will be done with a strong focus on applications in molecular biology, one of the most promising areas for automated causal inference from data.

## Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

In collaboration with other scientists, the CAFES team has published several papers on different aspects of causal inference.

The first paper (Van Ommen & Mooij, UAI 2017) studies algebraic constraints imposed by linear structural equation models. We showed how the half-trek criterion can be used to enumerate constraints imposed by a model, and to decide whether two causal graphs are observationally equivalent. We showed that taking into account the additional algebraic constraints may lead to improvements in model selection accuracy.

The second paper (Rubenstein et al., UAI 2017) addresses the notion of consistency of causal models that differ in their level of modeling detail. These theoretical considerations shed some new light on the interpretation of cyclic causal models and on the relationship between micro-level models versus macro-level models in which the macro-variables are linearly aggregated features of the micro-variables.

The third paper (Louizos et al., NIPS 2017) addresses adjustment for latent confounders in causal inference from observational data. Often, confounders cannot be measured directly, but noisy measurements of high-dimensional proxies for confounders may be available instead. We have proposed a new causal inference method based on Variational Autoencoders that estimates a low-dimensional encoding of the measured proxies to represent the latent confounders of cause and effect, and uses this for adjustment. We showed that the method is more robust than existing methods, and matches the state-of-the-art on previous benchmarks focused on individual treatment effects.

The first paper (Van Ommen & Mooij, UAI 2017) studies algebraic constraints imposed by linear structural equation models. We showed how the half-trek criterion can be used to enumerate constraints imposed by a model, and to decide whether two causal graphs are observationally equivalent. We showed that taking into account the additional algebraic constraints may lead to improvements in model selection accuracy.

The second paper (Rubenstein et al., UAI 2017) addresses the notion of consistency of causal models that differ in their level of modeling detail. These theoretical considerations shed some new light on the interpretation of cyclic causal models and on the relationship between micro-level models versus macro-level models in which the macro-variables are linearly aggregated features of the micro-variables.

The third paper (Louizos et al., NIPS 2017) addresses adjustment for latent confounders in causal inference from observational data. Often, confounders cannot be measured directly, but noisy measurements of high-dimensional proxies for confounders may be available instead. We have proposed a new causal inference method based on Variational Autoencoders that estimates a low-dimensional encoding of the measured proxies to represent the latent confounders of cause and effect, and uses this for adjustment. We showed that the method is more robust than existing methods, and matches the state-of-the-art on previous benchmarks focused on individual treatment effects.

## Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

"Currently, the CAFES team is working on several topics for which preprints are available.

One major achievement is the work ""Markov Properties for Graphical Models with Cycles and Latent Variables"" (arXiv:1710.08775) by Forré and Mooij. Here, the authors derive an extensive mathematical theory on the relationships between various possible generalizations of Markov properties for directed graphical models to the cyclic case. In contrast with the well-known DAG case, which corresponds to Bayesian networks, the various Markov properties that can be formulated have much subtler relationships in the cyclic, confounded setting. This work forms the basis for constraint-based causal discovery methods that will be developed in future work.

Another fundamental contribution to the theory of cyclic Structural Causal Models is available in draft form in ""Structural Causal Models: Cycles, Marginalizations, Exogenous Reparametrizations and Reductions"" (arXiv:1611.06221) by Bongers et al. In that paper, the basic theory of structural causal models with cycles and confounders is developed in a mathematically rigorous way. This involves measureme-theoretic considerations, solutions of SCMs, the definitions of direct and indirect causal relations, the notion of marginalization, the graphical representation, and different natural equivalence relations between SCMs.

The framework of Structural Causal Models can be extended in other ways as well. In ""From Deterministic ODEs to Dynamic Structural Causal Models"" (arXiv:1608.08028) by Rubenstein et al. we consider an extension that generalizes values of variables to trajectories. This is a novel approach to introduce temporal dependence into causal modeling frameworks, and allows for example to give concise causal models of systems that display periodic motion asymptotically.

Other expected future work regards developing algorithms for causal discovery in the presence of cycles and confounders, dealing with measurement error in causal discovery, addressing selection bias, refining the theory on the connection between ODEs and SCMs, extending the framework of SCMs to be able to deal with constraints, and addressing causal prediction and validation in settings where interventions are not perfect interventions with known targets. Furthermore, we hope to provide a large-scale and convincing demonstration of the usefulness of causal discovery methods in analyzing biological data."

One major achievement is the work ""Markov Properties for Graphical Models with Cycles and Latent Variables"" (arXiv:1710.08775) by Forré and Mooij. Here, the authors derive an extensive mathematical theory on the relationships between various possible generalizations of Markov properties for directed graphical models to the cyclic case. In contrast with the well-known DAG case, which corresponds to Bayesian networks, the various Markov properties that can be formulated have much subtler relationships in the cyclic, confounded setting. This work forms the basis for constraint-based causal discovery methods that will be developed in future work.

Another fundamental contribution to the theory of cyclic Structural Causal Models is available in draft form in ""Structural Causal Models: Cycles, Marginalizations, Exogenous Reparametrizations and Reductions"" (arXiv:1611.06221) by Bongers et al. In that paper, the basic theory of structural causal models with cycles and confounders is developed in a mathematically rigorous way. This involves measureme-theoretic considerations, solutions of SCMs, the definitions of direct and indirect causal relations, the notion of marginalization, the graphical representation, and different natural equivalence relations between SCMs.

The framework of Structural Causal Models can be extended in other ways as well. In ""From Deterministic ODEs to Dynamic Structural Causal Models"" (arXiv:1608.08028) by Rubenstein et al. we consider an extension that generalizes values of variables to trajectories. This is a novel approach to introduce temporal dependence into causal modeling frameworks, and allows for example to give concise causal models of systems that display periodic motion asymptotically.

Other expected future work regards developing algorithms for causal discovery in the presence of cycles and confounders, dealing with measurement error in causal discovery, addressing selection bias, refining the theory on the connection between ODEs and SCMs, extending the framework of SCMs to be able to deal with constraints, and addressing causal prediction and validation in settings where interventions are not perfect interventions with known targets. Furthermore, we hope to provide a large-scale and convincing demonstration of the usefulness of causal discovery methods in analyzing biological data."