Causal inference via string diagram surgery

Extracting causal relationships from observed correlations is a growing area in probabilistic reasoning, originating with the seminal work of Pearl and others from the early 1990s. This paper develops a new, categorically oriented view based on a clear distinction between syntax (string diagrams) and semantics (stochastic matrices), connected via interpretations as structure-preserving functors. A key notion in the identification of causal effects is that of an intervention, whereby a variable is forcefully set to a particular value independent of any prior propensities. We represent the effect of such an intervention as an endo-functor which performs 'string diagram surgery' within the syntactic category of string diagrams. This diagram surgery in turn yields a new, interventional distribution via the interpretation functor. While in general there is no way to compute interventional distributions purely from observed data, we show that this is possible in certain special cases using a calculational tool called comb disintegration. We demonstrate the use of this technique on two well-known toy examples: one where we predict the causal effect of smoking on cancer in the presence of a confounding common cause and where we show that this technique provides simple sufficient conditions for computing interventions which apply to a wide variety of situations considered in the causal inference literature; the other one is an illustration of counterfactual reasoning where the same interventional techniques are used, but now in a 'twinned' set-up, with two version of the world - one factual and one counterfactual - joined together via exogenous variables that capture the uncertainties at hand

Free gs-monoidal categories and free Markov categories

Categorical probability has recently seen significant advances through the formalism of Markov categories, within which several classical theorems have been proven in entirely abstract categorical terms. Closely related to Markov categories are gs-monoidal categories, also known as CD categories. These omit a condition that implements the normalization of probability. Extending work of Corradini and Gadducci, we construct free gs-monoidal and free Markov categories generated by a collection of morphisms of arbitrary arity and coarity. For free gs-monoidal categories, this comes in the form of an explicit combinatorial description of their morphisms as structured cospans of labeled hypergraphs. These can be thought of as a formalization of gs-monoidal string diagrams ($=$term graphs) as a combinatorial data structure. We formulate the appropriate $2$-categorical universal property based on ideas of Walters and prove that our categories satisfy it. We expect our free categories to be relevant for computer implementations and we also argue that they can be used as statistical causal models generalizing Bayesian networks.Comment: 35 pages. v2: references update

Active Inference in String Diagrams: A Categorical Account of Predictive Processing and Free Energy

We present a categorical formulation of the cognitive frameworks of Predictive Processing and Active Inference, expressed in terms of string diagrams interpreted in a monoidal category with copying and discarding. This includes diagrammatic accounts of generative models, Bayesian updating, perception, planning, active inference, and free energy. In particular we present a diagrammatic derivation of the formula for active inference via free energy minimisation, and establish a compositionality property for free energy, allowing free energy to be applied at all levels of an agent's generative model. Aside from aiming to provide a helpful graphical language for those familiar with active inference, we conversely hope that this article may provide a concise formulation and introduction to the framework

The d-separation criterion in Categorical Probability

The d-separation criterion detects the compatibility of a joint probability distribution with a directed acyclic graph through certain conditional independences. In this work, we study this problem in the context of categorical probability theory by introducing a categorical definition of causal models, a categorical notion of d-separation, and proving an abstract version of the d-separation criterion. This approach has two main benefits. First, categorical d-separation is a very intuitive criterion based on topological connectedness. Second, our results apply in measure-theoretic probability (with standard Borel spaces), and therefore provide a clean proof of the equivalence of local and global Markov properties with causal compatibility for continuous and mixed variables.Comment: 34 page

The process theory of causality: an overview

This article offers an informal overview of the category-theoretical approach to causal modeling introduced by Jacobs et al. (2019) and explores some of its conceptual as well as methodological implications. The categorical formalism emphasizes the aspect of causality as a process, and represents a causal system as a network of connected mechanisms. We show that this alternative perspective sheds new light on the long-standing issue regarding the validity of the Markov condition, and also provides a formal mapping between micro-level causal models and abstracted macro models

Finding and Analyzing de novo Mutations in the Exomes of Parent-Offspring Trios of Spontaneous Chiari Malformation Type 1 Patients

Chiari Malformation Type 1 (CM1) is a neurodevelopmental disorder that occurs when one of the cerebellar tonsils herniates past the foramen magnum causing headaches, motor or sensory deficits, sleep apnea, and difficulty swallowing. This disorder is estimated to affect 1% of the population but due to the need of neuroimaging for diagnosis and the presence of asymptomatic patients there is still uncertainty about the exact proportion of the population affected. CM1 often presents itself with other neurodevelopmental disorders such as syringomyelia, scoliosis, and known genetic syndromes such as Klippel-Feil and Marfan syndromes. Twin, family, and familial clustering studies have established a genetic component to CM1 etiology, but have failed to ascertain any specific causal gene. The difficulty ascertaining causal genes, its comorbidity with multiple different syndromes, and the complex symptomatology of its patients indicate genetic heterogeneity. Other neurodevelopmental disorders with genetic heterogeneity such as Autism Spectrum Disorder and Epileptic Encephalopathies have had success finding genes of interests by looking for de novo mutants (DNMs) from spontaneous patient trios. With this in mind, we sequenced the exomes of a cohort of 29 offspring-parent trios affected with CM1 in search of candidate causative DNMs. Using previously established methods that predict which variants in the exome are DNMs, we found 44 variants that passed multiple filtering steps for quality, likelihood of being real DNMs, and potential to be causative. Three of these variants were classified as stopgain which made them likelier to be detrimental. These three were validated and analyzed for their potential role in CM1 risk. From thousands of possible variants, we successfully obtained a shortlist of genes to further study in future studies

Causal Programming

Causality is central to scientific inquiry. There is broad agreement on the meaning of causal statements, such as “Smoking causes cancer”, or, “Applying pesticides affects crop yields”. However, formalizing the intuition underlying such statements and conducting rigorous inference is difficult in practice. Accordingly, the overall goal of this dissertation is to reduce the difficulty of, and ambiguity in, causal modeling and inference. In other words, the goal is to make it easy for researchers to state precise causal assumptions, understand what they represent, understand why they are necessary, and to yield precise causal conclusions with minimal difficulty. Using the framework of structural causal models, I introduce a causation coeffi- cient as an analogue of the correlation coefficient, analyze its properties, and create a taxonomy of correlation/causation relationships. Analyzing these relationships provides insight into why correlation and causation are often conflated in practice, as well as a principled argument as to why formal causal analysis is necessary. Next, I introduce a theory of causal programming that unifies a large number of previ- ously separate problems in causal modeling and inference. I describe the use and implementation of a causal programming language as an embedded, domain-specific language called ‘Whittemore’. Whittemore permits rigorously identifying and esti- mating interventional queries without requiring the user to understand the details of the underlying inference algorithms. Finally, I analyze the computational com- plexity in determining the equilibrium distribution of cyclic causal models. I show this is uncomputable in the general case, under mild assumptions about the distri- butions of the model’s variables, suggesting that the structural causal model focus on acyclic causal models is a ‘natural’ limitation. Further extensions of the concept will have to give up either completeness or require the user to make additional — likely parametric — model assumptions. Together, this work supports the thesis that rigorous causal modeling and inference can be effectively abstracted over, giving a researcher access to all of the relevant details of causal modeling while encapsulating and automating the irrelevant details of inference