Choosing DAG Models Using Markov and Minimal Edge Count in the Absence of Ground Truth

We give a novel nonparametric pointwise consistent statistical test (the Markov Checker) of the Markov condition for directed acyclic graph (DAG) or completed partially directed acyclic graph (CPDAG) models given a dataset. We also introduce the Cross-Algorithm Frugality Search (CAFS) for rejecting DAG models that either do not pass the Markov Checker test or that are not edge minimal. Edge minimality has been used previously by Raskutti and Uhler as a nonparametric simplicity criterion, though CAFS readily generalizes to other simplicity conditions. Reference to the ground truth is not necessary for CAFS, so it is useful for finding causal structure learning algorithms and tuning parameter settings that output causal models that are approximately true from a given data set. We provide a software tool for this analysis that is suitable for even quite large or dense models, provided a suitably fast pointwise consistent test of conditional independence is available. In addition, we show in simulation that the CAFS procedure can pick approximately correct models without knowing the ground truth.

Better Simulations for Validating Causal Discovery with the DAG-Adaptation of the Onion Method

The number of artificial intelligence algorithms for learning causal models from data is growing rapidly. Most ``causal discovery'' or ``causal structure learning'' algorithms are primarily validated through simulation studies. However, no widely accepted simulation standards exist and publications often report conflicting performance statistics -- even when only considering publications that simulate data from linear models. In response, several manuscripts have criticized a popular simulation design for validating algorithms in the linear case. We propose a new simulation design for generating linear models for directed acyclic graphs (DAGs): the DAG-adaptation of the Onion (DaO) method. DaO simulations are fundamentally different from existing simulations because they prioritize the distribution of correlation matrices rather than the distribution of linear effects. Specifically, the DaO method uniformly samples the space of all correlation matrices consistent with (i.e. Markov to) a DAG. We also discuss how to sample DAGs and present methods for generating DAGs with scale-free in-degree or out-degree. We compare the DaO method against two alternative simulation designs and provide implementations of the DaO method in Python and R: https://github.com/bja43/DaO_simulation. We advocate for others to adopt DaO simulations as a fair universal benchmark.

Causal Discovery for fMRI data: Challenges, Solutions, and a Case Study

Designing studies that apply causal discovery requires navigating many researcher degrees of freedom. This complexity is exacerbated when the study involves fMRI data. In this paper we (i) describe nine challenges that occur when applying causal discovery to fMRI data, (ii) discuss the space of decisions that need to be made, (iii) review how a recent case study made those decisions, (iv) and identify existing gaps that could potentially be solved by the development of new methods. Overall, causal discovery is a promising approach for analyzing fMRI data, and multiple successful applications have indicated that it is superior to traditional fMRI functional connectivity methods, but current causal discovery methods for fMRI leave room for improvement.

Fast Scalable and Accurate Discovery of DAGs Using the Best Order Score Search and Grow-Shrink Trees

Learning graphical conditional independence structures is an important machine learning problem and a cornerstone of causal discovery. However, the accuracy and execution time of learning algorithms generally struggle to scale to problems with hundreds of highly connected variables -- for instance, recovering brain networks from fMRI data. We introduce the best order score search (BOSS) and grow-shrink trees (GSTs) for learning directed acyclic graphs (DAGs) in this paradigm. BOSS greedily searches over permutations of variables, using GSTs to construct and score DAGs from permutations. GSTs efficiently cache scores to eliminate redundant calculations. BOSS achieves state-of-the-art performance in accuracy and execution time, comparing favorably to a variety of combinatorial and gradient-based learning algorithms under a broad range of conditions. To demonstrate its practicality, we apply BOSS to two sets of resting-state fMRI data: simulated data with pseudo-empirical noise distributions derived from randomized empirical fMRI cortical signals and clinical data from 3T fMRI scans processed into cortical parcels. BOSS is available for use within the TETRAD project which includes Python and R wrappers.

Py-Tetrad and RPy-Tetrad: A New Python Interface with R Support for Tetrad Causal Search

We give novel Python and R interfaces for the (Java) Tetrad project for causal modeling, search, and estimation. The Tetrad project is a mainstay in the literature, having been under consistent development for over 30 years. Some of its algorithms are now classics, like PC and FCI; others are recent developments. It is increasingly the case, however, that researchers need to access the underlying Java code from Python or R. Existing methods for doing this are inadequate. We provide new, up-to-date methods using the JPype Python-Java interface and the Reticulate Python-R interface, directly solving these issues. With the addition of some simple tools and the provision of working examples for both Python and R, using JPype and Reticulate to interface Python and R with Tetrad is straightforward and intuitive.

The m-connecting imset and factorization for ADMG models

Directed acyclic graph (DAG) models have become widely studied and applied in statistics and machine learning -- indeed, their simplicity facilitates efficient procedures for learning and inference. Unfortunately, these models are not closed under marginalization, making them poorly equipped to handle systems with latent confounding. Acyclic directed mixed graph (ADMG) models characterize margins of DAG models, making them far better suited to handle such systems. However, ADMG models have not seen wide-spread use due to their complexity and a shortage of statistical tools for their analysis. In this paper, we introduce the m-connecting imset which provides an alternative representation for the independence models induced by ADMGs. Furthermore, we define the m-connecting factorization criterion for ADMG models, characterized by a single equation, and prove its equivalence to the global Markov property. The m-connecting imset and factorization criterion provide two new statistical tools for learning and inference with ADMG models. We demonstrate the usefulness of these tools by formulating and evaluating a consistent scoring criterion with a closed form solution.

Greedy Relaxations of the Sparsest Permutation Algorithm

There has been an increasing interest in methods that exploit permutation reasoning to search for directed acyclic causal models, including the "Ordering Search" of Teyssier and Kohler and GSP of Solus, Wang and Uhler. We extend the methods of the latter by a permutation-based operation, tuck, and develop a class of algorithms, namely GRaSP, that are efficient and pointwise consistent under increasingly weaker assumptions than faithfulness. The most relaxed form of GRaSP outperforms many state-of-the-art causal search algorithms in simulation, allowing efficient and accurate search even for dense graphs and graphs with more than 100 variables.

Learning Latent Causal Structures with a Redundant Input Neural Network

Most causal discovery algorithms find causal structure among a set of observed variables. Learning the causal structure among latent variables remains an important open problem, particularly when using high-dimensional data. In this paper, we address a problem for which it is known that inputs cause outputs, and these causal relationships are encoded by a causal network among a set of an unknown number of latent variables. We developed a deep learning model, which we call a redundant input neural network (RINN), with a modified architecture and a regularized objective function to find causal relationships between input, hidden, and output variables. More specifically, our model allows input variables to directly interact with all latent variables in a neural network to influence what information the latent variables should encode in order to generate the output variables accurately. In this setting, the direct connections between input and latent variables makes the latent variables partially interpretable; furthermore, the connectivity among the latent variables in the neural network serves to model their potential causal relationships to each other and to the output variables. A series of simulation experiments provide support that the RINN method can successfully recover latent causal structure between input and output variables.

FASK with Interventional Knowledge Recovers Edges from the Sachs Model

We report a procedure that, in one step from continuous data with minimal preparation, recovers the graph found by Sachs et al. \cite{sachs2005causal}, with only a few edges different. The algorithm, Fast Adjacency Skewness (FASK), relies on a mixture of linear reasoning and reasoning from the skewness of variables; the Sachs data is a good candidate for this procedure since the skewness of the variables is quite pronounced. We review the ground truth model from Sachs et al. as well as some of the fluctuations seen in the protein abundances in the system, give the Sachs model and the FASK model, and perform a detailed comparison. Some variation in hyper-parameters is explored, though the main result uses values at or near the defaults learned from work modeling fMRI data.

A Comparison of Public Causal Search Packages on Linear, Gaussian Data with No Latent Variables

We compare Tetrad (Java) algorithms to the other public software packages BNT (Bayes Net Toolbox, Matlab), pcalg (R), bnlearn (R) on the \vanilla" task of recovering DAG structure to the extent possible from data generated recursively from linear, Gaussian structure equation models (SEMs) with no latent variables, for random graphs, with no additional knowledge of variable order or adjacency structure, and without additional specification of intervention information. Each one of the above packages offers at least one implementation suitable to this purpose. We compare them on adjacency and orientation accuracy as well as time performance, for fixed datasets. We vary the number of variables, the number of samples, and the density of graph, for a total of 27 combinations, averaging all statistics over 10 runs, for a total of 270 datasets. All runs are carried out on the same machine and on their native platforms. An interactive visualization tool is provided for the reader who wishes to know more than can be documented explicitly in this report.