Unmixing for Causal Inference: Thoughts on McCaffrey and Danks.

McCaffrey and Danks have posed the challenge of discovering causal relations in data drawn from a mixture of distributions as an impossibility result in functional magnetic resonance (fMRI). We give an algorithm that addresses this problem for the distributions commonly assumed in fMRI studies and find that in testing, it can accurately separate data from mixed distributions. As with other obstacles to automated search, the problem of mixed distributions is not an impossible one, but rather a challenge. 1Introduction2Background3Addressing the Problem4Discussion.

Estimating feedforward and feedback effective connections from fMRI time series: Assessments of statistical methods.

We test the adequacies of several proposed and two new statistical methods for recovering the causal structure of systems with feedback from synthetic BOLD time series. We compare an adaptation of the first correct method for recovering cyclic linear systems; Granger causal regression; a multivariate autoregressive model with a permutation test; the Group Iterative Multiple Model Estimation (GIMME) algorithm; the Ramsey et al. non-Gaussian methods; two non-Gaussian methods by Hyvärinen and Smith; a method due to Patel et al.; and the GlobalMIT algorithm. We introduce and also compare two new methods, Fast Adjacency Skewness (FASK) and Two-Step, both of which exploit non-Gaussian features of the BOLD signal. We give theoretical justifications for the latter two algorithms. Our test models include feedback structures with and without direct feedback (2-cycles), excitatory and inhibitory feedback, models using experimentally determined structural connectivities of macaques, and empirical human resting-state and task data. We find that averaged over all of our simulations, including those with 2-cycles, several of these methods have a better than 80% orientation precision (i.e., the probability of a directed edge is in the true structure given that a procedure estimates it to be so) and the two new methods also have better than 80% recall (probability of recovering an orientation in the true structure).

A million variables and more: the Fast Greedy Equivalence Search algorithm for learning high-dimensional graphical causal models, with an application to functional magnetic resonance images.

We describe two modifications that parallelize and reorganize caching in the well-known Greedy Equivalence Search (GES) algorithm for discovering directed acyclic graphs on random variables from sample values. We apply one of these modifications, the Fast Greedy Search (FGS) assuming faithfulness, to an i.i.d. sample of 1,000 units to recover with high precision and good recall an average degree 2 directed acyclic graph (DAG) with one million Gaussian variables. We describe a modification of the algorithm to rapidly find the Markov Blanket of any variable in a high dimensional system. Using 51,000 voxels that parcellate an entire human cortex, we apply the FGS algorithm to Blood Oxygenation Level Dependent (BOLD) time series obtained from resting state fMRI.