The deconfounder was proposed as a method for estimating causal parameters in a context with multiple causes and unobserved confounding. It is based on recovery of a latent variable from the observed causes. We disentangle the causal interpretation from the statistical estimation problem and show that the deconfounder in general estimates adjusted regression target parameters. It does so by outcome regression adjusted for the recovered latent variable termed the substitute. We refer to the general algorithm, stripped of causal assumptions, as substitute adjustment. We give theoretical results to support that substitute adjustment estimates adjusted regression parameters when the regressors are conditionally independent given the latent variable. We also introduce a variant of our substitute adjustment algorithm that estimates an assumption-lean target parameter with minimal model assumptions. We then give finite sample bounds and asymptotic results supporting substitute adjustment estimation in the case where the latent variable takes values in a finite set. A simulation study illustrates finite sample properties of substitute adjustment and shows that it can be a viable method for adjusted regression when the recovery error is small. Most importantly, we present clear assumptions about the data generating distribution that allow us to control the recovery error and ultimately the estimation error that can be attributed to the use of substitutes.

Constraint-based causal structure learning for point processes require empirical tests of local independence. Existing tests require strong model assumptions, e.g., that the true data generating model is a Hawkes process with no latent confounders. Even when restricting attention to Hawkes processes, latent confounders are a major technical difficulty because a marginalized process will generally not be a Hawkes process itself. We introduce an expansion similar to Volterra expansions as a tool to represent marginalized intensities. Our main theoretical result is that such expansions can approximate the true marginalized intensity arbitrarily well. Based on this, we propose a test of local independence and investigate its properties in real and simulated data.

In causal discovery, linear non-Gaussian acyclic models (LiNGAMs) have been studied extensively. While the causally sufficient case is well understood, in many real problems the observed variables are not causally related. Rather, they are generated by latent variables, such as confounders and mediators, which may themselves be causally related. Existing results on the identification of the causal structure among the latent variables often require very strong graphical assumptions. In this paper, we consider partially observed linear models with either non-Gaussian or heterogeneous errors. In that case we give two graphical conditions which are necessary for identification of the causal structure. These conditions are closely related to sparsity of the causal edges. Together with one additional condition on the coefficients, which holds generically for any graph, the two graphical conditions are also sufficient for identifiability. These new conditions can be satisfied even when there is a large number of latent variables. We demonstrate the validity of our results on synthetic data.

The partial copula provides a method for describing the dependence between two random variables $X$ and $Y$ conditional on a third random vector $Z$ in terms of nonparametric residuals $U_1$ and $U_2$. This paper develops a nonparametric test for conditional independence by combining the partial copula with a quantile regression based method for estimating the nonparametric residuals. We consider a test statistic based on generalized correlation between $U_1$ and $U_2$ and derive its large sample properties under consistency assumptions on the quantile regression procedure. We demonstrate through a simulation study that the resulting test is sound under complicated data generating distributions. Moreover, in the examples considered the test is competitive to other state-of-the-art conditional independence tests in terms of level and power, and it has superior power in cases with conditional variance heterogeneity of $X$ and $Y$ given $Z$.

The linear Lyapunov equation of a covariance matrix parametrizes theequilibrium covariance matrix of a stochastic process. This parametrization canbe interpreted as a new graphical model class, and we show how the model classbehaves under marginalization and introduce a method for structure learning via$\ell_1$-penalized loss minimization. Our proposed method is demonstrated tooutperform alternative structure learning algorithms in a simulation study, andwe illustrate its application for protein phosphorylation network reconstruction.

MOTIVATION: Contamination of a cancer tissue by the surrounding benign (non-cancerous) tissue is a concern for molecular cancer diagnostics. This is because an observed molecular signature will be distorted by the surrounding benign tissue, possibly leading to an incorrect diagnosis. One example is molecular identification of the primary tumor site of metastases because biopsies of metastases typically contain a significant amount of benign tissue. RESULTS: A model of tissue contamination is presented. This contamination model works independently of the training of a molecular predictor, and it can be combined with any predictor model. The usability of the model is illustrated on primary tumor site identification of liver biopsies, specifically, on a human dataset consisting of microRNA expression measurements of primary tumor samples, benign liver samples and liver metastases. For a predictor trained on primary tumor and benign liver samples, the contamination model decreased the test error on biopsies from liver metastases from 77 to 45%. A further reduction to 34% was obtained by including biopsies in the training data. AVAILABILITY AND IMPLEMENTATION: http://www.math.ku.dk/∼richard/msgl/. CONTACT: [email protected] SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

BACKGROUND: A central question in molecular biology is how transcriptional regulatory elements (TREs) act in combination. Recent high-throughput data provide us with the location of multiple regulatory regions for multiple regulators, and thus with the possibility of analyzing the multivariate distribution of the occurrences of these TREs along the genome. RESULTS: We present a model of TRE occurrences known as the Hawkes process. We illustrate the use of this model by analyzing two different publically available data sets. We are able to model, in detail, how the occurrence of one TRE is affected by the occurrences of others, and we can test a range of natural hypotheses about the dependencies among the TRE occurrences. In contrast to earlier efforts, pre-processing steps such as clustering or binning are not needed, and we thus retain information about the dependencies among the TREs that is otherwise lost. For each of the two data sets we provide two results: first, a qualitative description of the dependencies among the occurrences of the TREs, and second, quantitative results on the favored or avoided distances between the different TREs. CONCLUSIONS: The Hawkes process is a novel way of modeling the joint occurrences of multiple TREs along the genome that is capable of providing new insights into dependencies among elements involved in transcriptional regulation. The method is available as an R package from http://www.math.ku.dk/~richard/ppstat/.