Causal discovery beyond Markov equivalence

The focus of the dissertation is on learning causal diagrams beyond Markov equivalence. The baseline assumptions in causal structure learning are the acyclicity of the underlying structure and causal sufficiency, which requires that there are no unobserved confounder variables in the system. Under these assumptions, conditional independence relationships contain all the information in the distribution that can be used for structure learning. Therefore, the causal diagram can be identified only up to Markov equivalence, which is the set of structures reflecting the same conditional independence relationships. Hence, for many ground truth structures, the direction of a large portion of the edges will remain unidentified. Hence, in order to learn the structure beyond Markov equivalence, generating or having access to extra joint distributions from the perturbed causal system is required. There are two main scenarios for acquiring the extra joint distributions. The first and main scenario is when an experimenter is directly performing a sequence of interventions on subsets of the variables of the system to generate interventional distributions. We refer to the task of causal discovery from such interventional data as interventional causal structure learning. In this setting, the key question is determining which variables should be intervened on to gain the most information. This is the first focus of this dissertation. The second scenario for acquiring the extra joint distributions is when a subset of causal mechanisms, and consequently the joint distribution of the system, have varied or evolved due to reasons beyond the control of the experimenter. In this case, it is not even a priori known to the experimenter which causal mechanisms have varied. We refer to the task of causal discovery from such multi-domain data as multi-domain causal structure learning. In this setup the main question is how one can take the most advantage of the changes across domains for the task of causal discovery. This is the second focus of this dissertation. Next, we consider cases under which conditional independency may not reflect all the information in the distribution that can be used to identify the underlying structure. One such case is when cycles are allowed in the underlying structure. Unfortunately, a suitable characterization for equivalence for the case of cyclic directed graphs has been unknown so far. The third focus of this dissertation is on bridging the gap between cyclic and acyclic directed graphs by introducing a general approach for equivalence characterization and structure learning. Another case in which conditional independency may not reflect all the information in the distribution is when there are extra assumptions on the generating causal modules. A seminal result in this direction is that a linear model with non-Gaussian exogenous variables is uniquely identifiable. As the forth focus of this dissertation, we consider this setup, yet go one step further and allow for violation of causal sufficiency, and investigate how this generalization affects the identifiability

A study of covert queueing channels in shared schedulers

We study covert queueing channels (CQCs), which are a kind of covert timing channel that may be exploited in shared queues across supposedly isolated users. In our system model, a user modulates messages to another user via his pattern of access to the shared resource. One example of such a channel is the cross-virtual network covert channel in data center networks resulting from the queueing effects of the shared resource. First, we study a system comprising a transmitter and a receiver that share a deterministic and work-conserving first-come-first-served scheduler, and we compute the maximum reliable data transmission rate, i.e., the capacity, of this channel. Next, we extend the model to include a third user who also uses the shared resource and study the effect of the presence of this user on the information transmission rate. The solution approach presented in this extension may be applied to calculate the capacity of the covert queueing channel among any number of users. We also study a queueing covert channel between two users sharing a round robin scheduler. Such a covert channel can arise when users share a resource such as a computer processor or a router arbitrated by a round robin policy. We present an information-theoretic framework to model and derive the capacity of this channel for both noiseless and noisy scenarios. Our results show that seemingly isolated users can communicate at a high rate over the covert channel. Furthermore, we propose a practical finite-length code construction, which achieves the capacity limit

Counting and Sampling from Markov Equivalent DAGs Using Clique Trees

A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure--roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions

Identification and Estimation for Nonignorable Missing Data: A Data Fusion Approach

We consider the task of identifying and estimating a parameter of interest in settings where data is missing not at random (MNAR). In general, such parameters are not identified without strong assumptions on the missing data model. In this paper, we take an alternative approach and introduce a method inspired by data fusion, where information in an MNAR dataset is augmented by information in an auxiliary dataset subject to missingness at random (MAR). We show that even if the parameter of interest cannot be identified given either dataset alone, it can be identified given pooled data, under two complementary sets of assumptions. We derive an inverse probability weighted (IPW) estimator for identified parameters, and evaluate the performance of our estimation strategies via simulation studies.Comment: 21 pages, 4 figure

Partial Identification of Causal Effects Using Proxy Variables

Proximal causal inference is a recently proposed framework for evaluating the causal effect of a treatment on an outcome variable in the presence of unmeasured confounding (Miao et al., 2018a; Tchetgen Tchetgen et al., 2020). For nonparametric point identification, the framework leverages proxy variables of unobserved confounders, provided that such proxies are sufficiently relevant for the latter, a requirement that has previously been formalized as a completeness condition. Completeness is key to connecting the observed proxy data to hidden factors via a so-called confounding bridge function, identification of which is an important step towards proxy-based point identification of causal effects. However, completeness is well-known not to be empirically testable, therefore potentially restricting the application of the proximal causal framework. In this paper, we propose partial identification methods that do not require completeness and obviate the need for identification of a bridge function. That is, we establish that proxies of unobserved confounders can be leveraged to obtain bounds on the causal effect of the treatment on the outcome even if available information does not suffice to identify either a bridge function or a corresponding causal effect of interest. We further establish analogous partial identification results in related settings where identification hinges upon hidden mediators for which proxies are available, however such proxies are not sufficiently rich for point identification of a bridge function or a corresponding causal effect of interest