Feature importance metrics are critical for interpreting machine learning models and understanding the relevance of individual features. However, real-world data often exhibit missingness, thereby complicating how feature importance should be evaluated. We introduce the distinction between two evaluation frameworks under missing data: (1) feature importance under the full data, as if every feature had been fully measured, and (2) feature importance under the observed data, where missingness is governed by the current measurement policy. While the full data perspective offers insights into the data generating process, it often relies on unrealistic assumptions and cannot guide decisions when missingness persists at model deployment. Since neither framework directly informs improvements in data collection, we additionally introduce the feature measurement importance gradient (FMIG), a novel, model-agnostic metric that identifies features that should be measured more frequently to enhance predictive performance. Using synthetic data, we illustrate key differences between these metrics and the risks of conflating them.

Machine learning methods often assume that input features are available at no cost. However, in domains like healthcare, where acquiring features could be expensive or harmful, it is necessary to balance a feature's acquisition cost against its predictive value. The task of training an AI agent to decide which features to acquire is called active feature acquisition (AFA). By deploying an AFA agent, we effectively alter the acquisition strategy and trigger a distribution shift. To safely deploy AFA agents under this distribution shift, we present the problem of active feature acquisition performance evaluation (AFAPE). We examine AFAPE under i) a no direct effect (NDE) assumption, stating that acquisitions do not affect the underlying feature values; and ii) a no unobserved confounding (NUC) assumption, stating that retrospective feature acquisition decisions were only based on observed features. We show that one can apply missing data methods under the NDE assumption and offline reinforcement learning under the NUC assumption. When NUC and NDE hold, we propose a novel semi-offline reinforcement learning framework. This framework requires a weaker positivity assumption and introduces three new estimators: A direct method (DM), an inverse probability weighting (IPW), and a double reinforcement learning (DRL) estimator.

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 the 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 inverse probability weighted (IPW) estimators for identified parameters under both sets of assumptions, and evaluate the performance of our estimation strategies via simulation studies, and a data application.

Causal inference is made challenging by confounding, selection bias, and other complications. A common approach to addressing these difficulties is the inclusion of auxiliary data on the superpopulation of interest. Such data may measure a different set of variables, or be obtained under different experimental conditions than the primary dataset. Analysis based on multiple datasets must carefully account for similarities between datasets, while appropriately accounting for differences. In addition, selection of experimental units into different datasets may be systematic; similar difficulties are encountered in missing data problems. Existing methods for combining datasets either do not consider this issue, or assume simple selection mechanisms. In this paper, we provide a general approach, based on graphical causal models, for causal inference from data on the same superpopulation that is obtained under different experimental conditions. Our framework allows both arbitrary unobserved confounding, and arbitrary selection processes into different experimental regimes in our data. We describe how systematic selection processes may be organized into a hierarchy similar to censoring processes in missing data: selected completely at random (SCAR), selected at random (SAR), and selected not at random (SNAR). In addition, we provide a general identification algorithm for interventional distributions in this setting.

A common type of zero-inflated data has certain true values incorrectly replaced by zeros due to data recording conventions (rare outcomes assumed to be absent) or details of data recording equipment (e.g. artificial zeros in gene expression data). Existing methods for zero-inflated data either fit the observed data likelihood via parametric mixture models that explicitly represent excess zeros, or aim to replace excess zeros by imputed values. If the goal of the analysis relies on knowing true data realizations, a particular challenge with zero-inflated data is identifiability, since it is difficult to correctly determine which observed zeros are real and which are inflated. This paper views zero-inflated data as a general type of missing data problem, where the observability indicator for a potentially censored variable is itself unobserved whenever a zero is recorded. We show that, without additional assumptions, target parameters involving a zero-inflated variable are not identified. However, if a proxy of the missingness indicator is observed, a modification of the effect restoration approach of Kuroki and Pearl allows identification and estimation, given the proxy-indicator relationship is known. If this relationship is unknown, our approach yields a partial identification strategy for sensitivity analysis. Specifically, we show that only certain proxy-indicator relationships are compatible with the observed data distribution. We give an analytic bound for this relationship in cases with a categorical outcome, which is sharp in certain models. For more complex cases, sharp numerical bounds may be computed using methods in Duarte et al. [2023]. We illustrate our method via simulation studies and a data application on central line-associated bloodstream infections (CLABSIs).

Causality is a fundamental notion in science and engineering. It has attracted much interest across research communities in statistics, machine learning (ML), healthcare, and artificial intelligence (AI), and is becoming increasingly recognized as a vital research area. One of the fundamental problems in causality is how to find the causal structure or the underlying causal model. Accordingly, one focus of this Special Issue is oncausal discovery, i.e., how can we discover causal structure over a set of variables from observational data with automated procedures? Besides learning causality, another focus is on using causality to help understand and advance ML, that is, causality-inspired ML.

Graphical models such as Markov random fields (MRFs) that are associated with undirected graphs, and Bayesian networks (BNs) that are associated with directed acyclic graphs, have proven to be a very popular approach for reasoning under uncertainty, prediction problems and causal inference. Parametric MRF likelihoods are well-studied for Gaussian and categorical data. However, in more complicated parametric and semi-parametric settings, likelihoods specified via clique potential functions are generally not known to be congenial or non-redundant. Congenial and non-redundant DAG likelihoods are far simpler to specify in both parametric and semi-parametric settings by modeling Markov factors in the DAG factorization. However, DAG likelihoods specified in this way are not guaranteed to coincide in distinct DAGs within the same Markov equivalence class. This complicates likelihoods based model selection procedures for DAGs by “sneaking in” potentially unwarranted assumptions about edge orientations. In this paper we link a density function decomposition due to Chen with the clique factorization of MRFs described by Lauritzen to provide a general likelihood for MRF models. The proposed likelihood is composed of variationally independent, and non-redundant closed form functionals of the observed data distribution, and is sufficiently general to apply to arbitrary parametric and semi-parametric models. We use an extension of our developments to give a general likelihood for DAG models that is guaranteed to coincide for all members of a Markov equivalence class. Our results have direct applications for model selection and semi-parametric inference.

Unobserved confounding is a fundamental obstacle to establishing valid causal conclusions from observational data. Two complementary types of approaches have been developed to address this obstacle: obtaining identification using fortuitous external aids, such as instrumental variables or proxies, or by means of the ID algorithm, using Markov restrictions on the full data distribution encoded in graphical causal models. In this paper we aim to develop a synthesis of the former and latter approaches to identification in causal inference to yield the most general identification algorithm in multivariate systems currently known -- the proximal ID algorithm. In addition to being able to obtain nonparametric identification in all cases where the ID algorithm succeeds, our approach allows us to systematically exploit proxies to adjust for the presence of unobserved confounders that would have otherwise prevented identification. In addition, we outline a class of estimation strategies for causal parameters identified by our method in an important special case. We illustrate our approach by simulation studies and a data application.

Robins et al. (2008) introduced a class of influence functions (IFs) which could be used to obtain doubly robust moment functions for the corresponding parameters. However, that class does not include the IF of parameters for which the nuisance functions are solutions to integral equations. Such parameters are particularly important in the field of causal inference, specifically in the recently proposed proximal causal inference framework of Tchetgen Tchetgen et al. (2020), which allows for estimating the causal effect in the presence of latent confounders. In this paper, we first extend the class of Robins et al. to include doubly robust IFs in which the nuisance functions are solutions to integral equations. Then we demonstrate that the double robustness property of these IFs can be leveraged to construct estimating equations for the nuisance functions, which enables us to solve the integral equations without resorting to parametric models. We frame the estimation of the nuisance functions as a minimax optimization problem. We provide convergence rates for the nuisance functions and conditions required for asymptotic linearity of the estimator of the parameter of interest. The experiment results demonstrate that our proposed methodology leads to robust and high-performance estimators for average causal effect in the proximal causal inference framework.

We focus on causal discovery in the presence of measurement error in linear systems where the mixing matrix, i.e., the matrix indicating the independent exogenous noise terms pertaining to the observed variables, is identified up to permutation and scaling of the columns. We demonstrate a somewhat surprising connection between this problem and causal discovery in the presence of unobserved parentless causes, in the sense that there is a mapping, given by the mixing matrix, between the underlying models to be inferred in these problems. Consequently, any identifiability result based on the mixing matrix for one model translates to an identifiability result for the other model. We characterize to what extent the causal models can be identified under a two-part faithfulness assumption. Under only the first part of the assumption (corresponding to the conventional definition of faithfulness), the structure can be learned up to the causal ordering among an ordered grouping of the variables but not all the edges across the groups can be identified. We further show that if both parts of the faithfulness assumption are imposed, the structure can be learned up to a more refined ordered grouping. As a result of this refinement, for the latent variable model with unobserved parentless causes, the structure can be identified. Based on our theoretical results, we propose causal structure learning methods for both models, and evaluate their performance on synthetic data.

Cause-effect relationships are typically evaluated by comparing outcome responses to binary treatment values, representing two arms of a hypothetical randomized controlled trial. However, in certain applications, treatments of interest are continuous and multidimensional. For example, understanding the causal relationship between severity of radiation therapy, summarized by a multidimensional vector of radiation exposure values and post-treatment side effects is a problem of clinical interest in radiation oncology. An appropriate strategy for making interpretable causal conclusions is to reduce the dimension of treatment. If individual elements of a multidimensional treatment vector weakly affect the outcome, but the overall relationship between treatment and outcome is strong, careless approaches to dimension reduction may not preserve this relationship. Further, methods developed for regression problems do not directly transfer to causal inference due to confounding complications. In this paper, we use semiparametric inference theory for structural models to give a general approach to causal sufficient dimension reduction of a multidimensional treatment such that the cause-effect relationship between treatment and outcome is preserved. We illustrate the utility of our proposals through simulations and a real data application in radiation oncology.

Identification theory for causal effects in causal models associated with hidden variable directed acyclic graphs (DAGs) is well studied. However, the corresponding algorithms are underused due to the complexity of estimating the identifying functionals they output. In this work, we bridge the gap between identification and estimation of population-level causal effects involving a single treatment and a single outcome. We derive influence function based estimators that exhibit double robustness for the identified effects in a large class of hidden variable DAGs where the treatment satisfies a simple graphical criterion; this class includes models yielding the adjustment and front-door functionals as special cases. We also provide necessary and sufficient conditions under which the statistical model of a hidden variable DAG is nonparametrically saturated and implies no equality constraints on the observed data distribution. Further, we derive an important class of hidden variable DAGs that imply observed data distributions observationally equivalent (up to equality constraints) to fully observed DAGs. In these classes of DAGs, we derive estimators that achieve the semiparametric efficiency bounds for the target of interest where the treatment satisfies our graphical criterion. Finally, we provide a sound and complete identification algorithm that directly yields a weight based estimation strategy for any identifiable effect in hidden variable causal models.

The data drawn from biological, economic, and social systems are often confounded due to the presence of unmeasured variables. Prior work in causal discovery has focused on discrete search procedures for selecting acyclic directed mixed graphs (ADMGs), specifically ancestral ADMGs, that encode ordinary conditional independence constraints among the observed variables of the system. However, confounded systems also exhibit more general equality restrictions that cannot be represented via these graphs, placing a limit on the kinds of structures that can be learned using ancestral ADMGs. In this work, we derive differentiable algebraic constraints that fully characterize the space of ancestral ADMGs, as well as more general classes of ADMGs, arid ADMGs and bow-free ADMGs, that capture all equality restrictions on the observed variables. We use these constraints to cast causal discovery as a continuous optimization problem and design differentiable procedures to find the best fitting ADMG when the data comes from a confounded linear system of equations with correlated errors. We demonstrate the efficacy of our method through simulations and application to a protein expression dataset. Code implementing our methods is open-source and publicly available at https://gitlab.com/rbhatta8/dcd and will be incorporated into the Ananke package.

When data contains measurement errors, it is necessary to make modeling assumptions relating the error-prone measurements to the unobserved true values. Work on measurement error has largely focused on models that fully identify the parameter of interest. As a result, many practically useful models that result in bounds on the target parameter – known as partial identification – have been neglected. In this work, we present a method for partial identification in a class of measurement error models involving discrete variables. We focus on models that impose linear constraints on the target parameter, allowing us to compute partial identification bounds using off-the-shelf LP solvers. We show how several common measurement error assumptions can be composed with an extended class of instrumental variable-type models to create such linear constraint sets. We further show how this approach can be used to bound causal parameters, such as the average treatment effect, when treatment or outcome variables are measured with error. Using data from the Oregon Health Insurance Experiment, we apply this method to estimate bounds on the effect Medicaid enrollment has on depression when depression is measured with error.

Directed acyclic graphs (DAGs) with hidde variables are often used to characterize causal relations between variables in a system. When some variables are unobserved, DAGs imply a notoriously complicated set of constraints on the distribution of observed variables. In this work, we present entropic inequality constraints that are implied by e-separation relations in hidden variable DAGs with discrete observed variables. The constraints can intuitively be understood to follow from the fact that the capacity of variables along a causal pathway to convey information is restricted by their entropy; e.g. at the extreme case, a variable with entropy 0 can convey no information. We show how these constraints can be used to learn about the true causal model from an observed data distribution. In addition, we propose a measure of causal influence called the minimal mediary entropy, and demonstrate that it can concisely augment traditional measures such as the average treatment effect.

Causal analyses of longitudinal data generally assume that the qualitative causal structure relating variables remains invariant over time. In structured systems that transition between qualitatively different states in discrete time steps, such an approach is deficient on two fronts. First, time-varying variables may have state-specific causal relationships that need to be captured. Second, an intervention can result in state transitions downstream of the intervention different from those actually observed in the data. In other words, interventions may counterfactually alter the subsequent temporal evolution of the system.We introduce a generalization of causal graphical models, Path Dependent Structural Equation Models (PDSEMs), that can describe such systems. We show how causal inference may be performed in such models and illustrate its use in simulations and data obtained from a septoplasty surgical procedure.

In many applied fields, researchers are ofteninterested in tailoring treatments to unit-levelcharacteristics in order to optimize an outcomeof interest. Methods for identifying andestimating treatment policies are the subjectof the dynamic treatment regime literature. Separately, in many settings the assumptionthat data are independent and identically distributeddoes not hold due to inter-subjectdependence. The phenomenon where a subject’s outcome is dependent on his neighbor’s exposure is known as interference. These areasintersect in myriad real-world settings. Inthis paper we consider the problem of identifyingoptimal treatment policies in the presenceof interference. Using a general representationof interference, via Lauritzen-Wermuth-Freydenburg chain graphs (Lauritzen andRichardson, 2002), we formalize a variety ofpolicy interventions under interference andextend existing identification theory (Tian,2008; Sherman and Shpitser, 2018). Finally, we illustrate the efficacy of policy maximization under interference in a simulation study.

Missing data has the potential to affect analyses conducted in all fields of scientific study including healthcare, economics, and the social sciences. Several approaches to unbiased inference in the presence of non-ignorable missingness rely on the specification of the target distribution and its missingness process as a probability distribution that factorizes with respect to a directed acyclic graph. In this paper, we address the longstanding question of the characterization of models that are identifiable within this class of missing data distributions. We provide the first completeness result in this field of study – necessary and sufficient graphical conditions under which, the full data distribution can be recovered from the observed data distribution. We then simultaneously address issues that may arise due to the presence of both missing data and unmeasured confounding, by extending these graphical conditions and proofs of completeness, to settings where some variables are not just missing, but completely unobserved.

Causal parameters may not be point identified in the presence of unobserved confounding. However, information about non-identified parameters, in the form of bounds, may still be recovered from the observed data in some cases. We develop a new general method for obtaining bounds on causal parameters using rules of probability and restrictions on counterfactuals implied by causal graphical models. We additionally provide inequality constraints on functionals of the observed data law implied by such causal models. Our approach is motivated by the observation that logical relations between identified and non-identified counterfactual events often yield information about non-identified events. We show that this approach is powerful enough to recover known sharp bounds and tight inequality constraints, and to derive novel bounds and constraints.

Causal inference quantifies cause effect relationships by means of counterfactual responses had some variable been artificially set to a constant. A more refined notion of manipulation, where a variable is artificially set to a fixed function of its natural value is also of interest in particular domains. Examples include increases in financial aid, changes in drug dosing, and modifying length of stay in a hospital.We define counterfactual responses to manipulations of this type, which we call shift interventions. We show that in the presence of multiple variables being manipulated, two types of shift interventions are possible. Shift interventions on the treated (SITs) are defined with respect to natural values, and are connected to effects of treatment on the treated. Shift interventions as policies (SIPs) are defined recursively with respect to values of responses to prior shift interventions, and are connected to dynamic treatment regimes. We give sound and complete identification algorithms for both types of shift interventions, and derive efficient semi-parametric estimators for the mean response to a shift intervention in a special case motivated by a healthcare problem. Finally, we demonstrate the utility of our method by using an electronic health record dataset to estimate the effect of extending the length of stay in the intensive care unit (ICU) in a hospital by an extra day on patient ICU readmission probability.

The do-calculus is a well-known deductive system for deriving connections between interventional and observed distributions, and has been proven complete for a number of important identifiability problems in causal inference. Nevertheless, as it is currently defined, the do-calculus is inapplicable to causal problems that involve complex nested counterfactuals which cannot be expressed in terms of the "do" operator. Such problems include analyses of path-specific effects and dynamic treatment regimes. In this paper we present the potential outcome calculus (po-calculus), a natural generalization of do-calculus for arbitrary potential outcomes. We thereby provide a bridge between identification approaches which have their origins in artificial intelligence and statistics, respectively. We use po-calculus to give a complete identification algorithm for conditional path-specific effects with applications to problems in mediation analysis and algorithmic fairness.

Systematic discriminatory biases present in our society influence the way data is collected and stored, the way variables are defined, and the way scientific findings are put into practice as policy. Automated decision procedures and learning algorithms applied to such data may serve to perpetuate existing injustice or unfairness in our society. In this paper, we consider how to make optimal but fair decisions, which “break the cycle of injustice” by correcting for the unfair dependence of both decisions and outcomes on sensitive features (e.g., variables that correspond to gender, race, disability, or other protected attributes). We use methods from causal inference and constrained optimization to learn optimal policies in a way that addresses multiple potential biases which afflict data analysis in sensitive contexts, extending the approach of Nabi & Shpitser (2018). Our proposal comes equipped with the theoretical guarantee that the chosen fair policy will induce a joint distribution for new instances that satisfies given fairness constraints. We illustrate our approach with both synthetic data and real criminal justice data.

In this paper, we consider the problem of fair statistical inference involving outcome variables. Examples include classification and regression problems, and estimating treatment effects in randomized trials or observational data. The issue of fairness arises in such problems where some covariates or treatments are "sensitive," in the sense of having potential of creating discrimination. In this paper, we argue that the presence of discrimination can be formalized in a sensible way as the presence of an effect of a sensitive covariate on the outcome along certain causal pathways, a view which generalizes (Pearl 2009). A fair outcome model can then be learned by solving a constrained optimization problem. We discuss a number of complications that arise in classical statistical inference due to this view and provide workarounds based on recent work in causal and semi-parametric inference.

Causal understanding is essential for many kinds of decision-making, but causal inference from observational data has typically only been applied to structured, low-dimensional datasets. While text classifiers produce low-dimensional outputs, their use in causal inference has not previously been studied. To facilitate causal analyses based on language data, we consider the role that text classifiers can play in causal inference through established modeling mechanisms from the causality literature on missing data and measurement error. We demonstrate how to conduct causal analyses using text classifiers on simulated and Yelp data, and discuss the opportunities and challenges of future work that uses text data in causal inference.

The assumption that data samples are independent and identically distributed (iid) is standard in many areas of statistics and machine learning. Nevertheless, in some settings, such as social networks, infectious disease modeling, and reasoning with spatial and temporal data, this assumption is false. An extensive literature exists on making causal inferences under the iid assumption [12, 8, 21, 16], but, as pointed out in [14], causal inference in non-iid contexts is challenging due to the combination of unobserved confounding bias and data dependence. In this paper we develop a general theory describing when causal inferences are possible in such scenarios. We use segregated graphs [15], a generalization of latent projection mixed graphs [23], to represent causal models of this type and provide a complete algorithm for non-parametric identification in these models. We then demonstrate how statistical inferences may be performed on causal parameters identified by this algorithm, even in cases where parts of the model exhibit full interference, meaning only a single sample is available for parts of the model [19]. We apply these techniques to a synthetic data set which considers the adoption of fake news articles given the social network structure, articles read by each person, and baseline demographics and socioeconomic covariates.

Missing records are a perennial problem in analysis of complex data of all types, when the target of inference is some function of the full data law. In simple cases, where data is missing at random or completely at random (Rubin, 1976), well-known adjustments exist that result in consistent estimators of target quantities. Assumptions underlying these estimators are generally not realistic in practical missing data problems. Unfortunately, consistent estimators in more complex cases where data is missing not at random, and where no ordering on variables induces monotonicity of missingness status are not known in general, with some notable exceptions (Robins, 1997), (Tchetgen Tchetgen et al, 2016), (Sadinle and Reiter, 2016). In this paper, we propose a general class of consistent estimators for cases where data is missing not at random, and missingness status is non-monotonic. Our estimators, which are generalized inverse probability weighting estimators, make no assumptions on the underlying full data law, but instead place independence restrictions, and certain other fairly mild assumptions, on the distribution of missingness status conditional on the data. The assumptions we place on the distribution of missingness status conditional on the data can be viewed as a version of a conditional Markov random field (MRF) corresponding to a chain graph. Assumptions embedded in our model permit identification from the observed data law, and admit a natural fitting procedure based on the pseudo likelihood approach of (Besag, 1975). We illustrate our approach with a simple simulation study, and an analysis of risk of premature birth in women in Botswana exposed to highly active anti-retroviral therapy.

Bayesian networks are a popular representation of asymmetric (for example causal) relationships between random variables. Markov random fields (MRFs) are a complementary model of symmetric relationships used in computer vision, spatial modeling, and social and gene expression networks. A chain graph model under the Lauritzen-Wermuth-Frydenberg interpretation (hereafter a chain graph model) generalizes both Bayesian networks and MRFs, and can represent asymmetric and symmetric relationships together.As in other graphical models, the set of marginals from distributions in a chain graph model induced by the presence of hidden variables forms a complex model. One recent approach to the study of marginal graphical models is to consider a well-behaved supermodel. Such a supermodel of marginals of Bayesian networks, defined only by conditional independences, and termed the ordinary Markov model, was studied at length in (Evans and Richardson, 2014).In this paper, we show that special mixed graphs which we call segregated graphs can be associated, via a Markov property, with supermodels of a marginal of chain graphs defined only by conditional independences. Special features of segregated graphs imply the existence of a very natural factorization for these supermodels, and imply many existing results on the chain graph model, and ordinary Markov model carry over. Our results suggest that segregated graphs define an analogue of the ordinary Markov model for marginals of chain graph models.

There have been many efforts to identify causal graphical features such as directed edges between random variables from observational data. Recently, Tian et al. proposed a new dynamic programming algorithm which computes marginalized posterior probabilities of directed edge features over all the possible structures in O(n3n) time when the number of parents per node is bounded by a constant, where n is the number of variables of interest. However the main drawback of this approach is that deciding a single appropriate threshold for the existence of the directed edge feature is difficult due to the scale difference of the posterior probabilities between the directed edges forming v-structures and the directed edges not forming v-structures. We claim that computing posterior probabilities of both adjacencies and v-structures is necessary and more effective for discovering causal graphical features, since it allows us to find a single appropriate decision threshold for the existence of the feature that we are testing. For efficient computation, we provide a novel dynamic programming algorithm which computes the posterior probabilities of all of n(n – 1)/2 adjacency and n(n–1 choose 2) v-structure features in O(n3 * 3n) time.

Identity uncertainty is a pervasive problem in real-world data analysis. It arises whenever objects are not labeled with unique identifiers or when those identifiers may not be perceived perfectly. In such cases, two ob- servations may or may not correspond to the same object. In this paper, we consider the problem in the context of citation matching—the prob- lem of deciding which citations correspond to the same publication. Our approach is based on the use of a relational probability model to define a generative model for the domain, including models of author and title corruption and a probabilistic citation grammar. Identity uncertainty is handled by extending standard models to incorporate probabilities over the possible mappings between terms in the language and objects in the domain. Inference is based on Markov chain Monte Carlo, augmented with specific methods for generating efficient proposals when the domain contains many objects. Results on several citation data sets show that the method outperforms current algorithms for citation matching. The declarative, relational nature of the model also means that our algorithm can determine object characteristics such as author names by combining multiple citations of multiple papers.