We describe a method “Kg2Causal” for using a large-scale, general-purpose biomedical knowledge graph as a prior for data-driven causal network structure learning. Given a set of observed nodes in a dataset, and some relationship edges between the nodes derived from a knowledge graph, Kg2Causal uses the knowledge graph-derived edges to guide the data-driven inference of a causal Bayesian network. We tested Kg2Causal on several real-world biological datasets with known ground-truth networks and demonstrate improvement in network learning accuracy, relative to a baseline of an uninformative network structure prior. We also demonstrate the application of our method if data are collected under diferent experimental conditions including interventions on the observed variables.
where structured prior knowledge is available. Without expert knowledge, standard network
inference approaches, by default, assume uniform (uninformative) prior which can lead to
erroneous relationships or relationship orientations both due to (i) size of space of networks and
(ii) degeneracy of Markov-equivalent networks. Proper incorporation of informative priors can
enhance model eficiency [
For most applications of causal modeling, some prior knowledge is available. For example, in
medicine, most cases have prior knowledge about etiology, symptoms, and treatment of
underlying diseases or conditions which can be obtained from biomedical literature or
knowledgebases. Although there is in general large scale availability of structured prior knowledge (for
example, ontologies) in various scientific domains, these mostly comprise disparate
information sources in various standards and formats, which poses a challenge to integrate them into
single structure. These problems motivated building of large multi-graphs called knowledge
graphs (KG) [
In this work, we propose a method, “Kg2Causal”, for extracting relations as pairs of nodes from a knowledge graph, and for incorporating them as priors on corresponding edges in a score-based, data-driven causal network learning method. In this study, prior edges from knowledge graph are ac1https://github.com/RTXteam/RTX/code/reasoningtool/kg-construction that Kg2Causal had superior network learning accuracy to methods that do not use general knowledge-base as network structure prior. Finally, we demonstrate (Sec. 4.3) the application of Kg2Causal if data are collected under diferent experimental conditions including interventions on the observed variables. We implemented “Kg2Causal” in the R programming language (leveraging the bnLearn package [17]) and provide the code as open-source software 2.
In this section, we describe Kg2Causal’s conceptual foundations including CBNs, score-based causal modeling, interventions, and knowledge graph-based priors in network learning.
distribution over variables
as long as it satisfies two main assumptions:
set of variables (nodes) and ⊂
A causal Bayesian Network [
given the states of its parents Pa( ). A causal network represents a joint a) Causal Markov: Any given variable is independent of its non-descendants, conditioned on all of its direct causes. The assumption implies that the joint distribution ( ) can be factored as: ( ) = ∏ =1 ( ∣ Pa( )).
pendence relation in is entailed by the causal Markov assumption applied to [18].
b) Faithfulness: The joint distribution ( 1, … , ) is faithful to if every conditional inde
Let us assume, we have a dataset having observations over set of variables. One of the main
classes of causal learning approaches is the Score-based approach which is derived from classic
Bayesian method where a scoring function evaluates the fit of graph to data [
∣ ) ∝ ( ) ( ∣ ), where ( ) is prior distribution over space of . As described in Sec. 3, the Kg2Causal method incorporates a score-based approach. all possible DAGs reflecting prior knowledge and ( ∣ ) is marginal likelihood of the data learned
Interventions—external manipulations of nodes (“targets”) in a network—are important to detect causal relations that can help disambiguate Markov equivalent sub-networks [16]. Let represent set of target nodes that are altered in interventional experiment and = \ be 2https://github.com/meghasin/Kg2Causal the complementary set of observational variables. Each intervention can have one or more targets whose conditional probabilities are changed (so that, conditioned on intervention, target variable’s distribution may depend only on a (possibly empty) subset of its parent observables). Hence, each intervention results in deletion of arrows pointing towards the intervened nodes. The joint distribution of after intervention is ( 1, ..., ) = ( ))⋅∏ ∈
′ ( ∣ that
is not a target node, and ′( ∣
given its new set of parents
′
( )
( )), where ( ∣ ( )) is conditional probability similar to
′
( )) is post-intervention conditional probability of
. For a so-called “perfect” intervention, one would set
∏ ∈
( ∣
, given
′
( ) = ∅ [
In this subsection we introduce three types of uninformative priors on the network structure ( ), uniform prior, marginal prior, and Bayesian variable selection prior (VSP). We then describe the knowledge graph-based prior that we use in Kg2Causal method. In cases for lack of prior knowledge, default choice for prior ( ) is a uniform prior distribution, as follows: = ( ∪ { , } ∣ ) ( ∪ { , }) ( ∣ ∪ { , }) ( ∣ ) ( ) ( ∣ ) 1/3, since we know that ( ⇒V ) + ( ⇐ ) + ( ⇎ ) = 1 where nodes and can have three possible cases ⇒V (representing ( , ) ∈ ),V ⇐V So the probability for these edges are assigned as ( ⇒V ) = ( ⇐V ) = ( (representing ( , ) ∈ E) or ⇎V (no arc) and each have equal probability of occurrence. . This implies ⇎ ) = ( ⇒ ) + ( favouring the propagation of false positives in . Hence, its not always a good idea to use uniform prior specially for cases where data is not too supportive of the DAG learned and ⇐ ) = 2/3, which means a higher promotion for the inclusion of new arcs and where is large. A better version of uniform prior is to use marginal probabilities instead, where an independent prior can be assumed for each arc with same independent marginal probabilities as uniform priors, also called marginal uniform [20]. In this case, the probability of inclusion of each edge is assigned as ( ⇒ ) = ( ⇐ ) = 1/4 and ( ⇎ ) = 1/2.
Compared to the uniform prior, the marginal uniform prior is less prone to false-positive edges in the posterior-probability-maximizing graph. The Bayesian variable selection prior (VSP) assigns a probability of inclusion of possible parent nodes, with the default being 1/ .
The heart of Kg2Causal is the use of an informative prior based on a general-purpose
knowledge graph; for this purpose we use an edge decomposition technique described by Castelo
and Siebes [
general-purpose knowledge graph, we assign a prior probability ( ⇒V exists in the = 1/2) on those edges, ⇎V , since the later two are al ternate edges that have no corresponding edge in the general knowledge graph we use the uniform probability distribution as shown in Fig. 2. In this way we can create a complete prior probability (from partial knowledge) over the network ; on log scale, we define ( ) as log ( ) = ∑ ⇔ ∈, ≠ log ( ⇔ ) + ∑ ⋯ ∈, ≠ log ( ⇎ ) .
A “knowledge graph” [
We developed KG2Causal to leverage a general-purpose biomedical knowledge graph (see Sec. 2.5) in order to improve context-specific, data-driven network learning from multivariate observations; such observations could consist of gene expression measurements, proteomics measurements, or electronic health records. The key ideas of our approach are (i) mapping each variable in the dataset to a node in the knowledge graph, and querying relationships between them; (ii) extracting a subgraph containing the connected variables with edges between them; and (iii) use this edge set as our prior knowledge to guide the optimizing scoring step for inferring causal network. Mathematically, given a dataset , with a set of observable variables and given a general-purpose prior knowledge graph Γ as a multigraph, we want to learn a causal graph = ( , ) that approximately maximizes the posterior probability, i.e., argmax ( ( ∣ , Γ)), given a prior ( ∣ Γ). As a comparison, we used three uninformative prior distributions, namely uniform, marginal and Bayesian variable selection priors with each dataset in order to understand whether or not—and to what extent—using an informative network prior improves accuracy of causal network learning in a biomedical context. The Kg2Causal network discovery workflow, illustrated in Figure 3, consists of the following steps: • Map variables to nodes in Γ, and extract a list of edges from Γ among the nodes (collapsing same-direction multiedges to single edges).
3https://github.com/RTXteam/RTX/code/reasoningtool/kg-construction • Generate 100 random DAGs with nodes . We empirically determined, based on our previous study [21], that this number is adequate for the medium-to-large datasets 4. • In the score function, we include edge probability contributions from the prior knowledge graph (we assign probability 0.5 for every edge in ). For each DAG, we used the stochastic algorithm Tabu [14] to find a DAG that maximizes standard Bayesian Dirichlet equivalent uniform scoring function (BDeu) [15, 16]. • The previous step yields 100 optimized networks. Using these we compute the probability of each possible directed edge as its empirical frequency of occurrence among the DAGs. For example, if an edge ( , ) appears in 80 out of 100 optimized DAGs, we assign it an empirical probability of 0.80. We store the edge probabilities in a list. • We threshold the edge probabilities in order to obtain the set of edges for . Based on empirical studies, we chose a threshold of 0.85.
We chose Tabu for its robustness, simplicity (uses few parameters) and history-dependent (“memory”), although Kg2Causal is in principle compatible with any optimizing method.
In the case where the dataset is purely observational (i.e., no interventions) from a single experiment, Kg2Causal can be implemented algorithmically as described above; we provide a pseudocode description of the “observational” formulation of Kg2Causal in Algorithm 1.
With causal network learning based on a single observational dataset, it is dificult to diferentiate between compatible Markov equivalent models [22]. In the simple case of three variables , and , there are three possible causal models ⇒ ⇒ , ⇐ ⇐ , and ⇐ ⇒ ; all three structures are Markov equivalent. This ambiguity can be resolved by incorporating measurements from interventional experiments, causing the Markov equivalent structures to have diferent likelihoods. However, in real-world settings, it is dificult to obtain such interventional measurements as compared to observational measurement [23]. Even when interventional datasets are available, learning a causal network from mixed observational and interventional data is challenging, for two reasons: (i) datasets collected from diferent experiments under diferent environmental conditions or batches are not identically distributed, in which case their underlying causal structures may difer leading to errors if network inference is applied to the combined set of measurements; and (ii) in real-world settings interventions are not “perfect” but rather “uncertain” (i.e., “imperfect” or “fat-hand”), meaning that the interventions have other unknown targets, which if ignored would likely yield spurious interactions in network discovery. To deal with such cases, based on our previous study demonstrating the efectiveness of the Learn and Vote algorithm [21, 24], we extended Kg2Causal to include learning from a multi-experiment dataset using a voting-based integration method where experiment-specific causal networks are learned and combined by weighted averaging into a consensus causal network. The additional steps in Algorithm 2 are as follows: 1. Let there be experiments (can be observational and/or interventional) that produced datasets with observed variables as ( ) and known intervention targets as , if any. 2. Repeat steps 1-4 (from Sec. 3) for all experiments. 3. From the arc-weight lists, average arc strengths and directions over all the experiments in which the given arc is not intervened.
4. Per our earlier work [21], we used a threshold of 0.5 for the average arc probability.
In this section, we describe the observational datasets and ground-truth networks (Sec. 4.1) and the simulated mixed interventional-observational datasets (Sec. 4.2) that we analyzed. We present (Sec. 4.3) the results of empirical studies of network learning performance of Kg2Causal on these datasets in comparison to other types of network structure priors.
To assess performance of Kg2Causal on biological network inference problems, we empirically analyzed five real world datasets for which published ground-truth networks were available:
Hepatic encephalopathy: This is a clinical study about a serious liver complication called hepatic encephalopathy (HE) [25] with conditions like electrolyte disorders, infections, poor spirits. It is a categorical dataset with eight nodes and ground-truth containing ten edges.
Sachs et al. T cell signaling: This is a study on mixed observational and interventional experiments to infer causal connections between eleven protein and phospholipids in the intracellular signaling network of individual human CD4+ T-cells [26]. The dataset contains measurement of gene expressions with ground truth network containing twenty edges.
Hematopoietic Stem Cell Diferentiation (HSC): This is a real-world gene regulatory network to study underlying myeloid diferentiation from multipotent myeloid progenitors to megakaryocytes, erythrocytes, granulocytes and monocytes [27] in mammals [28]. The dataset contains measurement of gene expressions with ground-truth network having thirty edges.
Gonadal Sex Determination (GSD): This a real-world model which represents the gonadal diferentiation circuit which monitors the transformation of the bipotential gonadal primordium (BGP) into either female or male gonads [29]. The network consists of eighteen genes and one node for the urogenital ridge. The dataset contains measurement of gene expressions with ground-truth network containing seventy nine edges [28].
Yeast cell cycle: This is a dataset derived from a network model of thirty genes participating in cell-cycle regulation of yeast [30]. The dataset was created by integrating gene expression data with transitive protein-protein interaction. The ground-truth network has 317 edges.
We tested Kg2Causal using Sachs et al. interventional dataset and simulated observational and interventional measurement data from synthetic networks using the bnlearn package. For observational data, we drew random samples and for interventional data, we set some target nodes in the network to fixed values in order to create mutilated networks [31] before drawing samples from them. To simulate an uncertain intervention (or “fat-hand”) [32] we intervened one or more of child nodes of the intervention’s target node.
Cancer: This is a synthetic network [33] on causes and consequences of lung cancer. We simulated data from one observational and one interventional experiment with equal number of samples (500) from each experiment to avoid bias. For interventional experiment we generated a mutilated network: cancer_mut with one intervention (node Smoker).
Asia: This is a synthetic network [34], about occurrence of lung disease and their epidemiological connection a prior visit to Asia. We simulated one observational and two interventional ROC curve
PR curve
Accuracy HE Sachs HSC GSD
Yeast experiment from the synthetic network with equal number of samples (500) from each experiment to avoid bias. For the interventional experiments we conducted experiments to generate two mutilated networks: asia_mut1 with one intervention (node “Lung Cancer”) and asia_mut2 with two intervention (at nodes “Lung Cancer” and “Tuberculosis”).
In this section we present results of empirical studies of network learning performance on the ifve observational datasets (see Sec. 4.1) and three mixed observational-interventional datasets (see Sec. 4.2), for Kg2Causal in comparison with three other types of network structure priors. To quantify the performance, we considered presence of an edge in the ground truth network as a “true positive” and absence of an edge as a “true negative” causal arc. For the observational datasets, we used Algorithm 1 with indicated prior (KG, uniform, marginal, or Bayesian VSP) as described in Sec. 3. For mixed interventional-observational datasets, we used Algorithm 2 with the indicated prior. For each dataset, we found (Fig. 5) that using general knowledge graph as prior improves performance, by ROC, precision/recall, F1, and accuracy. Quantitatively, Kg2Causal had higher area under ROC curve (AUROC) and area under precision-recall curve (AUPR) scores than network learning with three non-KG priors tested, for the five observational (Table 1) and three mixed interventional-observational (Table 2) datasets. Moreover, the results of comparative analysis of Kg2Causal performance on mixed datasets (Table 2) show efect of pooling data from diferent experiments (Algorithm 1) as compared to voting (Algorithm 2) for such cases: pooling is better for small network (Cancer) (consistent with our previous findings [21]), whereas voting is better for medium-sized networks (Asia and Sachs).
A limitation of this study is that due to lack of availability large ground-truth causal networks,
all datasets analyzed in this work are for small to medium sized networks (8-30 nodes); due to
scalability issue of score-based methods, Kg2Causal method as described here would be
challenging to apply to larger networks (many hundreds to thousands of nodes and beyond), which
is an area of future work. Further, we plan to explore ways to incorporate a network structure
prior in constraint based algorithms (for example, PC algorithm [
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