Title: Bayesian Active Causal Discovery with Multi-Fidelity Experiments
Venue: NeurIPS 2023
Accepted: yes
Keywords: Causal Discovery

Active Learning

Multi-fidelity
Abstract: This paper studies the problem of active causal discovery when the experiments can be done based on multi-fidelity oracles, where higher fidelity experiments are more precise and expensive, while the lower ones are cheaper but less accurate. In this paper, we formally define the task of multi-fidelity active causal discovery, and design a probabilistic model for solving this problem. In specific, we first introduce a mutual-information based acquisition function to determine which variable should be intervened at which fidelity, and then a cascading model is proposed to capture the correlations between different fidelity oracles. Beyond the above basic framework, we also extend it to the batch intervention scenario. We find that the theoretical foundations behind the widely used and efficient greedy method do not hold in our problem. To solve this problem, we introduce a new concept called $\epsilon$-submodular, and design a constraint based fidelity model to theoretically validate the greedy method. We conduct extensive experiments to demonstrate the effectiveness of our model.

Bayesian Active Causal Discovery with Multi-Fidelity Experiments

Abstract

This paper studies the problem of active causal discovery when the experiments can be done based on multi-fidelity oracles, where higher fidelity experiments are more precise and expensive, while the lower ones are cheaper but less accurate. In this paper, we formally define the task of multi-fidelity active causal discovery, and design a probabilistic model for solving this problem. In specific, we first introduce a mutual-information based acquisition function to determine which variable should be intervened at which fidelity, and then a cascading model is proposed to capture the correlations between different fidelity oracles. Beyond the above basic framework, we also extend it to the batch intervention scenario. We find that the theoretical foundations behind the widely used and efficient greedy method do not hold in our problem. To solve this problem, we introduce a new concept called ϵ -submodular, and design a constraint based fidelity model to theoretically validate the greedy method. We conduct extensive experiments to demonstrate the effectiveness of our model.

1 Introduction

Causal discovery aims to learn the causal structure of a set of variables, which is fundamental for many real-world applications, including health caring [1], education [2] drug discovery [3] and protein synthesis [4]. In general, causal structure learning is an NP-hard problem [5], and purely based on the observational datasets, one cannot identify the unique causal structure, where the best result is discovering its Markov equivalence class [6].

To more accurately identify the unique causal structure, a promising direction is active causal discovery (ACD), where the model is allowed to actively intervene the causal structure to query key information for orienting the causal relations between different variables. For example, to study the causal relations between the drugs and diseases, one can conduct clinical tests via selectively administering the medicines to the patients. The key of active causal discovery is how to design effective experiments when the total cost ( e.g. , the number of experiments) is limited. To achieve this goal, recent years have witnessed many promising models. For example, Agrawal et al. [7] proposes to intervene on the variables which can orient as many as possible undirected edges. Tigas et al. [8]

Chaozhuo Li ∗

Xing Xie

designs a mutual information based method to determine the variables and values to be intervened, and study both single and batch intervention scenarios.

While the above models have achieved remarkable successes, they only allow to query a single oracle ( e.g. , the real causal structure) for the experiments 3 . However, in many real-world applications, the experiments can be done via different methods. For example, to investigate the drug-disease causal relations, in addition to the clinical tests, one can also build simulators to obtain the medicine effects on the patients [9]. Usually, each experimental method corresponds to a unique oracle, and different oracles have various fidelites. Higher-fidelity experiments are more accurate but expensive, for example, administering drugs to the real patients. Lower-fidelity experiments are cheaper but inaccurate, for example, using patient simulators. These different fidelity oracles may offer better cost-benefit trade-offs, which, however, cannot be handled by existing active causal discovery models.

To bridge the above gap, in this paper, we formally define the task of active causal discovery with multi-fidelity oracles, where the model has to actively select which variables and values to intervene at which fidelities. This task is non-trivial due to the following reasons: to begin with, because of the introduction of multi-fidelity oracles, the model has to strategically choose the lower-cost and informative enough experiments to uncover the real causal structure, which needs our special designs. Then, given the experiment results with different fidelities, how to infer the real causal structure is also not easy, since the experiment results can be not produced from the oracle corresponding to the real causal structure. In addition, in practice, an efficient experiment should allow simultaneously intervening multiple variables [10]. However, how to extend our model to the batch intervention scenario is still not clear.

To overcome the above challenges, we design a Bayesian active causal discovery model, which is composed of two components. The first one is a mutual information (MI) based acquisition function. It aims to select the interventional variables, values and fidelities which are more informative for the real causal structure. The second one is a cascading fidelity model. In specific, we first regard the highest fidelity oracle as the real causal structure, and then a cascading model is built to correlate different fidelity oracles, so that the experiment results at one fidelity can be leveraged to infer the oracle at another fidelity. To achieve more efficient experiments, we also extend our model to the batch intervention scenario. Previously, the greedy method is demonstrated to be an efficient and effective strategy for batch intervention [8]. However, we found that, by allowing multi-fidelity oracles, the theoretical foundations behind the greedy method do not hold. For alleviating this problem, we introduce a new concept called ϵ -submodular, and design a constraint-based fidelity model to theoretically validate the greedy method.

The main contributions of this paper are summarized as follows: (1) we formally define the task of active causal discovery with multi-fidelity oracles, which, to our knowledge, is the first time in the field of causal discovery. (2) To solve the above task, we propose a Bayesian framework, which is composed of a mutual information based acquisition function and a cascading fidelity model. (3) To extend our framework to the batch intervention scenario, we introduce a constraint-based fidelity model, which provides theoretical guarantees for the efficient greedy method. (4) We conduct extensive experiments to demonstrate the effectiveness of our model.

2 Preliminaries

2.1 Structure Causal Model

Structure causal model (SCM) is an effective language for describing and learning the causal relations between different random variables [11]. Usually, SCM is composed of a causal graph and a set of structure equation models (SEM).

For the causal graph, we denote it by G = ⟨ V, E ⟩ , where V is the node set, and E is the adjacency matrix. Each node in V corresponds to a variable. Suppose there are d variables in our studied problem, then we use X V = [ X 1 , X 2 , . . . , X d ] to denote the variable set. The adjacency matrix E ∈ { 0 , 1 } d × d describes the causal relations between different variables. E ij = 1 means that X i is a

parent of X j , and there exists an edge from X i to X j , while E ij = 0 indicates that there is no edge between X i and X j .

For the structure equation models, we denote them by g = { g 1 , g 2 , ..., g d } , where each g i quantitatively describes the relation between X i and its parents. Formally, we implement F with the commonly used additive noise models (ANM) [12], that is:

where γ i is the parameter set of g i , and the noise term ϵ i follows the Gaussian distribution with σ 2 i as the variance. We denote the complete parameter set of g as θ = { γ , σ } , where γ = { γ 1 , γ 2 , . . . , γ d } and σ = { σ 1 , σ 2 , . . . , σ d } . Noted that, given the above equation, we can easily derive the distribution of X V , that is, p ( X V ) = ∏ d i =1 N ( g i ( pa ( i ); γ i ) , σ 2 i ) .

Based on the above formulation, given an observational dataset D = { x k } N k =1 ∼ p ( X V ) , causal discovery aims to learn the adjacency matrix E , or more generally, simultaneously identify E and the SEM parameter θ . In this paper, we focus on the general case, and denote ϕ = ( θ , E ) .

2.2 Active Causal Discovery

Previous work has demonstrated that, purely based on the observational dataset, the real causal graph can only be identifiable to its Markov equivalence class. Active causal discovery holds the promise of identifying more accurate causal graph via designing interventional experiments.

Formally, an interventional experiment is represented by e = { ( j, v ) } , which means cutting all the causal relations pointing to X j , and fixing the value of X j as v . In causal learning, the experiment e can also be represented by do ( X j = v ) . Obviously, the distribution of X V is changed after the experiment e , and we denote the experiment-induced distribution by p ( X V | do ( X j = v )) . In practice, we cannot access the implementation of p , but can only observe the experiment result sampled from p ( X V | do ( X j = v )) . The key of active causal discovery is to design a series of experiments within limited budgets, such that the results can be better leveraged to identify ϕ .

2.3 Multi-Fidelity Active Causal Discovery

Existing ACD models mostly obtain the experiment results via interacting with the real SCM. However, in practice, the experiments can be done in different ways ( e.g. , real clinical tests or patient simulators). Each type of experiment corresponds to an underlying oracle, which produces the results of the experiments. Different oracles may provide better cost-benefit trade-offs for the experiment designs, which are failed to be considered by the previous work.

Formally, suppose we have M oracles, and the parameters of the i th oracle is denoted by ϕ i . Let the experiment cost of the i th oracle be λ i , and without loss of generality, we assume λ 1 ≤ λ 2 ≤ · · · ≤ λ M . Intuitively, if an oracle is more accurate ( i.e. , has higher fidelity), then it should be more expensive 4 . We regard the real SCM as the most accurate oracle, thus we set ϕ M = ϕ . We denote all the oracle parameters and costs as Φ = { ϕ 1 , ϕ 2 , ..., ϕ M } and Λ = { λ 1 , λ 2 , ..., λ M } , respectively.

Due to the introduction of multi-fidelity oracles, the experiment in traditional active causal discovery is extended to be a triplet e = { ( j, v ) , m } , where in addition to the intervention pair ( j, v ) , the fidelity m should also be considered in the experiment design. We define the dataset for model training as D = { e t , x t } T t =1 , where e t = { ( j, v ) , m } indicates the distribution for generating x t . In general, { ( j, v ) , m } means that x t is generated from p m ( X V | do ( X j = v )) , which is induced by ϕ m and the intervention ( j, v ) . If ( j, v ) = ∅ , then x t is an observational data, which is sampled from ϕ m without any intervention. Finally, we define the task of multi-fidelity active causal discovery as follows:

Definition 1 (Multi-Fidelity Active Causal Discovery (MFACD)) . Given M oracles with different costs Λ , we need to design a model f , which can strategically determine the intervention pair ( j, v ) and fidelity m to achieve better cost-benefit trade-off in terms of identifying the real SCM ϕ M .

3 The Licence Model

For solving the task of MFACD, we design a Bayesian framework called Licence (Mu l ti-f i delity a c tiv e lear n ing for c ausal discov e ry), which is composed of two components. The first one is an acquisition function based on mutual information, which is responsible for designing the experiments. The second one is a cascaded fidelity model, which is designed to capture the correlation between different ϕ i 's. The experiment results obtained from the first component are leveraged to update the fidelity model, and ϕ M in the fidelity model is the finally predicted result. At last, we introduce how to extend our model to the batch intervention scenario.

3.1 MI-based Acquisition Function

Intuitively, a better experiment should leverage little cost to reveal as much as possible information about the real SCM ϕ M . Thus, we design the following acquisition function:

where e = { ( j, v ) , m } is the experiment to be designed. D is the dataset already collected, which will be enlarged after each experiment. I ( x ; ϕ M | e , D ) is the mutual information, which indicates that if we conduct experiment e , then how much information the experiment result may share with the target parameter ϕ M . Obviously, we should select e which can lead to larger I ( x ; ϕ M | e , D ) . λ m is the cost of e . By dividing I ( x ; ϕ M | e , D ) with λ m , we trade-off the experiment informativeness and cost. To determine ( j, v, m ) , we derive an estimator for f ( j, v, m ) following the idea of Bayesian Active Learning by Disagreement (BALD) [13], that is:

where p ( x | e , D ) and p ( ϕ m | e , D ) correspond to the distributions of x and ϕ m after observing D under the intervention ( j, v ) . p ( ϕ M | D ) is the posterior of ϕ M after observing D . p ( x | e , ϕ m ) is the probability of x based on ϕ m intervened by ( j, v ) . We approximate the expectation operator based Monte Carlo sampling. The detailed derivation process can be seen in the appendix.

Obviously, determining the best ( j, v, m ) equals to solving the following problem:

In our task, the intervention value v is continuous. Following the previous work, we firstly learn the optimal v for each ( j, m ) pair based on Bayesian optimization (BO). Then, we compare all the results, and select the solution which can lead to the largest f ( j, v, m ) . We present the detailed Bayesian optimization process in the appendix. It should be noted that one can also leverage more advanced BO methods for jointly learning ( j, v, m ) [14]. However, we do not find significant performance improvements by these models.

3.2 Cascaded Fidelity Model

Intuitively, the experiment results from different oracles may share common information. The samples at one fidelity may help to infer the oracles at other fidelities. To capture the correlations between different oracles, we build a cascaded probabilistic model (see Figure 1), where the oracles with different fidelities are successively connected, and the observed samples are only determined by their corresponding oracles.

Formally, to achieve more robust optimization, we first regard the discrete adjacency matrix E as the samples from Bernoulli distribution. In specific, we let E ij ∼ Bernoulli ( σ ( S T · i T · j )) , where S , T ∈ R K × d ( K ≪ d ) are two continuous matrices. S · i and T · j are the i th and j th columns of S and T , respectively. By replacing E with S and T , we revise the parameter set ϕ as ( θ , S , T ) .

For each fidelity m , we assign a prior distribution of ϕ m given ϕ m -1 as follows:

Interventionvariableandvalue

(j,v)

m：

=

3

Fidelity-level

input

Oracle

1

Oracle

2

Oracle

3

Oracle

4

Oracle Φ

M

ndino

output

output

ndino

output

x

x

x

x

x

where we add subscript m to indicate different fidelities. f ( S m , T m ) = E p ( E | S m , T m ) [ λ 1 · { trace ( e E ) -d } + λ 2 · || E || 1 ] is a regularizer to encourage E to be a sparse and directed acyclic graph [15]. λ 1 , λ 2 , a , b and σ 2 are hyper-parameters.

Since the real SCM is ϕ M , we need to infer the posterior p ( ϕ M | D ) , where D is the initially observational dataset or experiment results. Directly computing p ( ϕ M | D ) is not easy, since the dataset D may contain samples from different fidelity oracles. To solve this problem, we first obtain the joint distribution p ( Φ | D ) , where Φ = { ϕ 1 , ϕ 2 , ..., ϕ M } is the collection of all the oracle parameters. Then we derive p ( ϕ M | D ) by marginalizing out { ϕ 1 , ϕ 2 , ..., ϕ M -1 } .

To efficiently compute and sample from p ( Φ | D ) , we introduce a variational approximator q ( Φ ) , which is specified as follows:

where ψ m = { c m , d m , σ m } is the set of learnable parameters, and we denote Ψ = { ψ m } M m =2 . According to the theory of variational inference [16], we maximize the following evidence lower bound (ELBO) [16] to learn Ψ :

where the likelihood p ( D | Φ ) can be easily obtained based on equation (1). More detailed derivation on the ELBO can be seen in the appendix. Once we have learned Ψ , the posterior p ( Φ | D ) is approximated by q ( Φ ) , and further, we have the following theory:

Theorem 1. If p ( Φ | D ) ≈ q ( Φ ) , then for any variable sets A,B ⊆ Φ , p ( A | B,D ) ≈ q ( A | B ) . As a special case p ( ϕ M | D ) ≈ q ( ϕ M ) .

The proof of this theory is immediate, since

Let A = ϕ M and B = ∅ , we have p ( ϕ M | D ) ≈ q ( ϕ M ) . Based on this theory, we leverage q to replace p in (2) for easy sampling.

Remark. According to the specification of q ψ m ( ϕ m | ϕ m -1 ) , we can easily demonstrate that q ( Φ ) follows Gaussian distribution. Such property enables us to use the reparameterization trick [17] to relate Φ with Ψ . Different from traditional variational models, in our objective, the adjacency matrix

E in f ( S m , T m ) is sampled from a discrete distribution, which cuts down the back-propagation signal. To solve this problem, we leverage gumbel-softmax to further associate E with ( S m , T m ) ∈ Φ . Since Φ can be further represented by Ψ , all the variables in (6) can be reparameterized by Ψ , which enables us to optimize it in an end to end manner. We present the detailed derivation process and the complete learning algorithm in the appendix.

3.3 Extension to Batch Intervention

In practice, simultaneously intervening multiple variables can be more efficient due to the lower frequency on interacting with the oracles. However, under the setting of batch intervention, the candidate intervention space exponentially increases with respect to the number of targets. Suppose we need to select c out of d variables for intervention, then the size of the candidate space is d c . Previous work found that the greedy strategy is a both efficient and effective method for the batch intervention scenario [8]. From the efficiency perspective, the greedy method only need to search in a cd -sized candidate space. From the effectiveness perspective, people demonstrate that the mutual information obtained by the greedy strategy is not worse than that of the optimal solution multiplied by (1 -1 e ) [18]. In the following, we introduce how to extend our model to the batch intervention scenario, and leverage the greedy strategy to solve our task.

Objective for batch intervention in MFACD . We use the following objective for batch intervention scenario:

where e i = { ( j i , v i ) , m i } is an experiment, x i is the observed sample from the experiment e i , that is, x i ∼ p m i ( X V | do ( X j i = v i )) . λ i is the cost of experiment e i . This objective aims to design a series of experiments with budget C, which can reveal the information about ϕ M as much as possible. The number of intervention targets n is not a fixed value, which is constrained by the total budget.

The greedy method for MFACD . The greedy method designs each experiment independently, and at each step, it selects the experiment which can maximize the average information gain. Following [19], the k th experiment is determined based on the following objective:

where { e i } k -1 i =1 is the previously designed experiments, and is fixed when learning e k .

What's wrong with the greedy method . The theoretical foundations of the greedy method is demonstrated by the previous work [19] as follows:

Theorem 2. If I ( x ; ϕ M | e , D ) is (1) submodular and (2) non-decreasing, then

where { e g i } n i =1 is the solution obtained from the greedy method, and { e ∗ i } n i =1 is the optimal solution.

However, due to the introduction of multi-fidelity, in our model, I ( x ; ϕ M | e , D ) is actually not submodular. More specifically, in the proof of submodular, two samples x s and x t has to be independent given ϕ M ( e.g. , see B.4 in [8]). In the single-fidelity setting, this requirement naturally holds, since ϕ M is exactly the parameter used to sample x s and x t . However, when we introduce multi-fidelity, x s and x t may not independent given ϕ M , since they are only directly influenced by their own oracles (see Figure 1).

An improved greedy method tailored to MFACD . To alleviate the above problem, in this section we design an improved greedy method tailored to our task. In specific, we first define several new concepts, and then build theories based on these concepts to inspire our model designs.

Definition 2 ( ϵ -independent) . For random variables A,B and C , if their mutual information satisfy I ( A ; B | C ) ≤ ϵ , then we say A and B are ϵ -independent given C .

Definition 3 ( ϵ -submodular) . Suppose f ( · ) is a set function on Ω . If for any A,B ⊆ Ω , A ⊆ B and x ∈ Ω \ B , f ( A ∪ { x } ) -f ( A ) ≥ f ( B ∪ { x } ) -f ( B ) -ϵ , then we say f is ϵ -submodular on Ω .

Based on the above two definitions, we have:

Theorem 3. For any two experiments e s and e t , if the corresponding samples x s and x t are ϵ -independent given ϕ M , { e s , e t } and D , then I ( · ; ϕ M |· , D ) is ϵ -submodular.

Theorem 4. If I ( · ; ϕ M |· , D ) is ϵ -submodular on X and non-decrease, for any i, j , λ i λ j ≤ B λ , then

Following this theory, we improve objective (6) to a constaint-based ELBO as follows to capture the degree of independence between different experiment results:

The proofs of the above theories are presented in the appendix, and similar to objective (2), we use Monte Carlo method to approximate the mutual information in objective (10).

4 Related Works

Bayesian Active Causal Discovery. Causal discovery [20-22] refers to recovering causality in a set of variables, especially trying to find a directed acyclic graph (DAG) that can represent the relationship between variables in a system. Active causal discovery was first proposed in [23, 24] with the assumption that the data is discrete-valued. In active causal discovery, the experimenter attempts to intervene on the variables in the system at each step, utilizes the interventional data to recover causal relation, and finally identifies the entire causal structure with minimal cost [25]. Many methods have been discussed in different settings over the past decades, including continuous linear Bayesian networks [26, 27], non-linear causal models [28], and large-scale causal models [8].

Multi-fidelity Settings. The fidelity commonly refers to how accurate the model or environment can be when providing information. Higher-fidelity models are more accuracy but cost much, while lower ones are less accurate but cheaper. In order to combine the strength of each model, multi-fidelity models are proposed to achieve an accurate model with lower costs. Multi-fidelity models can be divided into two main categories [29]: Multi-fidelity Surrogate Models (MFSM) [30-32] and Multi-fidelity Hierarchical Models (MFHM) [33-35]. In addition to optimization, multi-fidelity models can also be used for uncertainty propagation [36] and statistical inference [37]. Recent works also start to study the multi-fidelity settings when conducting Bayesian experiments [38-40], and adopt deep learning frameworks to solve corresponding problems.

Our paper makes a first step towards multi-fidelity active causal discovery, and solve the non-trivial challenges when combining the above two fields, which, to the best of our knowledge, is the first time in the causal inference domain.

5 Experiments

In this section, we conduct experiments to demonstrate the effectiveness of our model, where we focus on the following problems: (1) whether our model can achieve better performance than the previous ACD methods? (2) Whether the constraint in objective (10) in necessary? (3) How the DAG regularization coefficient influence the model performance? In the following, we first introduce the experiment setup, and then present and analyze the results.

ER Graph(single)

SF Graph(single)

ER Graph(batch)

SF Graph(batch)

30

30

30

30

25

SHD

20

20

10

10

15

10

10

203040

50

10

203040

50

10

203040

50

10

203040

50

totalbudget

totalbudget

totalbudget

totalbudget

ER Graph(single)

SF Graph(single)

ER Graph(batch)

SF Graph(batch)

0.4

0.4

0.4

0.4

AUPRC

AUPRC

AUPRC

AUPRC

0.3

0.3

0.3

0.3

0.2

0.2

0.2

0.2

10

20

30

40

50

10

203

3040

50

10

20

30

40

50

10

20

30

40

50

totalbudget

totalbudget

totalbudget

totalbudget

ER Graph(single)

SF Graph(single)

ER Graph(batch)

SF Graph(batch)

0.040

0.05

0.06

0.05

0.035

0.05

RMSE

0.030

0.025

0.03

0.03

0.03

0.020

10

20

30

40

50

10

20

30

40

50

10

20

30

40

50

10

20

30

40

50

totalbudget

totalbudget

totalbudget

totalbudget

1

AIT-REAL

AIT-RANDOM

I1

CBED-REAL

CBED-RANDOM

1

Licence

5.1 Experimental Setup

We experiment with three commonly used causal discovery datasets, including Erd˝ os-Rényi graph ( ER ) [41], Scale-Free graph ( SF ) [42] and DREAM [43]. To demonstrate the effectiveness of our model, we compare it with AIT [44] and CBED [8], which are the recent state-of-the-art models in this field. Since they cannot select different fidelities, we design two variants for each of the baseline, that is, X-REAL and X-RANDOM , which means that the model always interacts with the ground truth oracle ϕ M or randomly select the oracles. Here 'X' is AIT or CBED. For the evaluation metrics, we use SHD [45], AUPRC [46] and RMSE to evaluate different models. The first two metrics aim to evaluate the accuracy of the learned topological structure, and the last one measures the performance of functional relations. For single intervention, we first generate several observational samples to initialize the model. Then, we indicate a total intervention budget, and let the model interact with the causal graph with different fidelities until the budget runs out. For each interaction, the model will provide an intervention, and correspondingly obtain a sample from the oracles, which is used to update the model. Finally, the model outputs the estimated causal graph, which is leveraged to evaluate the performance. For batch intervention, we indicate the total budget for each intervention step, and the model determines n interventions simultaneously, which are delivered to the oracles to obtain the samples. We present more detailed settings in the appendix.

5.2 Overall Performance

In this experiment, we evaluate the models under different total budgets, and we present the results on ER and SF datasets with 10 graph nodes. The experiments on DREAM and more nodes are presented in the appendix. From the results shown in Figure 2, we can see: as the total budget becomes larger, the performances of all the models tend to increase on both datasets. This is not surprising, since more experiment budgets enable us to conduct more interventions or query more accurate oracles, which can reveal more information about the ground truth and facilitate more accurate causal discovery. In most cases, our model can perform better than the baselines across different datasets, evaluation metrics and intervention budgets. These results demonstrate the effectiveness of our model. In specific, on the metrics of SHD, AUPRC and RMSE, our model can on average improve performance of the best baseline by about 27.74%, 82.35% and 22.74% on ER, and 17.69%, 60.27% and 21.62%

on SF, respectively. If we look more carefully, we find that, for both AIT and CBED, randomly querying the oracles can sometimes perform better than always interacting with the ground truth oracles. This result suggests that lower fidelity oracles can be helpful to trade-off the performance and cost. However, the random method is still suboptimal, and designing more principled and tailored strategies to select the fidelities is necessary, which is evidenced by the lowered performance of 'X-RANDOM' than our model.

5.3 Necessity of the Mutual Information Constraint in Objective (10)

In our model, the mutual information constraint in objective (10) aims to make the greedy method validate. In this section, we study whether it is necessary by experiments. To achieve this goal, we introduce a variant of our model Licence (w/o reg) , where we remove the mutual information constraint. We evaluate the models based on the dataset of ER with 10 graph nodes, and the total budget is set as 30. The results are presented in Figure 3(a). We can see, in some cases, 'X-RANDOM' performs better than 'X-REAL', which is consistent with the above experiments, and demonstrates that always querying the ground truth oracle may not lead to better performance under limited budget. By comparing our model with the variant Licence (w/o reg), we find that our model can consistently achieve better performances on all the evaluation metrics. In specific, Licence can improve the performance of Licence (w/o reg) by about 2.05%, 16.57% and 14.92% on SHD, AUPRC and RMSE respectively. These results demonstrate that the mutual information constraint is necessary in our model, which empirically verifies the theories proposed in section 3.3.

5.4 Influence of the DAG regularization coefficient

In this section, we analysis the influence of the DAG regulation coefficient β in equation (4). The results are reported based on AUPRC. The coefficient β reflects the importance of DAG regulation when updating the model. As β increases, the DAGness is more emphasized for the causal graph. In this subsection, we conduct experiments for various β , ranging from 0 . 0 to 1 . 0 , and the results are shown in Figure 3(b). We can see as β increases, the performance of AUPRC improves as well, and peaks at β = 0 . 8 . That is probably because lower β will decrease the acyclic property of causal graphs, which is incompatible with the prior knowledge of true causal graphs. However, higher coefficient may also impact the optimization process, which leads to sub-optimal results. We think a trade-off between mild constraint for easy optimization and solid constraint for DAG property is supposed to take into consider for different tasks and settings.

6 Conclusion

This paper formally defines the task of active causal discovery with multi-fidelity oracles, which, to our knowledge, is the first time in the causal discovery domain. To solve this task, we propose a Bayesian framework, which is composed of a mutual information based acquisition function and a cascading fidelity model. We also extend our framework to the batch intervention scenario, and propose a constraint-based fidelity model to validate the efficient greedy method.

This paper actually makes an initial step toward considering different oracles in active causal discovery. There is much room left for improvement. To begin with, one can design more advanced batch intervention strategies, which can bypass the greedy method and does not need to introduce the mutual information constraint in the fidelity model. In addition, since the experiments in active causal discovery are conducted sequentially, and the former experiment results may influence the latter ones, it is interesting to consider the experiment designs as a Markov decision process, and leverage reinforcement learning to optimize the total information gains of all the experiments in a more principled manner.

ACKNOWLEDGMENTS

This work is supported in part by National Key R&D Program of China (2022ZD0120103), National Natural Science Foundation of China (No. 62102420), Beijing Outstanding Young Scientist Program NO. BJJWZYJH012019100020098, Intelligent Social Governance Platform, Major Innovation & Planning Interdisciplinary Platform for the "Double-First Class" Initiative, Renmin University of China, Public Computing Cloud, Renmin University of China, fund for building world-class universities (disciplines) of Renmin University of China, Intelligent Social Governance Platform.

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Appendix

A Monte Carlo Approximation for f ( j, v, m )

A.1 Derivation Process for f ( j, v, m )

Considering that the mutual information is not directly tractable, we approximate f ( j, v, m ) by:

where e = { ( j, v ) , m } is the experiment to be designed, ϕ ( c 1 ) m , ϕ ( k 1 ) m ∼ p ( ϕ m | D ) , x ( k 1 ,l 1 ) m ∼ p ( x | ϕ ( k 1 ) m , e ) , ϕ ( k 2 ) M ∼ p ( ϕ M | D ) , ϕ ( k 2 ,l 2 ) m ∼ p ( ϕ m | ϕ ( k 2 ) M , D ) and x ( k 2 ,l 2 ,c 2 ) m ∼ p ( x | ϕ ( k 2 ,l 2 ) m , e ) . We present the detailed approximation process as follows:

For part E , we can estimate it by

where for the first expectation on p ( ϕ m | e , D ) , we first sample ϕ ( k 1 ) m from ϕ ( k 1 ) m ∼ p ( ϕ m | e , D ) for K 1 times, and then for each ϕ ( k 1 ) m , we sample x ( k 1 ,l 1 ) m from x ( k 1 ,l 1 ) m ∼ p ( x | ϕ ( k 1 ) m , e ) for L 1 times. For the second expectation on p ( ϕ m | e , D ) , we sample ϕ ( c 1 ) m ∼ p ( ϕ m | e , D ) for C 1 times.

For part F , we have

It can be estimated by

where for the expectation on p ( ϕ M | D, e ) , we sample ϕ ( k 2 ) M from ϕ ( k 2 ) M ∼ p ( ϕ M | e , D ) for K 2 times. For the expectation on p ( ϕ m | ϕ M ) , for each ϕ ( k 2 ) M , we sample ϕ ( k 2 ,l 2 ) m from ϕ ( k 2 ,l 2 ) m ∼ p ( ϕ m | ϕ ( k 2 ) M ) for L 2 times. For the expectation on p ( x | ϕ m , e ) , for each ϕ ( k 2 ) M and ϕ ( k 2 ,l 2 ) m , we sample x ( k 2 ,l 2 ,c 2 ) m from x ( k 2 ,l 2 ,c 2 ) m ∼ p ( x | ϕ ( k 2 ,l 2 ) m , e ) for C 2 times.

Therefore, we can conclude that f ( j, v, m ) can be estimated by

where ϕ ( c 1 ) m , ϕ ( k 1 ) m ∼ p ( ϕ m | D ) , x ( k 1 ,l 1 ) m ∼ p ( x | ϕ ( k 1 ) m , e ) , ϕ ( k 2 ) M ∼ p ( ϕ M | D ) , ϕ ( k 2 ,l 2 ) m ∼ p ( ϕ m | ϕ ( k 2 ) M , D ) and x ( k 2 ,l 2 ,c 2 ) m ∼ p ( x | ϕ ( k 2 ,l 2 ) m , e ) .

Obviously, the above approximation of f ( j, v, m ) only depends on p ( ϕ m | D ) , p ( ϕ m | ϕ M , D ) and p ( x | ϕ m , e ) . In the next, we show how to sample from them in Section A.2, A.3 and A.4, respectively.

A.2 Sampling from p ( ϕ m | D )

Basically, sampling from the posterior of ' p ( ·| D ) ' is not easy. To solve this problem, as mentioned in the main paper, we introduce a variational probability ' q ' to approximate ' p '. In specific, in order to sample from p ( ϕ m | D ) , we first obtain a sample ϕ 1 from ϕ 1 ∼ N ( 0 , I ) , and then get ϕ m from the distribution q ( ϕ m | ϕ 1 ) .

Since and

we have q ( ϕ m | ϕ 1 ) is a Gaussian distribution, which is easy for sampling.

In our model, c i and σ 2 i I are diagonal matrices, which means that the dimensions in ϕ i are independent with each other. We denote ϕ i = [ ϕ i, 1 , ϕ i, 2 . . . ϕ i,d ] , where ϕ i,j is the j th element of ϕ i . Then, we have

So our target can be converted to calculate the probability q ( ϕ m,j , ϕ m -1 ,j , . . . , ϕ 2 ,j | ϕ 1 ,j ) for all dimensions ∀ 1 ≤ j ≤ d . Let c i,j , d i,j and σ 2 i,j be the j th element of c i , d i and σ 2 i , respectvely. We assume that σ i,j = √ c 2 i -1 ,j +1 · σ i -1 ,j ( i ≥ 4) and σ 3 ,j = σ 2 ,j = e , where e is the hyper-parameter. Suppose µ i,j is the mean of the Gaussian distribution for q ( ϕ i,j | ϕ i -1 ,j ) , that is,

then, the approximated joint distribution can be represented as

Then, we integrate ϕ 2 ,j , ϕ 3 ,j , . . . , ϕ m -1 ,j sequentially to obtain q ( ϕ m,j | ϕ 1 ,j ) . First of all, we integrate ϕ 2 ,j for the joint distribution, where we have:

Denote ¯ c 2 ,j = c 2 ,j and ¯ d 2 ,j = d 2 ,j , and because of σ 3 ,j = σ 2 ,j , we have

For S 1 , we have

Then we have

Therefore we have

The S 2 part is the integration form of ϕ 2 ,j ∼ N ( c 3 ,j ϕ 3 ,j +¯ c 2 ,j ϕ 1 ,j + ¯ d 2 ,j -d 3 ,j c 3 ,j c 2 3 ,j +1 , σ 2 3 ,j c 2 3 ,j +1 ) , which is equal to 1 , so we have

We denote ¯ c 3 ,j = c 3 ,j ¯ c 2 ,j and ¯ d 3 ,j = ¯ d 2 ,j c 3 ,j + d 3 ,j , and denote r 2 ,j = e d 2 3 ,j 2 σ 2 3 ,j · 1 √ c 2 3 ,j +1 , so we have

Similarly, then we integrate ϕ 3 ,j

The formulation is similar to the previous one, so we can utilize the process above to integrate succesively, and we finally obtain

which indicates

where we have the iterative calculation by

A.3 Sampling from p ( ϕ m | ϕ M , D )

To sample from the distribution p ( ϕ m | ϕ M , D ) , we first obtain a sample ϕ 1 from the prior distribution ( i.e., ϕ 1 ∼ N ( 0 , I ) ), then get ϕ m from a consecutive sampling process:

because of the Markov property in our cascaded model. So our target is obtaining the distributions p ( ϕ i -1 | ϕ i , ϕ 1 , D ) . For a certain p ( ϕ i -1 | ϕ i , ϕ 1 , D ) , according to the Bayes rule, we have

Similarly with the last section, we use non-bold symbols to represent one dimension of the multidimension parameters, where they are able to transfer independently, and finally construct the eventual parameters by concatenating, that is,

So our target can be converted to calculate the probability p ( ϕ i,j | ϕ i -1 ,j , ϕ 1 ,j , D ) for all dimensions ∀ 1 ≤ j ≤ d . According to the Markov property and the transportation probability, we have

According to the previous section, we have

Then we have

Then we calculate the part C as

where B does not include ϕ i , which indicates

A.4 Calculation of p ( x | e , ϕ m )

In this section, we will show how to calculate the graph probability p ( x | e , ϕ m ) . Remember the graph parameters ϕ m = [ θ m ; S m ; T m ] , so we have

According to Monte Carlo sampling, we have

where E l [ i, j ] ∼ Bernoulli ( σ ( S T m [ i ] · T m [ j ])) . In order to conduct intervention process, we change the j th column of E l to zeros, and represent it with ˜ E l . Moreover, we replace the j th element of x

with v , and get the result ˜ x . We change the j th element of ϵ m with zero, and get the result ˜ ϵ m . Then according the definition of causal graphs, we have

where f is the causal function that depends on the parameter γ m .

B Bayesian Optimization for Determining ( j ∗ , v ∗ , m ∗ )

We intend to find the best tuple for acquisition, that is,

We define the best interventional value v under interventional node j and fidelity m as

where f j,m ( v ) is rewritten from f ( j, v, m ) under given j, m . Therefore, our task is calculating v ∗ ( j, m ) for ∀ j ∈ [ d ] , m ∈ [ M ] with Bayesian optimization [47]. We utilize a Gaussian Process (GP) [48] to model surrogate function distributions for each v ∗ ( j, m ) . We denote f ∼ GP ( 0 , K ( v i , v j )) , and K ( v i , v j ) is the kernel of GP. We sequentially find v t and calculate f j,m ( v t ) to direct the process. According to GP, the previous t functions and the t +1 function are multivariate Gaussian distribution,

Given previous t steps, we have the posterior probability is

with the non-parametric means and variances

We acquire the next v t +1 with GP-UCB [49] function

where β ac is a hyper-parameter. Suppose the maximum of steps is T , the final output of function v ∗ ( j, m ) is

Then we choose the best interventional node j and fidelity m by their best values under O ( d · M )

where we define

C Detailed Training Process of ELBO

C.1 Derivation Process of ELBO

Because we use the distribution q ( ϕ m ) to approximate the distribution p ( ϕ m ) , then we intend to minimize the distance between these two distributions optimize the parameters of q ( ϕ m ) , where we utilize KL divergence to measure the distance, that is,

According to the variational inference, we have

Because log p ( D ) is not related to Ψ , minimizing KL [ q ( Φ ) || p ( Φ | D )] is equivalent to maximizing the ELBO part, and we have

Above all, we can conclude that

is equivalent to maximize evidence lower bound

C.2 Estimation of ELBO

We represent the equation of ELBO as

For the part A , we have

where N is the current number of samples in buffer. Then we have

Using Monte Carlo sampling [50], we can calculate the expectation by N S samples for each point.

where we sample ϕ ( j ) m ( i ) ∼ q ( ϕ m ( i ) ) with size N S .

Then we denote the distribution q ( Φ ) = N ( ˜ µ all , ˜ Σ all ) , and similarly, we have p ( Φ ) = ∏ M m =1 e -β · f ( S m , T m ) · N ( µ all , Σ all ) . Both the parameter ˜ µ all , ˜ Σ all can be represented by the parameters in Ψ , while µ all and Σ all are constant. Then we calculate part B

According to KL divergence of Gaussian distribution, we can calculate C in a close-form.

Then we calculate D by the following steps:

Using Monte Carlo sampling, we can calculate the expectation by N D samples for each point.

where we samples Φ ( i ) ∼ N ( µ all , Σ all ) with size N D . Using Monte Carlo sampling again, we can calculate the expectation by N E samples.

where we samples E ( j ) ∼ p ( E | S ( i ) m , T ( i ) m ) with size N E .

Finally, we obtain the estimation

C.3 Gaussian Reparameterization Trick

In the last section, we derive the objection function for optimizing the model parameters, where we can use methods of the gradient decent to solve it. However, a significant problem rises due to the sampling process, because the gradient of model parameters can not pass backward from the naive sampling process( i.e., untraceable). Therefore, we use Gaussian reparameterization trick to make the Gaussian sampling process traceable.

In specific, we will demonstrate the traceable calculation of ϕ by Gaussian reparameterization trick. In order to sample ϕ ∼ N ( µ , Σ ) , we first sample δ ∼ N ( 0 , I ) instead, and then obtain ϕ = µ + δ ⊙ Σ . Therefore, the gradient can be traced from ϕ to µ and Σ . In specific, both µ and Σ can be represented with the function of learnable parameter Ψ .

C.4 Gumbel-softmax Reparameterization Trick

Besides of the Gaussian sampling process, the Bernoulli sampling in our equation is not traceable either, so we utilize Gumbel-softmax reparameterization trick to make it traceable.

We demonstrate the traceable calculation of E ∼ p ( E | S , T ) by Gumbel-max reparameterization trick. According to Gumbel-max [51], we have

Instead of using unit step function, we utilize sigmoid function

where L i,j ∼ L (0 , 1) . Therefore, we sample L i,j ∼ L (0 , 1) instead, where L (0 , 1) is logistic distribution, and calculate E i,j = σ ( L i,j + S T i · T j ) to trace gradients. Specifically, both S i and T i can be represented with the function of learnable parameter Ψ .

Therefore, we have

C.5 Optimization of ELBO

With the estimation and reparameterization trick, we are able to conduct gradient descent methods to optimize our parameters with the objection function

The format of stochastic gradient descent (SGD) is

where γ is the learning rate.

D Training Process of Constraint based ELBO

We intend to optimize our parameter with

However, the objection has a constraint, which is hard to optimize with gradient descent methods. So we utilize Lagrange multiplier [52] to convert it to a constraint-free method:

where λ is the Lagrange multiplier. Then, we intend to calculate the constraint part.

First of all, we have

For the term H ( x s | ϕ M , e s , D ) , we have

We use Monte Carlo sampling to estimate H ( x s | ϕ M , e s , D ) , and we have

where we sample graphs ϕ k 1 m ∼ q ( ϕ m | e s , ϕ M ) , and obtain samples x ( k 1 ,k 2 ) ∼ p ( x | e s , ϕ k 1 m ) . Similarly, we can calculate

where we sample graphs ϕ k 1 m ∼ q ( ϕ m | e t , ϕ M ) , and obtain samples x ( k 1 ,k 2 ) ∼ p ( x | e t , ϕ k 1 m ) . And we have

where we sample graphs ϕ k 1 m ∼ q ( ϕ m |{ e s , e t } , ϕ M ) , obtain samples x ( k 1 ,k 1 2 ) ∼ p ( x | e s , ϕ k 1 m ) , and obtain samples x ( k 1 ,k 2 2 ) ∼ p ( x | e t , ϕ k 1 m ) .

Therefore, we add constraint on the original loss function to obtained the estimation of constraint based ELBO, that is,

E Proof of Theory 3

Proof. To begin with, we introduce two anchor variables x , e , indicating existing samples and experiments in the system, which are independent with the following experiments. Since x s , x t are ϵ -independent given ϕ M , { e s , e t } and D , we have:

Since

We have:

According to the basic mutual information property I ( A,B ; C ) -I ( B ; C ) = I ( A ; C | B ) , we have:

Thus, we have:

Since different experiments are independent, we have:

Thus, I ( · ; ϕ M |· , D ) is ϵ -submodular.

F Proof of Theory 4

For clear presentation, we denote g ( { e i } n i =1 ) = I ( { x i } n i =1 ; ϕ M |{ e i } n i =1 , D ) , then we need to solve the following problem:

Suppose S ∗ = { e ∗ i } n i =1 is the optimal solution for objective (14), and the results of the greedy method is S = { e i } n i =1 , where the experiments are sequentially determined from e 1 to e n . We denote S 1: j = { e i } j i =1 , and ∆( e | S 1: j ) = g ( S 1: j ∪ e ) -g ( S 1: j ) , according to the greedy method, we have:

where λ e is the cost of experiment e .

Based on all the above notations, we have:

where the first inequality holds because of the non-decreasing property, and the second inequality holds because of the ϵ -submodular property.

Since e j +1 = arg max e ∆( e | S 1: j ) λ e , we have ∆( e | S 1: j ) λ e ≤ ∆( e j +1 | S 1: j ) λ e j +1 for any e , thus ∆( e | S 1: j ) ≤ λ e λ e j +1 ∆( e j +1 | S 1: j ) ≤ B λ ∆( e j +1 | S 1: j ) . By bringing this result into the above equation, we have:

Let T j = g ( S ∗ ) -g ( S 1: j ) , we have:

Then

Let B = [(1 -1 nB λ )] n -1 ϵ B λ + ... + ϵ B λ = ϵ B λ ∑ n i =1 [(1 -1 nB λ )] i -1 , and considering that [(1 -1 nB λ )] n = e -1 B λ , we have:

Thus, we have g ( S 1: n ) ≥ (1 -e -1 B λ ) g ( S ∗ ) -B .

G Algorithm

The algorithm for Licence method for single interventiion scenario is shown in Algorithm 1. Moreover, the algorithm for Licence method for batch interventiion scenario is shown in Algorithm 2.

H More Experiments

H.1 Experimental Settings

H.1.1 Datasets

The details of our experimental datasets are presented as follows:

Erd˝ os-Rényi (ER) [41] graph is a random graph introduced by Paul Erd˝ os and Alfréd Rényi. For ER graph, a graph with n vertices is generated by connecting each pair of vertices with a probability p .

Scale-Free (SF) [42] graph is a type of random graph that has a degree distribution following power law. A small number of vertices in SF graph own a large number of edges, while the vast majority of vertices have relatively few edges.

DREAM [43] is the abbreviation for Dialogue for Reverse Engineering Assessments and Methods, which can estimate the reverse quality that causal discovery methods perform. Specifically, we use a biological graph generator GeneNetWeaver for our experiments, which is a real-word public dataset.

H.1.2 Baselines

The details of experimental baselines are demonstrated as follows. We utilize DiBS [53] as our basic graph representation component. For acquisition methods, we use AIT and CBED and obtain the query tuples of node and value.

AIT [44] is an active learning method that utilize f-score to select intervention queries.

CBED [8] is based on the calculation of mutual information (MI), which intend to select intervention queries with maximal MI scores after obtaining new samples under current queries.

For the batch intervention scenario, we extend above methods with greedy strategy, which can promise an lower bound for approximation with submodular property. For choosing the fidelities to query, we use two circumstances, i.e., REAL and RANDOM.

REAL fidelity means the model always choose the highest fidelity to conduct experiments. This strategy is aligned with classic causal discovery under active learning paradigm without multi-fidelity settings, which can just choose the most accurate samples to conduct discovery process.

RANDOM fidelity means the model choose different fidelities randomly with uniform probability.

H.1.3 Metrics

The details of experimental metrics are demonstrated as follows. We utilize SHD and AUPRC to reflect the topological structure discovering performance, and design RMSE to reflex the predicting performance of functional relations.

SHD [45] is the abbreviation for Structural Hamming Distance, and it estimate the topological structure by counting the number of different edges on adjacency matrix. We calculate the expectation of SHD under multiple graph samplings.

AUPRC [46] is the area under precision-recall curve, where we consider entities on the adjacency matrix as binary classification problem. The AUPRC is also under the expectation for multiple graph sampling.

RMSE is designed for estimating the performance of grasping functional relations. We obtain several samples from the true causal graph, and let our model and the true causal function to conduct forward process respectively, then calculate the RMSE between the two results. We calculate RMSE by sampling graphs for multiple times.

H.2 Simulation of Oracles with Different Fidelities

For a given intervention ( j, v ) , suppose we have M oracles { ϕ 1 , ϕ 2 , ..., ϕ M } , then the experiment results { x j,v, 1 , x j,v, 2 , ..., x j,v,M } are specified as follows:

where x j,v,M is the ground truth, which can be directly obtained from the datasets. Since x j,v,m is correlated with x j,v,M by the first line, their underlying oracles ϕ m and ϕ M are correlated in our simulation. In our experiment, we set δ 1 > δ 2 > ... > δ M = 0 . Suppose the cost of ϕ m as λ m , then we set λ 1 < λ 2 < ... < λ M .

H.3 Details of Configurations and Computation

The details of the configurations of device and platform are demonstrate in Table 1(left). We will show the details of the time cost on computation. We measure the time cost on the generation of each intervention per fidelity for all models, and the results are shown in Figure 1(right). We find that our method cost a little more than the baselines, which is probably due to the more complex sampling process in our model.

We also show the details of experimental settings for our overall experiments in Table 2. We carefully tune the hyper-parameters for baselines and our model, and the final values can be obtained in the configuration file in our codes.

H.4 Experiments on DREAM Dataset

We conduct experiments on a real-world biological dataset, called DREAM. Note that, DREAM does not support the calculation of RMSE, because of the lack of interface in this real-world dataset. We use two sub-datasets Ecoli and Yeast as our true causal graphs. The results are shown in Figure 4. We find that our model outperforms that other baselines on both Ecoli and Yeast , and both single and batch intervention scenario.

H.5 Experiments on More Nodes

In this section, we conduct further experiments on datasets with more nodes. We extend the number of nodes from 10 to 20, and experiment on the ER graph. The results are shown in Figure 3. We find that our model is still effective on the scenario of more nodes, and is better than baselines.

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H.6 Supplementary Experiments on MAE Metric

We further compare our model with the baselines based on the Mean Absolute Error (MAE) metric. The experiments are conducted based on ER with different total budgets. The results are presented in Table 4.

The results indicate that our model surpasses the baselines in terms of MAE. This further provides evidence that the superior performance of our model is a general conclusion.

H.7 Supplementary Experiments on Different Oracle Settings

To demonstrate that our model is generally effective for different cost- and noisy-levels. We conduct experiments based on different sets of oracles. In specific, the experiments are conducted based on the following settings in Table 5. The results are presented in Figure 5.

From the results, we can see that our model can always perform better than the baselines on different sets of oracles.

H.8 Supplementary Experiments on Regularization Coefficient λ

In our model, the ϵ -independent constraint in Equation 10 is an important contribution. In the optimization process, we convert it to the objective. We study the influence of the coefficient λ by tuning it in the range of [ 10 -5 , 10 -6 , 10 -7 , 10 -8 , 10 -9 ]. The results are presented in Figure 6.

From the results, we can see the performances of our model varies as we set different λ 's. In most cases, the best performance is achieved when λ is moderated (not too large or too small).

H.9 Supplementary Experiments on Ablation Studies

To study whether the correlation modeling between different oracles are necessary, we first build a variant of our model by regarding different oracles as independent components, that is, removing the links between different ϕ 's in Figure 1, and then compare our model with such variant. The results are presented in Table 6.

We can see, in most cases, our model can achieve better performance than its variant without modeling the correlations between different oracles.

I Potentially Negative Social Impact

Causal discovery focuses on understanding causal relationships between variables. While causal discovery has the potential to bring about positive social impacts, it is important to consider both the positive and negative implications of its applications. In this response, I will focus on the negative impact of causal discovery.

· Reductionism and Oversimplification. Causal discovery techniques often aim to identify simple cause-and-effect relationships. However, complex social phenomena often involve a multitude of interconnected factors, making it difficult to capture the full complexity of the system. Relying solely on causal discovery may lead to oversimplification and reductionism, neglecting the nuanced interactions between variables.

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Ethical Concerns. Causal discovery can involve analyzing sensitive data, such as personal information or medical records. If not handled carefully, the use of this data can raise significant ethical concerns related to privacy, consent, and potential discrimination. Improper handling of data could lead to violations of privacy and unfair treatment of individuals or groups.

Overreliance on Correlation. Causal discovery often relies on identifying statistical correlations between variables. However, correlation does not imply causation, and there is a risk of mistakenly inferring causal relationships based solely on correlation. Overreliance on such methods can lead to erroneous conclusions, leading to misguided decision-making and ineffective interventions.

Social Bias and Inequality. Causal discovery relies on the data used for analysis, which can reflect existing biases and inequalities present in society. If the data used is biased, the causal relationships discovered may perpetuate or exacerbate existing social inequalities. Causal discovery methods need to be sensitive to potential biases and strive for fairness and inclusivity in both data collection and analysis.

In conclusion, while causal discovery holds promise in understanding complex systems, it is crucial to consider its potential negative impacts. Oversimplification, ethical concerns, overreliance on correlation, and social bias are all factors that need to be addressed to ensure responsible and beneficial applications of causal discovery. It is essential to approach this field with caution and incorporate broader societal considerations to mitigate the negative impacts and harness its potential for positive social change.

J Limitations

In this section, we analyze the limitations of our work, including sub-optimum of greedy method, estimation of mutual information, and scale of causal graph.

Sub-optimum of greedy method. For the whole process of active causal discovery, the interventional data will be acquired successively in the greedy manner. Therefore, even if the strategy for acquisition is the optimal for each current step, the entire trajectory of causal discovery is sub-optimal. A possible solution is finding the best acquisition trajectory by reinforcement learning.

Estimation of mutual information. For different circumstances, the costs, accuracy and data scale can be various. Therefore, the scale of mutual information can be affected as well. So it is important to adjust hyper-parameters accordingly.

Scale of causal graph. It is a classic problem for causal discovery that most existing methods suffer from the difficulty in extending to large-scale graphs. The efficiency and effectiveness are supposed to be further improved, and we will optimize our model as well.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce the task of active causal discovery with multi-fidelity oracles which they go onto show is superior to many state-of-the-art methods for active causal discovery. They demonstrate their method in multiple settings and compare it alongside the aforementioned methods, to show superior performance almost across the board.\n\nSTRENGTHS:\nPlease see the Questions section for the main review.\n\nWEAKNESSES:\nPlease see the Questions section for the main review.\n\n- The paper is very hard to follow for a number of reasons. First, the language is lacking in a lot of places and complex sentence construction makes following the arguments presented, somewhat harder still (this sentence for example on line 256: \"Higher-fidelity models are more accuracy but cost much\" should presumably be 'higher-fidelity models are more accurate but they cost more'). To help the authors I recommend Grammarly which is a great free tool that can pickup many of the language mistakes that are present in the manuscript. Second, the authors jump a lot in their narrative thus making quite difficult to understand the point they are trying to make. The manuscript would do well to be proof-read a few more times by different people. Most all the manuscript lacks a red thread, taking the reader through the paper one coherent argument at time, thus building a narrative as it goes along. At the end of section three, I still do not quite understand where the authors are heading with their method.\n\nQUESTIONS:\n## Abstract\n\n- A very good abstract which I enjoyed reading but it is too long, much too long. The abstract is merely meant to summarise what you are doing in the paper, and pick out some main results and then leave the rest for the paper. I suggest about half the length of what you currently have.\n\n## Introduction\n\n- Typo: Markov equivalence class (not ‘equivalent’) - line 26.\n- The paragraph starting on line 37 is excellent, very interesting. I would be interested to know if the authors have more information on this particular topic discussed in the paragraph i.e. do clinicians actually take these considerations intom account? References to that effect would be very informative.\n- Do you know if there are more sources on multi-objective (multi-target), line 54, studies which you mention? This is an open problem for much of causal inference as you rightly know. It would be effectual to add such sources if you have them.\n\n## Preliminary\n\n- Typo / language: the section tile should be ‘Preliminaries’ as it is plural as you are describing multiple ideas here.\n- Reference Pearl when you first introduce the SCM on line 78. It was his idea after all and he deserves credit whenever it is raised in academic context.\n- You will need to give more information on the SCM. It is a strict definition which consists of a four-tuple. See Def 7.1.1 of Pearl’s book on causality. At present you are not giving the correct definition as there are pieces missing w.r.t. your description of the SCM.\n- Your language for describing the adjacency matrix is a bit confusing. If i and j are denoted by a 1 in the matrix that means that there is an arc from i to j in the causal diagram or that there is an edge which exists i and points to j. Phrase it in this relation instead of saying a ‘causal relation’ as you have done, as you are fundamentally describing a DAG.\n- Line 102: you cannot say ‘using causal language’ to define the do-operator because the do-operator is not unique for modelling interventions, there are many other frameworks such as the potential outcome framework. Please revise and provide adequate references.\n- You have not discussed identifiability in section 2.2 which seems somewhat important because at present you are assuming that all your interventional distributions can be calculated from the observational distribution which is not always the case.\n\n## The license model\n\n- Explain what $H$ is in equation 2.\n- Figure 1 does not make sense. What does the dashed arrow represent? Why does it point to another arrow? You are going to have to redo that figure with much more detail and clarification because this reviewer cannot make heads or tails of it. It looks like a first-order Markov model but I am sure that that has nothing to do with the problem you are studying but equally I do not quite understand what you are trying to show with that figure.\n\n## Experiments\n\n- Line 276: you cannot say “state-of-arts”. That phrasing is used as so: ‘the state-of-the-art model’ - it is an adjective that refers to a noun, in this case ‘model’. \n- This section could be improved by adding the experimental topologies that are being targeted by the different causal discovery methods, to give the reader an idea of the complexity involved.\n\nLIMITATIONS:\nPlease see the Questions section for the main review. I would be happy to increase my score but my concerns above would have to adequately addressed in order to do so. This is not nitpicking (which it surely will be construed as from the authors) but the paper is genuinely very difficult to follow for the reasons listed but could be much improved with some simple proof-reading and revisions.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a method for active causal discovery with \"multi-fidelity\" oracles. \nHere multi-fidelity refers to the option to request outcome labels for a given experiment from a set of oracles with different quality levels. \nThe method extends causal experimental design methods where experiments are defined by a given variable and value to this (to my knowledge) novel problem setting. \nAs such, their method views an experiment as consisting of a triple: (variable, value, fidelity) and defines a mutual information objective for experiment selection.\nThe method also includes a \"cascading fidelity model\" accounting for information shared between fidelity models.\nThey propose an \"$\\epsilon-$submodular\" method for multiple-intervention experiments.\nThey empirically validate their results.\n\nSTRENGTHS:\nThe paper makes novel non-obvious contributions, which I have summarized above. To my knowledge this is the first exploration of multi-fidelity active causal discovery, and they have provided a clever and principled solution. I cannot speak to the correctness of the improved greedy method, but the rest of the paper does not seem to have major flaws.\n\nWEAKNESSES:\n## Minor Concerns\n\nI really only have minor concerns, which I hope will help to improve an already excellent paper.\n\n1. **Writing Quality**. As an example, \"which is fundamental for many real-world applications, ranging from health caring, education, to drug discovery, and protein synthesis,\" is a little awkward and could be improved as, \"which is fundamental for many real-world applications, including health care, education, drug discovery, and protein synthesis.\" Another example would be  that \"In specific\" can be replaced by \"Specifically\" or \"In particular.\" Another would be starting a sentence with a citation should use the \\citet{} function rather than \\cite{} or \\citep{}, such as on line 34. There are other examples and I would suggest using a tool like Grammarly. I do not find this to be a deal breaker, though, and some sections are written very well (e.g., 2.1).\n\n1. **Motivating Example**. The example of a high-fidelity experiment as an actual patient outcome vs. a low-fidelity experiment as a  simulated patient outcome is a little weak. At least need to explain why the simulator could be lower-fidelity. If the simulator has domain knowledge, why not incorporate that knowledge into the causal model rather than try to infer that knowledge? Is it a black box? As alternatives, maybe consider spacial or temporally measured outcomes as examples of low vs. high-fidelity. For example, perhaps a lower spatial or temporal resolution measurement vs. a higher spatial or temporal measurement? A satellite image at 10-mile resolution vs. 1-mile resolution. An MRI from a 1T machine vs. a 7T machine? Maybe alternatively, a radiologist's assessment of malignancy vs. a biopsy result?\n\n1. **line 91**. covariance -> variance\n\n1. **f**. You use $f$ to define the SCM functions and the acquisistion function, which is a little confusing. \n\n1. **Active Causal Discovery**. Since you work on identifying the adjacency matrix and the  SEM parameters, I'd use causal experimental design. Only because causal discovery has focused more on just identifying the adjacency matrix. But, active causal discovery does have a nice ring to it.\n\n1. **Duplicated references**. There are references with multiple but distinct entries.\n\n1. **Licence** is a bit arbitrary as a name.\n\nQUESTIONS:\nWhat issues would arise if the cost $\\lambda$ and the mutual information $I$ are not on similar scales?\n\nCan you know the scale of the mutual information? Is your mutual information estimator reliable?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this manuscript, the authors propose a novel method for active causal discovery (ACD) in a multi-fidelity setting, where experiments with different cost and accuracy can be designed and performed for network intervention for the purpose of accurately learning the causal structure.\nFor this multi-fidelity ACD (MFACD), the manuscript proposes License - a Bayesian framework for \"Multi-fidelity\nactive learning for causal discovery\".\nLicense adopts an information-theoretic acquisition function motivated by the popular Bayesian Active Learning by Disagreement (BALD) for the prediction of the best experiment, which consists of selecting the node for intervention, its value, and the fidelity.\nFurthermore, a cascaded fidelity model is proposed to capture the correlations between the experimental outcomes across different fidelities.\n\nSTRENGTHS:\nThe problem of ACD is widely studied in a number of fields, especially so in life science to uncover the regulatory relations among genes.\nNetwork interventions are routinely performed in labs to unveil the causal structure of the network, hence designing experiments that can accurately uncover the causal relations among nodes and minimizing the overall cost of designing and carrying out the experiments is an important problem.\nMulti-fidelity experiments may often be considered in practical settings to maximize the value of information acquired by the experiments based on a given experimental budget, and the MFACD problem tackled in this manuscript is therefore of practical importance as well as theoretical interest.\n\nThe proposed acquisition function for predicting the best experiment is reasonable, as its acquisition function is motivated by the widely popular active learning scheme - BALD - which is normalized by the experimental cost to assess the cost-normalized value of information that may be attained by a given experiment.\n\nThe evaluation results in the manuscript shows that the proposed License method may have potential advantages over some alternative schemes.\n\nWEAKNESSES:\nWhile the problem being tackled in this current study is very interesting and is both of practical importance as well as of theoretical significance, there are several major concerns regarding the current manuscript as summarized below.\n\nAlthough the concept of designing multi-fidelity experiments to maximize the accuracy of ACD while minimizing the overall experimental cost is important, the type of multi-fidelity experiments considered in this work is not well motivated.\n\nFor example, the authors assume that there are multiple SCMs with different fidelities.\nHowever, it is unclear how the accuracy of the SCM at a specific fidelity is defined and what parameters govern this accuracy.\n\nAnd in practice, how does one construct or obtain multiple SCMs at different fidelities?\nWhere does the different experimental cost arise, if all SCMs are of the same form with different fidelities?\nSince the manuscript doesn't give any practical example of how such multi-fidelity models may be constructed or accessed and how (and why) they may differ in terms of the acquisition cost (of the experimental outcomes), it is unclear whether the proposed method and the derived results may be applicable in any practical setting.\n\nFor example, in drug discovery, it is typical to assess the efficacy of a drug candidate against a specific target using multi-fidelity models, starting from a fast ML surrogate model, and then to a high-throughput docking model, then use molecular dynamics simulations to assess the binding affinity between the drug candidate and the target.\nNaturally, there are differences in terms of the computational cost as well as the accuracy, hence decisions need to be made as to which model should be incurred to maximize the value of information on the computational investment made.\nUnfortunately, the current manuscript does not provide any real example to motivate the proposed method and its problem setting, making it difficult to see how the proposed approach might be applied.\n\nFurthermore, it is natural that multi-fidelity experimental outcomes will be correlated to each other (at least to a certain extent) and it is a good idea to explore this property when learning/constructing the multi-fidelity models and utilizing them in MFACD experimental design.\nHowever, in the current study, it is unclear how leveraging such correlations across different fidelities actually translates to the efficacy of the designed experiments and/or savings in terms of the experimental cost.\n\nThe description of \"extension to multi-target intervention\" is also somewhat confusing.\nBased on the technical descriptions in 3.3. and the equations (7)-(9), it appears that while the experiments (or interventions) are designed in sequence (hence referred to as \"a series of experiments\"), the interventions are being applied to the network \"simultaneously\".\nHowever, as the actual \"multi-target intervention\" setting being investigated in the current study is not clearly described (or described in a potentially misleading manner), there is some ambiguity in the experimental setting which needs to be clarified.\n\nTheorem 3 states that \"For any two experiments es and et, if the corresponding samples xs and xt are\nε-independent given φM, {es, et} and D, then I(·; φM |·, D) is ε-submodular.\"\nHowever, how can one actually guarantee whether a given pair of samples xs and xt are indeed ε-independent, considering the unknown and complex causal structure underlying these samples?\nThis needs to be elaborated more clearly.\n\nFinally, the experimental results shown in the current study are very limited, and far from convincing to demonstrate the general applicability of the proposed License method to general MFACD problems.\nFurthermore, the multi-fidelity SCMs used in these evaluations, the number of fidelities considered, and the respective experimental/intervention costs are not clearly described, which makes it difficult to understand how the overall evaluations have been carried out.\n\nQUESTIONS:\nPlease see the questions and concerns in the above section.\n\nLIMITATIONS:\nThe broader impacts of the current study and its limitations are not discussed in the current manuscript.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an approach for Bayesian active causal discovery with multi-fidelity observations. This approach has two main components: (1) a cascade probabilistic model handling the correlation between fidelity levels, and (2) a cost-aware information-theoretic acquisition function, which quantifies the mutual information (per unit of cost) between the causal graph and an observation at a given input location and fidelity level. An extension to the multi-target setting, where multiple nodes can be intervened simultaneously, is considered. In such a setting, a seemingly natural choice is to select the nodes to be intervened in a greedy fashion by iteratively maximizing the mutual information. However, submodularity does not hold, so the classical approximation guarantee is not obtained. To alleviate this issue, the notions of $\\epsilon$-independence and $\\epsilon$-submodularity. The high-level idea is to select points with low mutual information so that submodularity holds approximately. The proposed approach is shown to significantly outperform various baselines across three test problems.\n\nSTRENGTHS:\n1. The problem considered by this paper is of significant practical relevance.\n2. The proposed approach is technically sound.\n3. This paper is very well written overall.\n\nWEAKNESSES:\nMy only significant concern about this paper is its empirical evaluation, as there are several details I could not figure out. See questions 1 and 2 below.\n\nQUESTIONS:\n1. How is $\\epsilon$ chosen? More generally, how were all the algorithm hyperparameters chosen?\n\n2. I believe some of the datasets used are, in principle, not multi-fidelity. How was this addressed?\n\n3. It would be helpful to include figures showing the causal graph structure of the test problems considered.\n\n4. In the context of causal optimization, the term \"target\" is often used to denote the node to be optimized. Thus, I believe the term \"multi-target\" is misleading in its use here. Perhaps \"batch\" would be more appropriate.\n\nLIMITATIONS:\nThe potential negative societal impact was adequately discussed (in the supplement). The authors also mention some interesting ways to improve their method. However, I believe two important limitations were not addressed:\n\n1. Computational cost of the proposed approach vs. the standard approach.\n2. Robustness to the choice of the algorithm hyperparameters.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-14"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper addresses the problem of active causal discovery with multi-fidelity oracles, where experiments can be done based on different costs, precisions, and reliabilities. The paper formally defines the task of multi-fidelity active causal discovery and proposes a Bayesian framework consisting of a mutual information-based acquisition function and a cascading fidelity model. The paper also extends the framework to the multi-target intervention scenario and introduces a constraint-based fidelity model to validate the greedy method. The effectiveness of the proposed model is demonstrated through extensive experiments.\n\nSTRENGTHS:\n(1) The paper addresses an important and practical problem of active causal discovery with multi-fidelity oracles, which is more realistic than previous single-fidelity settings.\n(2) The proposed Bayesian framework, including the mutual information-based acquisition function and the cascading fidelity model, provides a novel and practical solution to the multi-fidelity active causal discovery problem.\n(3) The extension to the multi-target intervention scenario and the introduction of the constraint-based fidelity model further enhance the applicability of the proposed model.\n\nWEAKNESSES:\n(1) The evaluation and ablation studies for the proposed method are limited, and more comprehensive experiments could be conducted to strengthen the empirical findings.\n(2) Have you considered other evaluation metrics besides SHD, AUPRC, and MSE? How does the proposed model compare to other state-of-the-art methods in terms of these metrics?\n\nQUESTIONS:\nHave you considered other evaluation metrics besides SHD, AUPRC, and MSE? How does the proposed model compare to other state-of-the-art methods in terms of these metrics?\n\nLIMITATIONS:\nNo potential negative societal impact of their work.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-26"}