Causal Modelling Agents: Causal Graph Discovery through Synergising Metadata- and Data-driven Reasoning
Authors: Ahmed Abdulaal
adamos hadjivasiliou
Nina Montana-Brown
Tiantian He
Ayodeji Ijishakin
Ivana Drobnjak
Daniel C. Castro
Daniel C. Alexander
Venue: ICLR 2024 poster (2024)
Keywords: Causal Reasoning
Causal Discovery
Structural Causal Models
Large Language Models
Primary area: causal reasoning

Abstract: Scientific discovery hinges on the effective integration of metadata, which refers to a set of 'cognitive' operations such as determining what information is relevant for inquiry, and data, which encompasses physical operations such as observation and experimentation. This paper introduces the Causal Modelling Agent (CMA), a novel framework that synergizes the metadata-based reasoning capabilities of Large Language Models (LLMs) with the data-driven modelling of Deep Structural Causal Models (DSCMs) for the task of causal discovery. We evaluate the CMA's performance on a number of benchmarks, as well as on the real-world task of modelling the clinical and radiological phenotype of Alzheimer's Disease (AD). Our experimental results indicate that the CMA can outperform previous data-driven or metadata-driven approaches to causal discovery. In our real-world application, we use the CMA to derive new insights into the causal relationships among biomarkers of AD.

Published as a conference paper at ICLR 2024
CAUSAL MODELLING AGENTS: CAUSAL GRAPH DIS- COVERY THROUGH SYNERGISING METADATA- AND DATA-DRIVEN REASONING
Ahmed Abdulaal
Adamos Hadjivasilion'
Nina Montaña-Brown' Tiantian He*
Ayodeji Ijishakin!
Ivana Drobnjak!
Daniel C. Castro² Daniel C. Alexander
Centre for Medical Image Computing, UCL, London, United Kingdom
2 Microsoff Research, Cambridge
ABSTRACT
Scientific discovery hinges the effective integration of metadata, which refers to a set of conceptual operations such as determining what information is relevant for inquiry, and data, which encompasses physical operations such observation and experimentation. This paper introduces the Causal Modelling Agent (CMA), a novel framework that synergizes the metadata-based reasoning capabilities of Large Language Models (LLMs) with the data-driven modelling of Deep Struc- tural Causal Models (DSCMs) for the task of causal discovery. We evaluate the CMA's performance on number of benchmarks, as well as on the real-world task of modelling the clinical and radiological phenotype of Alzheimer's 's Disease (AD). Our experimental results indicate that the CMA can outperform previous purely data-driven or metadata-driven approaches to causal discovery. In our real-world application, we use the CMA to derive new insights into the causal relationships among biomarkers of AD.
INTRODUCTION
Scientific discovery the output of successful scientific inquiry, and its objects include events, causes, processes, and hypotheses. Arguably, the main mechanism of scientific reasoning is the generation novel hypotheses that align with or explain observed datasets, or that allow for the derivation of testable consequences (Schickore) 2014). The 'pragmatic logics of discovery' refer to a set of reasoning strategies which are used in knowledge generation (Letseka & Zireva 2013). and include both mental operations such determination of relevant information for given inquiry, and physical operations such as observation and experimentation (Schickore 2014 Schiller 1917). We refer the former operations as 'metadata" and the latter 'data', Both metadata and data foundational to producing, communicating, and validating conjectures, and are important aspects of several theories of the scientific method (Hanson) [965] Godfrey-Smmi Smith 2009). Whilst we can Directed Acyclic Graphs (DAGs) in several fields (Spirtes) al. 2000 Sachs et al., 22005) Zhang et al. (2033). we are often faced with the challenging problem of inferring causal structure from its empirical implications, which is known the causal discovery problem (Peters et al. 2017).
represent conjectures
hypotheses
There number of algorithms which attempt solve the causal discovery problem by identifying the correct DAG given a dataset (Zheng et al. 2018 Yu et al.) 2019 Nauta et al., 2019). However, in attempting to identify the generative process by leveraging asymmetries in the numerical data alone, even state-of-the. -ottte--rttttt--- causal discovery algorithms can be ineffective on real-world datasets (Tu et all)
2019 Huang al.] 2021b Kaiser Sipos (0022,, and can struggle in the setting where the data consists of as few as five synthetic variables, where each pair of variables can at most have a single confounder (Ashman al. 2023).
More recently, it was demonstrated that Large Language Models (LLMs) establish new state-of-the art performance on multiple causal benchmarks including counterfactual reasoning, actual causality, and causal discovery [Kıcıman et al. 2023 Lampinen et al. 2023). is hypothesized that LLMSs are able to capture domain knowledge (as encoded in natural language), which can then be translated into causal graphs or used to identify background causal context. In essence, this describes metadata- based reasoning; a task previously assumed to restricted to humans (Sahu et al. 2022 Trott al.)
2023
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Coextensive the causal discovery problem, causal modelling has seen numerous innovations in creating increasingly flexible models capable of causal reasoning on complex or even multi-modal data. For example, recent advances in probabilistic generative modelling have led to the emer- gence of Deep Structural Causal Models (DSCMs) (Pawlowski et al.] 2020; Khemakhem et al.) 202T (Sanchez & Tsaftaris [022]] Dash al., (202)) which combine modular Deep Learning (DL.) elements with Structural Causal Models (SCMs). DSCMs can perform interventional and counter- factual queries in high-dimensional data settings, including in the imaging space (Pawlowski al.) 2020). However, they are limited in that they require the causal relationships between the variables (i.e., the causal graph) to be known a priori, which is rarely the case for real- world problems. An- other limitation is the assumption of no unmeasured confounding, which requires yet more flexible types of graphical models (such as chain graph models) to account for hidden variables.
In this work, we investigate the effectiveness of combining LLM--ased methods with generaliza- tion of the DSCM framework by proposing the Causal Modelling Agent (CMA). The CMA com- bines the data-based modelling from DSCMs with the complementary and distinct metadata-based reasoning that LLMs utilise for the task of causal discovery, including for multi-modal datasets. Combining LLM- --ssedd methods with data-driven causal modelling approaches is beneficial in two main ways: 1) The LLMs can act as proxies of human knowledge and allow for the efficient explo- ration of causal graph space, and 2) data-driven causal methods can allow LLMS to formalize, com- municate, and ground their reasoning (Kıcıman et al. (002)). We assess this framework on a number of synthetic experiments and causal discovery benchmarks, before applying to the real- world task of modelling the clinical and radiological phenotype of Alzheimer's disease (AD). Our main con- tributions are: 1) A unified framework for causal discovery which combines LLMs with deep prob- abilistic graphical models, and in which the LLMs act as priors, critics, and post-processors over the training of such models; 2) generalised approach for the automatic construction of DSCM from an experimenttdescriiiion file, which allows for flexible experimentation; 3) a novel modelling strategy which parametrises chain graphs with deep learning elements to account for unmeasured confounding; 4) in our real-world application, use the CMA to derive new insights into the causal relationships among biomarkers of AD.
RELATED WORK
Deep Structural Causal Models A multi-modal DSCM framework is proposed by Pawlowski et al. (2020). Whilst previous work has extended DSCMs and similar causal models to number of application areas (Reinhold et al., 202T, Li al., 2023a previous DSCMs require that the causal graph be known priori. As a general model class for counterfactual inference in multi- data, DSCMs do not directly attempt to solve the causal discovery problem and have not previously been used as part of a causal discovery algorithm. Additionally, there is no method with which to handle unmeasured confounding/associaiiee relationships.
LLMs and Causality Data-driven causal reasoning methods are well-established (Li et al., 2023b), but LLMs* application in this field is recent (Willig et al.] 2022). īZečević et al. (2023) Tound that LLMS may recite causal knowledge instead of reason per se, with improved performance noted under Chain of Thought (CoT) prompting (Wei et al., 2022). ) Yadlowsky et al. (2023) showed LLMs' capacity to learn new tasks deteriorates with tasks increasingly divergent from their pretrain- ing data. Contrarily, (Lampinen et al.) 2023) observed that transformer- based agents can passively learn and apply generalizable causal strategies if allowed intervention during tests. In light of this nascent research field, a number of encouraging results have been demonstrated for the task of causal discovery. For example, Long al. (2023b) demonstrated that LLMs can construct correct 3-4 vari- able graphs. TTu et al. (2023) considered causal discovery for a neuropathic pain dataset (Tu et al. 2019) using LLMs alone. Choi et al. (2022) demonstrated that LLMs can produce a prior hypothesis which improved the accuracy of data- based causal discovery algorithms, and Zhiheng et al. (2022) attempted to partially redefine the causal discovery problem such that included relevant metadata. In these cases, LLMs were used in one stage of the causal modelling process, centring mostly around producing a causal graph alone or creating a causal graph which is used as a prior, for example in Ban et al. (2023). Long et al. (2023a) demonstrated that LLMs can reduce the size of Markov equivalence class assuming an optimal output from a discovery algorithm; their work viewed LLMs as post-processing step alone. Kıeıman et al. (203) investigated the graph discovery capabilities of LLMs over a broader set of real-world datasets; however, they did not assess combinations of L.LMs with existing causal methods.
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Agents and Reasoning In our setting, an agent is an LLM which has access or more 'tools', such as an internet search engine. We briefly summarise recent agent frameworks for reasoning tasks and compare them with our agent, whose tool is a causal modelling framework. The ReAct framework (Yao et al. 2022) aimed to combine reasoning and acting capabilities in language models by using reasoning traces design and update action plans. Reflexion (Shinn et al., 2023) built on ReAct by adding a 'self-reflection mechanism to aid in inferring future actions, and AutoGPT (Firat& & Kuleli 2023) a framework which decomposed overarching goals into sub-goals, carrying them out in a ReAct-like loop. DERA (Nair et al., 2023) attempted to enhance task completion using dialogue between two LLM agents, and Generative Agents (Park et al.] (023) stored agent experiences as memories which can be retrieved for future planning, however actions here were not executable. An extension of these ideas was Voyager (Wang et al., 2023), an LLM- embodied agent which combined an automatic curriculum with an iterative prompting mechanism. These frameworks did not consider LLMs for the task of causal modelling, or indeed for inferring causal structure from its empirical implications.
Additional related topics include works by Feder et al. which introduced the sta- tistical challenge of estimating causal effects with text (Feder et al., 2022). and investigated how language representation models can effectively learn a counterfactual representation for given con- cept (Feder et al. 202T)_ Veitch et al.[ (202T) investigated counterfactual invariance in the context of text classification, and Abraham et al. (2022) introduced benchmark for assessing concept-based explanation methods in natural language processing (NLP). Mind's 's Eye (Liu et al. (022) attempted to ground language model reasoning through simulation. The MuJoCo physics engine was used to simulate outcomes for given physics questions, and the results were passed to the LLM to improve model reasoning ability. However, this process did not iterate nor consider a causal modelling task. To the best of our knowledge, this is the first work which combines I.LM-based methods with gen- eral deep causal modelling approach in which the I.L acts as prior, critic and hypothesis engine, as well as post-processor. Additionally, in contrast to previous work (Long et al. [202aa] Ban al., 2023), the CMA does not make any explicit assumptions about the initial graph structure, and is assessed on complex data scenarios beyond synthetic cases.
Related topics
3 CAUSAL MODELLING AGENT (CMA)
We introduce the CMA framework (Figure D which unifies metadata- and data-based modelling paradigms to reason over a dataset and its associated data-generating process. Conceptually, the framework is defined as function that takes input a dataset D and associated metadata M, which represents the knowledge in the LLM's training corpus, and outputs a trained causal model and associated causal graph. The framework makes use of four concepts: 1) Hypothesis generation, 2) Model fitting, 3) Post-pree Post-processing, and 4) Hypothesis amendment.
Hypothesis generation In hypothesis generation, we produce a causal graph by use of an LLM. The only required input is set of variable names. In our case, during the first iteration (t 0), the language model function LLM acts as a prior which proposes a representation of a causal graph I:-n of the current variables given metadata M and an empty graph G (that is, a causal graph with no relationships between variables): Gı=0 LLM M). Graph G==0 can be encoded into an appropriate structured format Gi = e(G:=o), where e(-) is an encoding function and Gi is a structured causal graph (Figure (h))
Model fitting At iteration t, the model fitting stage uses a data-driven approach to calculate metric of fit F.. As an example, the metric of fit could represent the log-likelihood of a dataset D under the model, given structured graph Gj: F, log Poş (D). Indeed, this will be the metric of fit considered in this work. In our instantiation of a CMA, we use a modelling approach which requires that account for the order in which modules misspecifications. Therefore, let I. be a function which takes as input Gi and transmutes it into a (correctly specified) computational graph CG,, that is, CG. K(G)). Then, the model fitting stage is defined:
constructed to avoid errors and model
F, == log Po;
ME CGt),
where ME E is 'Modelling Engine' function which is described in more detail (alongside the moti- vations and specific choices for the function XC) in Appendix I. (FigureTe) The model fit F, is stored at this stage.
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Figure A schematic of the CMA. The CMA designs hypothesis which is represented in ISON file (Panels A and B), which is then transmuted into a computational graph and trained on data (Panel C). The data can be a set of scalar variables or a combination of scalar variables and images. The CMA then post-processes the results, producing a 'memory'. which keeps track of previous amendments and their impact on model fit (Panel D). This process then iterates. The CMA can critique hypotheses on whole-graph basis (Panel A), or per-edge basis (Panel B). During the local hypothesis amendment phase, an optional Retrieval Augmented Generation (RAG) pattern can be used (Panel B; dashed black) to provide additional context for decision decision-making. The red rectangles indicate LLM-based modules, whilst the blue rectangle indicates data-driven modelling.
nottpproesssig The post-pnn post-processing stage used produce a 'memory' μ.. This stage requires as input the graph at the current iteration Gj. the graph at the previous iteration G and their associated metrics of fit, Fi and Fi-1, respectively. The function LLM,, produces a memory at iteration He
( Fi, Fi-1). Memories encode information about changes to the causal graph, their implications, and their impact on model fit, and are stored at this stage.
Hypothesis amendment Hypothesis amendment is divided into two phases, which call the 'global' and 'local' phases. In the global phase, the function LLM acts as a critic which proposes set of amendments Agish by taking input the (structured) graph at t 1, metadata M, and an 1t-1;; 1 M). This phase is expected to keep track of
optional memory ft-1: Agtsh = LLM high-level relationships and account for the overarching structure of the DAG, and can be guided by previous amendments through the memory system (Figure [p. It should be noted that the hypothesis generation stage can be seen as a special case of the global amendment phase, which takes an empty graph as input (with no memory).
The local phase considers pairwise comparisons between vertices u, U. V, u o U, where V the set of vertices in the structured causal graph. This phase makes amendments Are as:
Are v) (ADJUS((,, u; M) if edge exists between u and v AASESS((, v; M) otherwise,
where ADJUST(U, v; M) is an LLM--ased function that outputs one of three actions: (keep, remove, reverse), and AsSESS o; M) is similar function which outputs one of the fol- lowing options: (No direct causality, + v. u). We can augment the metadata of the LLM to UC, where C is additional domain-specifie context. This is known as a Retrieval Augmented Generation (RAG) pattern (Lewis et al.) 2020) and be used to encourage enforce specific relationships according to domain expertise (Figure[Tp) This is expected to introduce edges (under specific domain considerations), and otherwise to critique, validate, and/or prune the edge outputs from the global phase. Hypothesis amendment at iteration I, HA,, can be seen as compositional set of amendments the structured graph from iteration 1:
HA, Pt-1; M) (Agtoh) U (Atoc(u, u) Vu, u V, V.u # w).
(3)
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The hypothesis amendment, model fitting, and post-processing stages (Figure L I and It) iterate throughout a given experiment. The order iteration is shown in Algorithm I our implementation of the CMA, the model fitting stage creates DSCM (Pawlowski et al. 2020), which enables reasoning over multi-modal datasets. We additionally propose novel modelling approach, whereby parameterise chain graph models with DI elements represent associative but not necessarily causal links (for example, due to unmeasured confounding between the endogenous variables). Additional background DSCMs and chain graphs, as well as further implementation details, can be found in Appendix A.1
Algorithm Iterative procedure of the CMA Framework
Require: Empty graph Go, data D and metadata M Iteration
1: Go == eLLLM 0; M))
Hypothesis Generation (no memory) b. Model Fitting
2: Fo = log Pag (D)
Iteration 1
1: G == c(HA, (Go, 0; M))
Hypothesis Amendment (no memory)
2: Fi == log po. (D)
Model Fitting Post-processing
3: = (9 9ū: Fo)
Iteration 2 2
1: while Early stopping criterion do
Early stopping criterion
2: g = e(HA, (G--1, 11-1; M))
Hypothesis Amendment (with memory)
3: F. == log Psy (D)
b Model Fitting Post-processing
4: Me := LL Gi-1, Fr, Fr-1)
5: end while
4 EXPERIMENTS
Our overarching hypothesis is that the CMA effective at the task of causal discovery. The ex- perimental logic is as follows: First, we assess the constituent elements of the CMA to ensure they are appropriate for reasoning over causal graphs. The synthetic neuropathic protein experiment assesses the data-driven module of the CMA (described in more detail in Appendix A..144). We hypothesise that correctly defined causal graphs lead to higher data likelihoods under the model (up to Markov equivalence; see Appendix (.2.3) than misspecified graphs using simple DAG setting. Whilst we focus on the data-driven module of the CMA, we present additional results on LLM behavioural patterns in Appendix A.2.4
Second, we wish to assess the CMA relative to other data- or metadata-driven approaches for causal discovery. We assess the performance of various methods on three causal discovery benchmarks where a ground-truth graph is known a priori: 1) The Arctic Sea Ice benchmark; 2) the Sangiovese benchmark: 3) an Alzheimer's Disease (AD) benchmark. We develop the AD benchmark in collab- oration with 5 domain experts, with expertise in either clinical neurology (with specialist interest in AD) or neuroradiology, with specialist interest in neurodegenerative diseases. Additional details can be found in Appendix A.5.I
Finally, assess the CMA in real-world setting with multi-modal data, we apply data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), where the task is to jointly model the clinical and radiological phenotype of the disease. contrast to the AD benchmark. the ADNI dataset contains noisy real-world data representing the complex aetiological process of the disease, for which the ground-truth ultimately remains contentious (Herrup) 2015) Gulisano et al. 2018). Additionally, we demonstrate the CMA's M ability to perform causal discovery task in a multi-modal data setting (tabular data and images), which is not possible for any of the other baseline methods. We assess the implications of the causal graph proposed by the CMA and extract various insights relating to biomarkers of the disease.
4.1 SYNTHETIC NEUROPATHIC PROTEIN EXPERIMENT
Experimental setup To validate the modelling engine for the model fitting stage (Appendix A.1.4.. consider simple DAG whose node description relates to the health sciences.
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Phosphorylated-tau (P-tau) and B-amyloid are proteins which are involved in neurodegenerative processes in the human brain, and both are affected by the ageing process. We consider the DAG
'Age + AV45', 'Age- + P-tau', and 'AV45 + P-tau', as per the classical amyloid cascade hypothesis (Hardy & Higgins 1992). Here, AV45 refers to f-amyyoid levels as measured by the Florbetapir F 18
tracer following Positron Emission Tomography (PET) scanning (Varghese et al. 2013 M & Yassa 2019). Based on the synthetic distributions, we generate 10,000 independent and identically
distributed (i.i.d.) data points from the resulting graph using ancestral sampling and test the hy- pothesis that correctly defined DAG can appropriately induce the observational distribution. We train each model 100 times on the data points and calculate an average data likelihood. Additional implementation details as well as results relating to Markov equivalence in graphs and the CMA's behavioural patterns can be found in Appendix A.2
Ground Truth
DAG Perfect
DAG
DAG 3: Chain Graph
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Figure 2: Observational samples for different modelling strategies in the synthetic neuropathic pro- tein experiment. DAG represents the correct data-generating process (a perfect map). DAG 2 adds an independence statement not present in the ground-truth distribution, and DAG 3 considers a chain component between the neuropathic proteins, which represents an associative link (represented as a bidirectional edge). Row 1 illustrates the causal graphs, and Row 2 illustrates observational samples from the trained models (or ground-truth function) representing those graphs.
Results Figure 2 illlustrates the ground-truth DAG alongside three modelling strategies. We assess the three strategies by comparing the data likelihoods under each model. The first DAG (DAG 1) is perfect map of the data-generating process, which is defined an equivalence of independence statements in the graph and the ground-truth distribution. In other words, the graph matches the data-generating process. This model has an average data likelihood of 4402.68, with standard deviation of +22.700, and this acts as the reference ground-truth likelihood. The second DAG (DAG 2) removes a causal link which exists in the data-generating process, thereby adding graphical independence statement not present in the ground-truth distribution. The second model is, therefore, a deficient representation of the ground-truth distribution, and has an average data likelihood of 3857.53 + 18.98. The third DAG (DAG 3) adds chain component between AV45 and P-tau, which represents an associative but not necessarily causal link. This model has an average data likelihood of 3757.55 = 60.80. The Tukey HSD test (Abdi & Williams) 2010) is used for pairwise model comparisons whilst ensuring the overall error rate for the family of comparisons is kept to the 0.05 threshold by control of the Familywise Error Rate (FWER). There is a statistically significant difference in likelihood between all three models, with the model representing the data-generating process having the highest data likelihood. Detailed results of the comparisons are shown in Table [ As can be seen, the results validate our initial hypothesis: The model that aligns most closely with the true data-gent generatinenngggggge process produces the highest data likelihood, As expected, we find that this is only valid up to the Markov equivalence class of the ground ground-truth DAG (see Appendix A.2.3
4.2 BENCHMARKING EXPERIMENTS
Experimental setup We consider the task of recovering causal graphs on a number of causal dis- covery datasets. The Arctic sea ice dataset (Huang al. (021b) is from the field of atmospheric science and is an increasingly popular dataset for the task of full causal graph discovery (Kıcıman
et al. 2023). This dataset considers the relations of several geophysical variables to sea ice thick-

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Table Pairwise comparisons between three different graphical hypotheses for the neuropathic pro- tein experiment, using the Tukey HSD test. DAG represents the data-genen generrting process. DAGs and represent missing edge and an associative edge where a causal one should exist, respectively.
Lower Bound
Mean
Reject Null
Model A
Model
Upper Bound
P-Value
Difference
Hypothesis
DAG 1
DAG
-558.499
-531.799
-545.149
<0,001
True
DAG 1
DAG
-658.479
-631.780
-645.130
<0,001
True
DAG 2
DAG 3
-113.330
-86.630
-99.980
<0.001
True
ness (12 nodes; 48 true edges). full description can found Appendix [A.3.] The Sangiovese dataset is from the field of agricultural science and is conditional lincar Gaussian Bayesian Net- work from the popular bnle onlearn R package (Magrini et al., 2017). The DAG considers several variables that relate to grape quality in Sangiovese vineyards in Tuscany (15 nodes; 55 true edges). Additional information can be found in Appendix [A.4.] The Alzheimer's dataset is another condi- tional linear Gaussian Bayesian Network that developed in collaboration with 5 domain experts. The synthetically generated dataset considers demographic, clinical, imaging-based biomarker, and cognitive assessment variables, and how they relate the aetiology of AD (11 nodes; 19 true edges). We construct the ground-trun -ruth graph based on consensus heuristic, for which full details can be found in Appendix [A.5.1] We compare the CMA with commonly used, soate-of-the-asse art. data-driven algorithms including NOTEARS (Zheng et al., 2018). DAG-GNN (Yu et al.) 2019), and TCDF (for the Arctic sea ice dataset; more details in A.3.2) (Nauta et al. 2019). We also compare the CMA with metadata-based benchmarks as per Kiciman et al. (2023).
The experimental setup for the
metadata-based (LLM) benchmarks given in Appendix|A.33 A.3.2
Results The results of the benchmarking experiments can be seen Table 2 We report the nor- malised Hamming distance (NHD), which is defined for a predicted graph g' and ground-truth g as 1/m² E [[ + Gt,y The edges in one graph and not the other, normalised by the total num- ber of possible edges. It should be noted that the NHD depends on the number of edges reported by causal discovery algorithm. Consequently, whereby we report the ratio of the NHD with a Baseline Hamming distance (BHD). The BHD is defined as graph that contains the same number of edges as G. but all of them are incorrect. A lower NHD BHD ratio is the multiple by which the causal discovery algorithm outperforms the 'floor' baseline graph.
the approach described in Kicuman al. (2023),
Table 2: A comparison of CMA performance against a number of data-driven or metadata-driven approaches to causal discovery on our three benchmark datasets. NHD: Normalised Hamming Dis- tance, BHD: Baseline Hamming Distance, No. Edges: Total number of predicted edges. Ratio:
NHD BHD. A lower ratio (L) is better. Extended results can be found Appendix A.3.3 A.4.3
and A.5.3 respectively.
Algorithm
Arctic sea ice
Alzheimer's 's disease
Sangiovese
NHD No. Edges
Ratio (L)
No.
Ratio (1)
No. Edges
Ratio (1)
BHD
NHD
BHD
NHD
BHD
Edges
TCDF
0.33
0.37
0.89
NOTEARS (Static)
0.30
0.38
0.82
0.21
7
0.26
0.74
0.26
0.27
0.96
NOTEARS Temporal)
0.31
8
0.38
0.82
DAG-GNN (Static)
0.32
14
0.38
0.85
0.37
28
0.44
0.83
0.27
27
0.34
0.79
DAG-GNN (Temporal)
0.32
0.43
0.74
gpt-3.5-turbo gpt-4
0.42
57
0.53
0.80
0.20
18
0.35
0.57
0.41
89
0.56
0.73
0.35
58
0.51
0.68
0.11
23
0.41
0.27
0.32
67
0.49
0.65
CMA
0.25
36
0.54
0.46
0.07
16
0.35
0.21
0.23
24
0.36
0.63
The CMA outperforms causal discovery techniques which rely exclusively on data- or metadata (LLM)-driven approaches. In multiple instances, the CMA is capable of proposing additional causal relations within 'internal' variables, which we define as variables for which we have direct access to data. We give illustrative example from the Arctic sea ice dataset. The CMA proposes that there

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is 70% probability that 'net longwave flux at the surface' (LW) should have causal relationship to "sensible plus latent heat flux' (HFLX). The CMA's reasoning trace is shown in Appendix Figure Observational samples well counterfactual inference under the causal model which en- codes this edge suggest a positive relationship between the variables (Appendix [A.3.4). Whilst this relationship is not present in the original causal graph, nor are its dynamics present in the domain literature used to construct it (Huang et al. 202 increased absorption of longwave radiation (LW) produces tropospheric warming, leading an increase in sea surface temperature (SST), until SST- dependent cooling increases to establish equilibrium, which occurs through LW's s effect on latent and sensible heat fluxes (HFLX) (Bates et al., 2012). We therefore contend this is an example of missing edge from the ground-truth graph detected by the CMA. Extended results for the Arctic sea
12
ice, Sangiovese, and AD benchmarks can be found Appendix sections A.3.3 A.4.3 and A.5.3 respectively.
4.3 CASE STUDY: ALZHEIMER'S ' DISEASE NEUROIMAGING EXPERIMENT
Experimental setup In this experiment, a CMA is applied to a challenging real-world dataset from the Alzheimer's ' Disease Neuroimaging Initiative (ADNI) (Petersen al. 2010). We consider the same variables as in the AD benchmark experiment above (Section 4.2) with the exception that we link the clinical and radiological phenotypes of AD by considering how the variables might impact a brain Magnetic Resonance Image (MRI) for a given participant. This allows us to assess the effects of interventions and counterfactual queries in the imaging space. For example, if we intervene on brain volume by reducing it, we would expect reduction in brain size on the MRI image in reasonable model. We assess the counterfactual implications of the model proposed by the CMA and illustrate the framework's 's utility for identifying potentially useful external variables for inclusion in the causal graph. Additional experimental setup details can be found Appendix A.6
A: Proposed Caunal Graph
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360
Figure 3: Outputs of the CMA on the Alzheimer's Disease Neurolmaging (ADNI) dataset. The CMA proposes high-quality, biologically plausible causal graph (A), which contains insightful causal relationships which are not currently considered conventional domain knowledge (B). The model is multi-modal and produces plausible counterfactuals of brain MRIs. Best viewed zoomed in.
Results The CMA outputs trained SCM that encodes functional causal relationships between the disease covariates and the presentation of the disease in MRI images. Figure (3p illustrates the CMA's proposed causal graph, whilst illustrates the biologically plausible output of counterfac- tual queries in the imaging space for qualitative evaluation. For example, increasing age produces visible cortical degeneration in the counterfactual image (Figure 3 row 2). It should be noted that increasing the age value not only demonstrates cortical neurodegenerration but the ventricles are also expanded, which is the expected neuro-radiological result of increasing age (Dinsdale et al. 2021). Directly intervening on whole brain and ventricular volumes (Figure Be; rows and 4, respectively) leads expected morphological effects in both instances.
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As with the Aretic sea ice dataset, we note interesting edges proposed for internal variables in the causal graph. In particular, we note the relationship 'biological sex-+tau pathology'. This relation- ship was not proposed by any of the domain experts for the AD benchmarking task in Section 4.2 The graphs proposed by the experts can be seen in Appendix A.55 Nonetheless, counterfactual infer- ence using the model trained by the CMA (under the ADNI dataset) provides evidence of increased P-tau protein levels in females (Figure Bb). Counterfactually intervening on sex led to a statistically significant shift in P-tau levels following Welch's t-test of unequal variances. This was observed when intervening on both males and females (t = - < 0.001, and t 6.05. p < 0.001, respectively). Despite not reflecting current expert consensus, there is emerging evidence that such mechanism exists. In recent work by Yan et al. (2022). it was shown in both in-vitro and in- vivo models that the X chromosome-linkee linked protein ubiquitin-speciin tau aggregation patterns by deubiquitination initiated at lysine-218. Deubiquitination (removal of ubiquitin, a 76 amino acid protein), enhances tau buildup, which produces a damaging effect on brain tissue. USPIL "escapes' complete X-inactivation (Yan al. (022), meaning females exhibit greater levels than males, and delimits the causal relationship between having an XX genotype and increased tau pathology.
peptidase 11 (USPI1) alters
As the CMA leverages LLMs to produce hypotheses, it is capable of proposing potentially con- founding modulating variables for which we do not have direct access to data ('external' vari- ables). For example, whilst assessing a potential relationship between the APOE4 gene and tau pathology (APOE4->ta pathology). the CMA proposes that the TREM2 gene may be potential confounder. The reasoning trace is shown in full in Appendix [A.7 We accrue data for the soluble form of TREM2 (sTREM2) from the ADNI dataset and include it in the causal graph with directed edge to P-tau (sTREM2->Hn -tau pathology). By counterfactual inference, we find a statistically sig- nificant positive relationship between sTREM2 and P-tau (t 6.55, < 0.001). Whilst several associational studies exist with inconclusive results (Suárez-Calvet et al. 2019; Zhao et al.) 2022). we believe this analysis to be the first wtwperrraacttal--fneen------eeee
approach to analysing this re- lationship. TREM2 is an immune receptor expressed by support cells of the central nervous system Bouchhn et al. 2001). and is broken down by ADAM metalloprotease (Kleinberger et al. 2014) to produce its soluble form. Whether this breakdown (and therefore reduction of TREM2 cell surface membrane) leads to increased tau pathology, or whether sTREM2 itself exerts a specific biological function is unclear (Filipello et al. 2022). Whilst our work proposes a potential relation from STREM2 to tau deposition, the underlying mechanism is currently unknown (Filipello et al., 2022
the
5 CONCLUSION
We introduce the CMA, modular framework for causal discovery with large language models and deep SCMs (DSCMs), enabling causal discovery for mudti-modal data. Through a diverse set of experiments synthetic and real-world datasets from a range of scientific fields (agriculture, geophysics, and health), we demonstrate that the CMA outperforms existing data- or metadata- driven approaches for causal discovery. The CMA is capable of proposing insightful relationships between variables internal and external to the DSCMs, and we provide a number of illustrative examples in the geophysies and health domains.
Future work should address several important limitations. Whilst discrete variables with parents currently require continuous relaxation, DSCMs could support discrete mechanisms with a Gum- bel-max parametrisation (Pawlowski et al. (2020). Appendix C). In our AD case study, our model used 2D axial slices of the brain, which may not be optimal for producing counterfactuals based on 3D volumes; hence, incorporating 3D volume modelling in DSCMs is a potential extension. Al- though the observational distribution is determined by graph's 's Markov Equivalence Class (MEC), there is evidence that LLMs can effectively reduce the MEC by ruling out implausible graphs (Long jtt al., 2023a). We discuss these ideas in more detail in Appendix A.2.4 Finally, The CMA relies on DSCMs to fit data, which assumes a Markovian DAG. Whilst we extend the framework to in- clude chain graph elements, in practice, we found that LLMS struggle to represent chains in a fully automated manner- instead, the CMA proposes potentially bidirectional relationships and human intervention is necessary to allow appropriate Deep Chain Graph modelling. Investigating tech- niques to enable fully automated chain graph modelling, and indeed more generally extending deep SCMs to represent more flexible, non-Main Man-Markovinn causal graphs (for example, models which allow for feedback loops) represents natural avenue of future research.
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REPRODUCIBILITY STATEMENT
To increase reproducibility, we have included all implementation details in Appendix A. We also
include implementation and prompting code http::ttp//// htttps://anonymoono htwp://wmouuuuuoooso..oooo
4open. wcw.sciiesssoo science///
causa Laddsll_nodell modell aing
ACKNOWLEDGMENTS
This work is supported by an EPSRC Industrial Case grant [EP/W522077/1] and the EPSRC- funded UCL Centre for Doctoral Training in Intelligent, Integrated Imaging in Healthcare (i4health) (EP/S021930/1). AA is supported by a Microsoft Research PhD Scholarship. Wellcome Trust award 221915/2/20/Z, JPND and MRC award MR/T046422/1 and the NIHR ULCH Biomedical Research Centre support DCA's work on this topic.
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Lyu Zhiheng, Zhijing Jin, Rada Mihalcea, Mrinmaya Sachan, and Bernhard Schölkopf. Can large language models distinguish cause from effect? In UAI 2022 Workshop on Causal Representation Learning. 2022.
A APPENDIX
Summary of Appendices.
A.I
Causal Modelling Agent
A.2
Synthetic neuropathic protein experiment
A.3
3
Benchmark: Arctic Sea Ice
A.4
Benchmark: Sangiovese
A.5
Benchmark: Alzheimer's Disease
A.6
Case Study: Alzheimer's Disease Neuroimaging Experiment
A.1 CAUSAL MODELLING AGENT
In this section, we provide additional background on DSCMs and chain graph models, before intro- ducing novel parameterisation of chain graph with deep learning elements. We then provide implementation details for our global and local hypothesis amendment phases, the modelling engine, and the post-procet processccnsing phase of the CMA.
A.1.I ADDITIONAL BACKGROUND
Deep Structural Causal Models A structural causal model (SCM) e (s) consists of collection = ff......fp of mechanisms X, := (PAj, E,), j = 1, 1....,,, D, where PA, S (Xi,..., XD) (X,) assumed be direct causes (parents) of X,, and jointly independent distribution over exogenous noise (c) variables P. II P(Ea). where D is the number of vari- ables for which there is associated exogenous noise. An SCM e entails unique distribution {X denoted by Pe. SCM satisfies the Markov condition (Geiger & Pearl 1990), whereby every node is independent of its non-descendents given its parents. Bach conditional probability
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can be seen being defined by its corresponding mechanism and noise P(X.(PAa)) = fa(es: PAa)
(Pawlowski et al. 2020; Peters et al., 2017). We perform interventions by altering the mechanism for variable Xx such that X5 == f(PAs, IPPAR.CA Ex). An intervention induces a new interventional distri- bution PX Px PC &c))
(Peters et al. 2017). Interventions which place point mass on real value m are called atomic interventions and are written as _x pedooX In the scenario where we already have some observations,
define SCM in which the distribution of noise variables has been modified as ex=x == (S, p_ where Pe p=|x=x == Pexx=xx and X = x refers to specific observation of random variable X. Performing an intervention in this setting amounts to performing a counterfactual query (Pearl, 2011; Peters et al.] 2017). Computing coun- terfactual queries be performed in three steps known as abduction, action, and prediction (Pearl) 2011]: 1) In abduction, we infer the exogenous noise which is compatible with the observations x, i... compute
xx 2) Next, we perform intervention of interest, such as do(æ),1 = xX,1) 3) Finally, it is possible to infer a quantity of interest. If we are interested in quantity z, we could {[x=xxdo(rr
compute Pzz
(Peters et al. 2017: Pearl 2011 2018
For DSCMs, mechanisms parameterized by invertible DL elements that the exogenous noise variables can be computed as per the abduction step. Conditional normalizing flows (Rezende & Mohamed 2015 Trippe & Tumer 2018) are used to learn bijective mappings between the observed variables and their exogenous noise. The mappings operate in the data space, which means they are costly for modelling high-dimensional data (e-g., images). Pawlowski et al.] (2020) propose instead to decompose such a mechanism fu into invertible hg and non-invertible
functions. The noise is correspondingly decomposed e = (uk, 2k). with p(ex) = =(u))p(2z)) The non-invertible noise term 24 is computed by the recognition model of conditional variational autoencoder (CVAE) (Sohn et al., 2013). This approach has been termed an 'amortized, explicit-likclihoo mechanism'.
Additional details can be found in Pawlowski et al. (2020).
Deep Chain Graph Models Chain graphs (CGs) are a form of probabilistic graphical model which contains both directed and undirected edges. A CG can be more expressive than either SCMs or Markov networks by introducing conditional independence statements not possible by either model type alone (Barber) 2012 Lauritzen (996). CG can be interpreted as Directed Acyclic Graph (DAG) over chain components (Frydenberg) 1990), where the chain components of a graph G can be identified by first removing all directed edges from G to produce graph G'. The remaining connected components in G' (i.e., those vertices with undirected edges connecting them) are chain components т. Each component represents a distribution over the variables within it, conditioned on parental components (Barber 2012):
Px II p( | p(x+ (PA,) x II p(aa PPA)) II ((X.), eEC,
(4)
JED,
where D, the set of variables in component T with directed terms, and C, denotes the union of the cliques in r. Unlike a fully specified DAG on singleton nodes, CGs can also be used to represent potential unmeasured confounding between two or more variables. We can take advantage of this increased flexibility by operationalising them using modular DL elements. Namely,
use (condi- tional) multivariate normalizing flows (Lu & Huang 2019 Winkler et al.) 2019) as the mechanisms fi S maximal cliques. We call this model Deep Chain Graph Model (DCGM). Training objective as the DSCM. Whilst previous work has focussed on interpreting neural networks elements within them as CGs (Shen & Cremers 20200 Hwang et al. 2022), as far as we know, this is the first time that a CG is explicitly defined and then parameterized with DL modules such that it remains compatible with the counterfactual inference procedure described in A.I.I
DCGM proceeds with the same likelihood-
A.1.2 GLOBAL HYPOTHESIS AMENDMENT
The global phase implementation requires two inputs, input JSON and optional memory. During the first two iterations of the CMA, the memory is empty, as can be seen in the main text. The output of the call is a JSON with the same format as the input JSON. To ensure an appropriately structured output, we leverage 4pt-4-05 -4-0-3''''s 'function calling' capability (OpenAI, 2023b a), which represents the encoding function e(-) that operates the representation of a given graph G
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to produce structured graph G*, in the main text. The parameters of the input JSON the nodes of the imput graph, which have as properties an array called parents'. The LLM is instructed to fill the 'parents' array of each node. Where memory is included, it is embedded into the prompt directly. The algorithm and system prompt for the global phase are shown in Algorithm and Listing respectively.
Algorithm Global Hypothesis Amendment
Require: An input JSON G representing the initial graph and a Memory μ.
1: procedure GLOBAL PHASE(C(V, E))
2: if First iteration of CMA then
Initialize Il as empty 4: end if
5: if Second iteration of CMA then
6: Initialize μ. as empty
7: end
8: Extract nodes and their properties from g
9:
if y exists then
10:
Embed the memory μ into the prompt directly
11:
end
12:
for each node in G do
13:
Make API call to gpt-4-0613 with n to get updated edges Update G with the response from the API call
14:
15:
end for
return modified JSON (VV, with updated edges for each node.
17:
end procedure
Listing 1: System prompt and schema for the global phase.
Schema and system prompt for the global hypothesis amendment phase
n*m
Schema for function calling output
schema ('name': "changeDAAG' 'description': "The new improved DAG json', "parameters': {' l'type': 'object', 'properties': ('nodel'; {'type': object', 'id': 'nodel', 'parents'; l'type': 'array', 'description' List of parents of nodel. Think about what variables of ['node2', node3', ....] can affect "}}, 'node2': ('type': 'object', 'id': node2', 'parents': ...).
Base prompt
prompt f"*"Consider the follewing JSCN object: (input_ sioonddett... Are there any links you'd like to add? Any you think should be removed? Any whose direction should be reversed? Consi der directed changes of the variables in the JSON ONLY. Do not propose changes variables not in the given JSON. Output be an improved representation of the causal relationships between the variables. "nn
valid JSON that you think would
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nodes list: [nodes]
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13
14
Do not give an explanation. Only cutput JSON. Background knovledge: This JSCN describes directed acyelic graph (DAG), which describes the causal relationships between nodes. Each node may be an ancestor (have an empty parent array), or may be the child of some other nodes (where you can see the parents in the JSON)."M """
15
if memory
prompt E" [ppronpp)] in (memory)
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A.1.3 LOCAL HYPOTHESIS AMENDMENT
This section outlines the details of the local hypothesis amendment phase. As described in the main text, this phase considers pairwise comparisons between the variables. Let V be the set of vertices in the structured causal graph G For each pairwise comparison between the vertices in the graph, (u, v). V u, E V, u # v, the local phase outputs set of probabilities over all possible actions. The action space varies based on two types of pairwise comparisons: 1) 'Relationship assessment', and 2) 'Relationship adjustment". Relationship assessment assumes that there is currently no causal link between variables u and v. We ask the L.M to output three probability values based on the assumption that and are not related causally. The probabilities reflect three possible decisions:
1. 'No direct causality": There is no direct causal relationship between the variables.
2. 'u causes v²: Changing u causes a change in v in a causal manner.
3. causes u': Changing e causes change in u in a causal manner.
If the probability values for 'u causes v' and 'v causes u' within a 5% threshold of each other, and are both individually greater than the probability that there is 'No direct causality', then the CMA saves these variables potentially having 'bidirectional' causal relationship. The CMA randomly chooses a direction in this case (as long as the condition that the output is a Markovian DAG is satisfied), and implements causal edge. Following training, user can assess whether the flagged 'bidirectional' relationships warrant additional modelling with a Deep Chain Graph model. An example output of a relationship assessment is shown below:
("NO DIRECT CAUSALITY": 90, "U CAUSES v 5, "V CAUSES U": 5)
Relationship adjustment, on the other hand, assumes that a causal relationship already exists in either direction between variables u and v. The LLM outputs probability values which reflect three different possible decisions:
1. 'Keep' The causal relationship is correct in the current direction and should be kept.
2. 'Remove:: Remove this relationship completely; there is no causal relationship between and v.
3. 'Reverse': Flip the direction of this relationship. There is evidence of a causal relationship in the opposite direction to the currently proposed direction.
An example output of a relationship adjustment is shown below:
("KEEP": "FLIP": "REMOVe": 95
Naively, one could make API calls to the LLM for each CMA iteration (after each global phase) to obtain the relevant probabilities for each edge (and non-edge). However, this could be less ef- ficient and may incur significant costs. Instead, to obtain probabilities for use in decision-making from the LLM, an initial (two-stage) 'pre-computation" phase occurs before the first CMA itera- tion. We gather all relevant probabilities as a preliminary step and apply them as appropriate for each subsequent iteration of the CMA. First, for each pairwise comparison (v, u), two API calls are made to gpt-4-0611 4-0613 (OpenAl 2023b). Specifically, one call is made which assumes that the variables are independent (relationship assessment), and one call made assuming relationship already exists (relationship adjustment); i.e., + u. A run is defined as complete iteration of all pairwise comparisons in the graph (two API calls are made for each comparison). Multiple runs are made to mitigate potential variability or inconsistency in the outputs. We use gpt-4-00 3 with temperature setting of 0.6 for experiments.
In addition, the local phase can optionally utilise a Retrieval Augmentation Generation (RAG) pat- tern, whereby each edge in the DAG is described using natural language (for example, for the edge 'precipitation The edge descriptions are embedded into a vector space using the text-embedee wdweddddnnnn-aadaeeeeaaaaaaaaaaa model, where all embeddings semantic text search with a vector store to retrieve the most relevant information which relates to the current edge. Retrieval occurs by calculating the cosine similarity between the edge description and
humidity', we can write 'precipitation has causative effect on humidity'). )
normalized to unitary length. The embeddings used to perform
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the document embeddings in the vector store cos(w) (x) which simplifies to dot- product due to the normalisation. The retrieved documents can be passed into the "system prompt' additional context. Aside from the (optional) additional context, the system prompt (Listing 2 contains instructions to consider the monotonicity, temporality, subjectivity, and potential spillover effects for each relationship, well potential confounders mediating factors when outputting probability values.
The output probabilities are stored JSON format. An optional explanation which represents the reasoning trace of the model can also be saved. The output is a JSON file for every pair of variables, with the name:
pair(0) [0]pair[1] temp prompt loop (if exisss) In the second stage of the pre-computttion phase, the probabilities stored in the JSONs for every combination are grouped, averaged over, and saved into two tables, one for "empty" relationships (when there is no prior causal link assumed) and another for "existing' relationships (assuming prior causal relationship in either direction). During the CMA iterations, for any given pairwise comparison, if an edge already exists (e.g. from previous iteration), the CMA looks into the "existing" relationships table, otherwise into the 'empty" table. A final decision is made by sampling from uniform distribution and then mapping the value to the probabilities output by the LLM. Algorithms3aandpe an exposition the pre-computation and local amendment operations, respectively.
Listing 2: System prompt for the local phase. Includes output options for both existing and empty relationships.
System prompt for the local hypethesis amendment phase n**
"You are an expert {INSERT DOMAIN) with a sub-specialist interest in { INSERT SUB-FIELD). Your task is to explore the hypothesis space of the relationships that might exist between number of variables relating to (INSERT SPECIFIC TOPIC)."
7 The opticns when assuming a pre-existing causal relationship: Please decide the probaoilistic nature of the relationship using the following options:
KEEF: "The causal relationship is correct in the current direction and should be kept."
10 PLIP: "FLIP the direction of this relationship. There is evidence to suggest that it is plausible that changing (pair[1]} causes change to (pair[0])."
REMOVE: "Remove this relationship completely; that is, no causal relationship exists between [pair[0]] and (pair[1]]."
12
The options when assuming no prier causal relationship: Please decide the probabilistic nature of the relationship using the follewing options:
15
16 NO
DIRECT CAUSALITY: "There is no direct causal relationship between the (pair[01} and (pairl:.)."
changing (pair101} causes changes to [pair|1l): "Changing [pair[0]] causes a change in (pair[ll) in a causal manner." changing [pair[l]} causes changes te [pair[0]}: "Changing [pair[1]) causes change in (pair(0]} in causal manner."
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20
21
22 To make your decision, consider the following criteria: The first criterion applies if edge already exists.
Reversal and Causality: "Could the direction this relationship be inverted, such that the [pair[1]} might cause changes to [pair[0]] instead? If yes, the probability of option (FLIP) should be high." Confounding "Could this relationship be confounded by any other vari ables??
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Mediation Modulation: "Are there any additional variables that might either mediate or moderate this relationship?"
27
Monotonicity: "Do you think the relationship is monotonic?"
28 Temporality: "Is there a time-lag between causation?"
Subjectivity: "Could this relaticnship be subjective?"
Spillover: "Could this relationship be spillover?"
32
33
You may want consider the following contextual Information about {pai: [ and (pair(1)):\n\n
35
{CONTEXT)
36
Algorithm Local Hypothesis Precomputation Phase
1:
for each node V do for each node u E where u f v do
2:
3: Formulate the API requests with pair (u, u)
4: Make API call to
assuming no causal link between and
5: Store output probabilities in ISON file in ASSESSED-FOLDER
6: Make API call to 9 613 for potential causal relationship in direction ->
7: Store output probabilities JSON file in ADIUSTED-FOLDER
8: end for
9: end for
10: Process JSON files in ASSESSED-FOR DFEESEE--FOLDRR group and average probabilities
11: Store averaged probabilities in ASSESSED- TABLE CSV
Process JSON files in DOJUSTED-FOLDDER to group and average probabilities
13: Store averaged probabilities in ADJUSTED- CSV
Algorithm Local Hypothesis Amendment
Require: An input JSON & representing the output graph from the global phase. (V,
1: procedure LoCAL
2:
for each node E V do for each node u E V where u * v do directed edge exists between v and u then
3:
4:
5:
Retrieve probabilities from ADJUSTED-TABLE CSV for relationship
6:
else
7:
Retrieve probabilities from ASSESSED- -ABELE CSV for the pair (v, u))
8:
end if
Decide relationships based retrieved probabilities and threshold adding or flipping a directed edge creates a cycle then Skip this amendment 12: end if
9:
10:
end for end for
14:
return modified ISON GV(V, E') with amended hypothesis.
16:
end procedure
A.1.4 MODELLING ENGINE
use the term Modelling Engine (M to refer to software abstraction of both DSCMs and DCGMs, whereby either model type can be specified using a single JavaScript Object Notation (JSON) file which includes the name of each variable, its parent nodes, its type (binary/discrete/com tinuous/image),, and whether exists in chain component r. Because SCMs are specified by DAGs, and CGs can be interpreted as DAGs over chain components (Barber) 2012). is important to ac- count for the order in which we define the modules, they must reflect the child- parent pathways
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in the DAG. For example, attempting construct conditional network for a child node without first having constructed the network(s) for its parent(s) can lead to errors/model misspecification.
We implement the function K. from the main text using Kahn's topological sorting algorithm (Kahn 1962). In this case, the purpose of I is to sort the nodes and order network construction. As previ- ously discussed, structured graph 9 is taken input K. to produce computational graph CG2- Let V be the set vertices in the structured causal graph, and be the set of edges. The in-degree of vertex a E V defined as the number of incoming edges; in-degree(a) 0. These nodes are removed from . alongside any out-going edges they have and added to list This process iterates until there are no nodes in N, leaving sorted list in L. The computational graph is then defined by iteratively constructing the appropriate neural networks, hypernetworks, normalizing flows, and CVAE architectures where appropriate for each variable in the order given by L.
H&6 E V (b, a) E}--
We identify a list A of nodes with in-degree(n)
A.1.5 POST-PROCESSING
The post-processing phase consists of three steps:
1. In the first step, we compare the graph JSONs from the current and previous iterations. This allows us to identify differences in relationships, which we then convert into textual descriptions.
2. In the second step, compare the average data likelihoods from the DSCM models of both the current and previous iterations. This comparison allows us to gauge the impact of changes on data likelihood.
3. The final step involves using the identified differences in relationships and likelihood fit to make an API call to gpt-4-0 1111-113 using the prompt Listing 3 We instruct the LLM to generate explanation about the potential causes of the observed differences in fit, This explanation constitutes a 'memory', and is saved as a text file. The memory file can be used guide the global phase of the subsequent iteration.
The post-processing phase summarised in Algorithm
5
Listing 3: System prompt for the memory generation.
"You are an expert (INSERT DOMAIN} with a sub-specialist interest in { INSERT SUB-FIELD). Your task is to explore the hypothesis space of the relationships that might exist between number of variables relating [INSERT SPECIPIC TOPIC)."
am going to present you two JSONs in <XMI>> tags that represent di rected acyelic graphs which relate to [INSERT DOMAIN).
For each node, you see its parents in its relevant 'parents' array. Note that if two nodes have non-zero chain number, they are chain', which means they have an associative but NOT a causal relationship.
<PREVIOUS GRAPH> { (preerious_rraphttopooooo
</PREVIOUS GRAPH>
8 <CURRENT GRAPH>
</CURRENT GRAPH>
In experiment where fit these graphs onto real data,
11
12
Based on your expert knowledge, try to provide an explanation for why this mi might be the case by considering all the individual di differences between both graphs. Here is aummary of the differences you should focus en:
13 <GRAPH DIFFERENCES> (string diferences) DIFFERENCES>
14
Contextualize your analysis by thinking about the current literature on adverb}. Provide specific advice for improving the current graph using these variables, given the literature. IF there are chain diffcrences, you recommend making the relationships between the variables causal or associative? Explain your reasoning.
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Algorithm Post-processing and memory generation
Require: 2 input JSONs Gt Gi-1 and numbers Fr. Fi-1 representing the graphs and likelihood fits produced in current and previous iterations.
procedure POST-PROCESSINGCCC, 9r-1, Pt, F(-1)
2: Find differences D3 between graphs Gt, Gt-1
3: Convert D, into text
4: Calculate difference D, between Fr, Fi-1
5: Embed Dj and Dy into an API call to gpt-4-0613 to get a memory μ4
6: Save memory (4 in text file
7: end procedure
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A.2 SYNTHETIC NEUROPATHIC PROTEIN EXPERIMENT
A.2.1 DATA GENERATION
As described in the manuscript, we consider the following DAG structure: 'Age + AV45', 'Age + P-tau', and 'AV45 - P-tau'. The data-generating process is defined as follows:
age: Sge (Eige) 55 + Eage
(),
Fage
av45 Jivas (Gv45:: age) = (age* + 2v4 + 900)/00
Gv45 (.3,,.3),
(5)
tau: = füu (iiu av45, age) = (age² + av45 + e + 110)/6,
mmu (((,0..2)
A.2.2 EXPERIMENTAL SETUP
For mechanisms S = (fi...,, , fa). all continuous singleton nodes are represented by (conditional) rational spline normalizing flows (Reinhold et al. [021] Pawlowski et al. 2020), whereas continuous nodes in chains are represented by (conditional) multivariate normalizing flows (Lu & Huang 2019 Winkler et al., 2019). The flows are composed of components that fit the distribution as well as constrain the support of the output distribution. We consider two variables K and V, where K an ancestor with no parents, and V is a child of K. Their mechanisms are defined as:
K == fk(ek) = (exp AffineNormalisation o Spline,)(<x),
(6)
; fvlev; (exp o AffineNormalistionn Condiiiionl/Tannstomm. (E)))
(7)
where the subscript 0 denotes elements with learnable parameters. The Spline, transforma- tion is a first-order neural spline flow (Durkan et al. 2019), and ConditionalTransfomnn conditional transformation which might represent conditional affine transformation in the case of univariate nodes, or a combination of spline coupling and conditional spline transformations [SplineCoupling, ConditionalSllineg] in the case of chain components (Lu & Huang 2019 Win- kler et al. 2019). We require conditional flows for singleton nodes or chains with parents. Condi- tional flows use hyper-networks. which are multi-layer perceptrons (MLPs) that predict the trans- formation parameters of otherwise invertible functions. The MLPs are composed of two hidden layers, with 10 and 16 nodes in the first and second layers, respectively. All variables are singly bounded and therefore an exponential transform is applied to constrain the unbounded values fol- lowing a fixed AffinNNormalisation transformation. The AffineNormallsation has the location and scale parameters set to the logarithm of the mean and variance of the training data, respectively, and is equivalent to a whitening operation in unbounded log-space (Pawlowski et al. 2020). Note the hat in equation (8), which refers to unconstrained value (i.e., we invert the constraint transforms of exponentiation and AffineNormalisatioon)) For all exogenous noise variables ~ Pe. we use unit Gaussians as base distributions. All learnable flow parameters were optimized by maximizing the likelihood using the AdamW optimizer You et al. 2019) with a learning rate of 3 x 10-3 for 300 epochs.
is a
A.2.3 ADDITIONAL RESULTS - MARKOV EQUIVALENCE
Ground Truth
DAG 1: Perfect Map
DAG 2: Markov Equivalent
DAG 3: Depandence Map
DAG 4: Chain graph
Navi
Pins
Ape
Agp
Figure 4: Figure illustrating the Directed Ayclic Graphs (DAGs) used in synthetic neuropathic protein experiment. DAG 2 Markov equivalence to the data generating DAG (DAG 1).
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In this section, explicitly define the notion of equivalence classes for SCMs and present further numerical results as they relate to such a class.
Markov equivalence in graphs Given DAG 9 and distribution P, the distribution is said to be Markovian if it satisfies the local (or global) Markov property. The local Markov property states that all variables are independent of their non-descendants given their parents. The global Markov property with respect to G is defined as:
X L = X
for all disjoint vertex sets X, Y, Z ( denotes d-separation in the graph) (Peters et al. 2017). Let M(G) be the set of all distributions that are Markovian with respect to G. Two DAGs G; and Gj are Markov equivalent if M(G;)
MM(G). The set of all DAGs which are Markov equivalent to
a DAG g is called the Markov Equivalence Class (MEC) of g (Pearl 2011). This implies that a MEC for given DAG is the set of DAGs which entail the same set of (conditional) independence statements.
DAGS Markov equivalent if they have the same skeleton and y-structure. A skeleton is defined as the set of edges in the graph after all directed edges are converted into undirected ones, and v-structure is any structure of the form A C +- B; here, C is a collider, and A and B are not connected.
From graphical perspective,
MEC experiment We consider an expanded DAG from Figure 2 which includes a causal parent (CP) and causal ancestor (CA) of AV45. We perform a similar experiment as in section[4.]] however, we additionally consider graph from the MEC of the expanded DAG. Figure iillustrates all DAGs considered for analysis. We conducted an ANOVA test for group differences and a Tukey HSD test for pairwise assessments. The ANOVA indicates a statistically significant difference between model likelihoods (p < 0.01). Table shows the results of the pairwise Tukey HSD test. There is a statistically significant difference in data likelihood between all pairs of DAGs except the perfect map graph and its Markov equivalent graph. This would indicate that the reward signal produced by the Modelling Engine (the 'data-driven' element of the CMA; section 1.1.4) is valid only up to the MEC of the data-generating process.
Table 3: Pairwise comparisons between four different graphical hypotheses using the Tukey HSD test. Mean differences in data likelihood, p-values, lower and upper bounds, and whether to reject the null hypothesis are reported.
Mean Difference
Lower Bound
Upper Bound
Reject Null Hypothesis
Model A
Model B
P- P-luuee
Chain
D-Map
-1 124.3537
0.001
1108.7990
1139.9084
True
Chain
MEC
466.7370
0.001
-48222917
-451.1824
True
Chain
Perfect Map
466.0028
0.001
-481.5575
-450.4481
True
D-Map
MEC
1591.0907
0.001
-1606.6454
-1575.5361
True
D-Map
Perfect Map
1590.3565
0.001
-1605.9112
-1574.8018
True
MEC
Perfect Map
-0,7342
0.900
-14.8204
16,2889
False
A.2.4 ADDITIONAL RESULTS II LLM BEHAVIOURAL PATTERNS
Here, we discuss the use of LLMs in the context of MEC and highlight early evidence of the utility of using LLMS to reduce the size of the MEC by excluding implausible graphs. Finally, we present additional analyses which delimit the behavioural patterns of the LLM portion the CMA.
The role of LLMs in the context of Markov Equivalence Due to any given DAG having MEC, it is generally not possible to learn the correct graph for a given dataset of observational data alone (Peters et al. 2017). Learning the correct graph given dataset is known as the identifiability prob- lem. Classically, there have been two approaches to this issue (Kıcıman et al. 2023). First, one could assume that the data-generating process follows specific functional form under which identifiabil- ity of a single graph might be possible (Glymour et al.) (019), for example by assuming a linear functional form and adding non-gaussian noise (Shimizu al. 2006), assuming non-linear
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functional form with additive noise (hhang & Chan 2006 Zhang & Hyvärinen 2009). However, even the simple setting of a dataset with linear equations and Gaussian noise is non non-identifiable (Peters et al., 2017). The second approach is to jointly model all variables using deep learning techniques. This approach is not empirically more effective on many real-world datasets and does not resolve the identifiability issue. Kıcıman et al.] (2023) hypothesise that LLMs might represent a powerful utility to alleviate this issue by leveraging metadata-based reasoning to construct causal graphs. Indeed, there early empirical evidence that LLMs can reduce the size of the MEC of given DAG with a high probability of leaving behind the ground-trute graph when the LLMs are used as a post-processing step (Long et al.] (023a). To better understand this phenomenon, we investigate LLM behavioural patterns for reasoning over edges in the causal graph space.
Behavioural experiment Figure 5 illlstrates hypothesis amendments across three different types of relationship. We begin by considering the relationship between the degenerative proteins B- amyloid and P-tau. If the starting point is an 'empty' relationship, the CMA is inclined to output the edge Amyloid-;Tau the majority of the time. This is also the case where an edge already exists in this direction (Figure [ top-left and top-centre, respectively). This is in line with the 'amyloid cascade" hypothesis. which states that amyloid aggregation is upstream, and may also be causative of tau protein deposition (Hardy & Higgins] 1992; Gulisano et al. 2018). On the other hand, if the proposed edge is Tau Tau-+Amyloid, the CMA either removes this relationship or 'flips' to align with the amyloid cascade hypothesis a majority of the time. Nonetheless, it should be noted that this edge is 'kept' in many instances (Figure [ top-right). Overall, the CMA can be seen to prefer the causal relationship Amyloid-Taau however will consider the opposite relationship Tau--Amyloid a non- non-trivia con-trivial proportion of the time, irrespective of the initial orientation and/or existence of the edge. This likely reflects recent doubt in the literature about the veracity of the amyloid cascade hy- pothesis (Herrup 2015). and whether there might be a feedback loop running from Tau Amylodd, or whether both proteins may be confounded by other variables (Gulisano al., 2018). However, this is unlikely to be the full story. In recent work, Berglund et (2023) described a phenomenon dubbed the 'reversal curse', which exposes a failure of generalization in auto-regressive LLMs. In brief, if model is trained on 'A is B', it will not automatically generalize to 'B is A', The authors do not specifically assess the implications of this phenomenon for causal questions. Nevertheless, we hypothesize that the differences in model behaviour may in part be explained by the order in which a causal relationship is presented. Additional work is required investigate the 'reversal curse' in the context of causal reasoning.
The second row of Figure considers the relationship between biological sex and P-tau. The re- lationship Sex->-ttau has found some empirical support in recent work (Yan et al.) 2022). Where the initial relationship is empty, or already in the direction Sex-P-tau, the CMA will consider a potential causative path from sex to P-tau majority of the time, however, will also posit that there may not be a causal relationship in many cases. This relationship is interesting because one would expect that P-tau should not have causal effect on sex, and indeed if the relationship P-tau is proposed to the CMA (e.g. from a previous iteration), the model will either flip or remove it in almost all instances.
The third row of Figure 5 illustrates results for a relationship which should not plausibly exist in either direction: Biological sex and chronological age. Indeed, we see the CMA does not introduce this edge if it does not exist, and if the edge already exists (e.g. from a previous iteration), it is nearly always removed. The LLM modules of the CMA thus behave in a broadly expected manner; where relationships are not fully understood, there is often a default to current hypotheses, however, the CMA can make proposals which do not necessarily align with expert consensus in a conventional manner. Relationships which should not exist altogether are not suggested and/or removed.
Finally, we assess the effect of Retrieval Augmented Generation (RAG) patterns on causal graph construction. We use gpt-4-0613 (with no RAG pattern) for all our main experiments, which has cut-off date of September 2021. Utilizing a RAG pattern could allow it to account for more recent scientific literature. We consider the relationship between biological sex and P-tau once more, this time allowing the CMA to utilise a RAG pattern which accesses a vector store that contains embeddings of recently published abstracts to PubMed, including the works by (Yan et al.) 2022). Results are illustrated in Figure 6 As can be seen, the additional context strongly enforces the new edge Sex Sex->P-tan Pttau this underscores the large effect additional in-context learning can provide.
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Empty: amylold and Tau
Existing: amyloid -> Tbu
Existing: Tau -> amyloid
100
100
100
of resulted grapits
% of resulted graphs
resulted graphs
60
Chagpo
causality
remove
amylold causing
causing amyloid
fllp
2a
04
04
044
Temperature
Temperatune
Temperature
RAG: Biological
Biological ssx
RAG: Biological
% of resuited graphs
% of resulted graphs
% resuited
Oeiges No direct causality
Change
Owrge
remove
remove
Biological sex causing Tau
20-
keep
20-
keep
Tau causing Blological sex
flip
flip
Temperature
Temperature
Temperature
Empty: Biological Age
Existing: Biological Age
Existing: Age Biological
% of resulted grophs
% resulted graiphs
% resulted
Charges
Chargea
Change
No direct causality
remove
remove
20-
Biological sex causing Age
20-
keep
keep
Age causing Biological
flip
flp
64
Temperature
Temperature
Temperature
Figure 5: Hypothesis amendments across three different types of relationship; Relationships which could plausibly exist in either direction (Amyloid and Tau), relationships which plausibly exist in single direction (Sex and Tau), and variables which have no biologically plausible causal link (Sex and Age).
Whilst the CMA represents relationships in a probabilistic manner, a strong enforcement can be made nonetheless based on RAG pattern.
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Biological sex
Bialogical sox
Biological
resuited graphs
% of resulted graphs
% resuited
62
Charger No direct causality
Oarga
Charges
remove
remove
Bi Blological sex causing Tau
20-
keep
23
keep
Tau causing Bi Bloogiical sex
flip
flp
0.4
Temperature Biological
Temperature
Temperature
Biological <>
Tau Biological sex
of resulted graphs
% of resulted graphs
% resuited
Charger
Osrgo
Charger
No direct causality
remove
remove
Bi Blological sex causing Tau Tau causing Bl Bloogiical sex
20-
keep
25
keep
flip
flpp
0.4
Temperature
Temperature
Temperature
Figure 6: Hypothesis amendments with and without a Retrieval Augmented Generation (RAG) pat- tern for in-context learning. The additional context contains recently published information that provides evidence for a causal link between biological sex and tau pathology.
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A.3 BENCHMARK: ARCTIC SEA ICE
A.3.1 DATASET
The dataset describes Arctic sea ice concentration in relation to number of atmospheric variables such as total cloud water path, relative humidity, wind components, and geopotential height (Huang et al. 202 The full dataset was originally collected by analysing ice concentration levels provided by the National Snow and Ice Data Center, and the atmospheric data was obtained via the European Centre for Medium-Range Weather Forecasts' Integrated Forecast System (IFS) (Huang graph with 12 nodes and 48 edges was proposed to describe causal relationships between the atmospheric variables and the sea ice concentration. Full details of the dataset are provided at (Huang et al. (021a). We summarise variables in the dataset in
et al. 2021a). Additionally, a domainnkkowwed
Table 4
Table 4: Description of variables in Arctic Sea Ice dataset
Variable Name
Description of Variable
HFLX
Sensible plus latent heat flux
SW
Net shortwave flux at the surface
LW
Net longwave flux at the surface
SLP
Sea level pressure
Precip
Total precipitation
RH
Relative humidity
ul0m
Zonal (u-component) wind at 10m Meridional (v-component) wind at 10m
v10m
sea ice
Sea ice extent in the Northern Hemisphere Total cloud cover
CC
CW
Total cloud water path Geopotential heights
GH
A.3.2 EXPERIMENTAL SETUP
Data-driven methods We compare the CMA to the Non-combinatorial Optimization via Trace Exponential and Augmented lagrangian for Structure learning (NOTEARS) (Zheng et al.) (018)) DAG Structure Learning with Graph Neural Networks (DAG-GNN) Yu al. 2019), and the Tem-
poral Causal Discovery Frameworks (TCDF) (Nauta et al. 2019).
the principal hyperparameters of NOTHARS. and t. (the Ll penalty parameter and the threshold parameter, respectively). We consider the (0.001, 0.01, 0.1} and {0, 0.1, 0.3). This contains the default hy- perparameter set 10.1, 0.3] (Zheng et al. 2018). We use the implementation available at litps:: We use the default hyperparameters for DAG-GNN, and report results over two threshold parameters, == 0.3).
We perform grid hyperparameter search
hyperparameter sets:
//github. com/ wwwnnzhhnnnnottaaae
Whilst Huang et (021a) and KıcIman et al. (2023) both report the performance of 'temporal' versions NOTEARS and DAG-GNN, it should be noted that both of these techniques assume i.i.d. observations of the variables. For the 'static" models, all observations different time points are considered i.i.d., and fed directly into the algorithms. The 'temporal' versions require that the data augmented by adding lagged versions of each variable, then considering the newly created lagged versions as a set of i.i.d. observations, before running NOTEARS and DAG-GNN on the augmented datasets. The TCDF algorithm requires time-series data and is therefore given temporal data from the Arctic sea ice dataset directly.
We report LLM benchmarking results based on the approach devel- oped by KICIman et al. (2023). Specifically, we use the 'single prompt' approach ((ııııan et al. 2023 Appendix A.1, Table 14). which has been modified for the task of full graph discovery
Metadata-driven methods
(KKcmmann et al. 2023); ; section 3.2.1; p13). The full prompt alongside an example is shown in table
We define two types of error for given LLM output: 1) A missing answer tag, and 2) incor-
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Table 5: Prompt asking a single question to establish whether a causal relationship should exist between two variables, and to orient the direction of the edge as appropriate, as per Kıciman et al. 2023).
Single prompt
Template:
Which aause-and-effect relationship is more likely?
A. changing {A] causes change in (B).
B. changing {B} causes a change in {A].
C: No causal relationship exists
Let's work this out step by step way to be sure that have the right answer. Then provide your final answer within the tags ABB/swrr>A//CC /Answer>.
Example:
Which cause-and-effed relationship is more likely?
A. changing the altitude causes a change temperature.
B. changing the temperature causes change in altitude.
C: No causal relationship exists
Let's work this out in step by step way be sure that we have the right answer. Then provide your final answer within the tags <Aswer>>A/B/CC Answer>>
rectly formatted answer tag. For example, the output "<Answer>A and B</Answer>" is not correct, because the model should choose single output.
A.3.3 ADDITIONAL RESULTS BENCHMARKS
Results for the data-driven approaches on the Arctic Sea Ice dataset are reported Table The orig- inal graph contains 48 edges; the maximum and minimum number of edges found by the NOTEARS benchmark was 15 and O, respectively, with the maximum and minimum number of true positives being and O. respectively. An illustration of the best-performing NOTEARS output can be in Figure 7 The DAG-GNN algorithm discovered a higher number of total and true-positive edges with threshold of 0.1, with similar result over the threshold of 0.3. The best performing DAG- GNN output shown in Figure g Table 7 shows the results of the LLM benchmark for the Arctic Sea Ice dataset. The minimum number of predicted edges is 47, and the maximum number is 64. Error rates were as high as 16.6% with gpt 3. 5 S-tur at a temperature of 1.0. Results show that gpt-3. 3. turbo (Figure 9 has worse performance in comparison gpt 4 (Figure [TO,, and that the metadata-dii driven (LLM-based) approaches outperform the data-driven techniques. The CMA (Figure TT) outperforms both metadata- or data-driven approaches alone, with fewer false positives in particular than the LLM benchmarks.
is interesting to note that these results vary from the metadata- based results reported by (Kıciman et al. 2023). We believe this to be down to two reasons. First, Kicmman et al.] (2023) do not explicitly describe their error handling procedure, which may differ from ours (described above). Second, appears that gpt- 5-turbo and gpt-4 performance on multitude of tasks can change over time, as demonstrated by (Chen et al. 2023), which may partly explain the observed differences.
A.3.4 ADDITIONAL RESULTS II LW AND HFLX
The CMA identifies a high probability that net longwave flux at the surface (LW) has direct effect on the sensible and latent heat flux (HFLX).. The reasoning trace is given Figure 12] Whilst an observational relationship exists between the variables (as shown in Figure 13 4.72, p 0.01), to investigate this relationship from a causal perspective, the output of the model trained by the CMA shown Figure 14 This illustrates that by counterfactual inference, increase in LW leads to an increased measurement of HFLX. There is a borderline statistically significant difference between the observational and counterfactual distributions (t - 11889, p 0.059). We contend that this likely represents a missing edge from the ground-truth graph in Huang et al. (2021a).
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Table 6: Results for data-driven benchmarks on Aretic Sea Ice dataset. graph was predicted for the corresponding parameters.
Dashes indicate a cyclic
Lambda
Threshold Edges
NHD
BHD
Ratio
TP
Prec.
Recall
F1
0.001
0.001
0.1
0.340
0.382
0.891
0.304
0.091
0.092
NOTEARS
0.001
0.3
8
0.301
0.376
0.815
0.600
0.045
0.084
0.01
0.01
0.1
0.319
0.347
0.920
3
0.600
0.021
0.042
0.01
0.3
0.333
0.333
0.333
0.007
0.014
0.1
0.333
0.333
0
0
0
0.1
0.1
0.333
0.333
0.333
0.333
NOTEARS
0.001
0.306
0.375
0.815
0.750
0.125
0,214
poral)
DAG. GNN
N/A
0.424
0.396
0,211
0.132
0.162
N/A
0.3
14
0.319
0.375
0.852
8
0.4
0.063
0.108
DAG-
GNN
N/A
0.3
0.319
0.431
0.742
0.562
0.188
0.281
(tem-
poral)
МMHC
N/A
N/A
16
0.319
0.389
0.821
9
0.562
0.188
0.281
GES
N/A
N/A
43
0.4
0.479
0.826
17
0.395
0.354
0.374
PC
N/A
N/A
0.326
0.381
0.855
0.529
0.188
0.277
LiNGAM
N/A
0.375
0.389
0.864
0.286
0.083
0.129
TCDF
N/A
N/A
0.326
0.368
0.887
0.556
0.104
0.175
Table 7: Large language model benchmarks for the Arctic Sea Ice dataset.
Model
Temp.
Error (%)
Edges
NHD
BHD
Ratio
TP
Prec.
Recall
F1
gpt-3.5
0.0
0.0
57
0.38
0.49
0.77
25
0.44
0.52
0.48
gpt-3.5
0.2
60
0.42
0.47
0.88
0.4
0.5
0.44
gpt-3.5
0.4
6.1
56
0.4
0.44
0.91
0.41
0.48
0.44
gpt-3.5
0.6
9.1
54
0.4
0.47
0.85
22
0.41
0.46
0.43
gpt-3.5
0.8
4.5
57
0.38
0.53
0.71
25
0.44
0.52
0.48
gpt-3.5
1.0
16.6
47
0.44
0.44
1.0
16
0.34
0.33
0.34
gpt-4
0.0
1.5
64
0.39
0.57
0.68
28
0.44
0.58
0.5
gpt-4
0.2
3.0
56
0.39
0.54
0.72
24
0.43
0.5
0.46
gpt-4
0.4
54
0.35
0.51
0.68
0.48
0.54
0.51
gpt-4
0.6
3.0
60
0.4
0.46
0.88
25
0.42
0.52
0.46
gpt-4
0.8
6.1
51
0.37
0.42
0.87
23
0.45
0.48
0.46
gpt-4
1.0
3.0
59
0.35
0.48
0.74
28
0.47
0.58
0.52
CMA
0.6
36
0.25
0.54
0.46
24
0.67
0.5
0.57
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True Binery, Acjacency Matrix
Predicter Binary Adjacency Metrix 0000.0 1= 0.31
Confusion Matrix
HELX
9V
SU
Predp
TN
610m
Y20m
CM
-T
50
Precio
INLX
R
RH
Mia
8
Figure 7: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for the NOTEARS predicted DAG for the Aretic Sea Ice dataset (A=0.001, f=0.3). We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
True Binery Acjacency Matrix
Predicted Birary Acjacency Matrix DAG-GNM
Confusion Matrix
inx
IN
51
200m
FR
200 100
64
181x
50
Reeio
100m
C
ITXX
R
MILX
reci
Figure 8: Comparison between ground-trui truth adjacency matrix and predicted adjacency matrix for
the DAG-GNN predicted DAG for the Arctic Sea Ice dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
True Binary Adjacency Matrix
Predicted Binary Adjacency Matrix 5-t-b.
Confiusion Matrix
reLx
SUP
Rec
=30m
200m
CC
oV
SA
UN
SUP
INLX
u
a
pi
com
MILK
moce
Figure 9: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for
the gpt-3.5-turbo predicted DAG for the Aretic Sea Ice dataset. We additionally provide an illus- tration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) the third panel.
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True Binary Adjacency Matrix
Predicted Binary Adjacercy Mazrik gpt-4
Confusion Matrix
mn..c.
EL
1W
R1-
420m
-19
01-
5R
L
SUP
IF
UN
Mots
mm
Figure 10: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for the gpt-4 predicted DAG for the Arctic Sea Ice dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
True Binary Adjacency Matrix
Predicted Binary Adjacency Matrix CMA
Confusion Matrix
NELX
HFLX E
Predp
v10m
FP
SR
LI
SUP
Precip
HILX
LK SP
Presi
RI
ut0n
10m
08
50
Pecio
uten
10m
Figure 11: Comparison between ground-trutt adjacency matrix and predicted adjacency matrix for the CMA predicted DAG for the Arctic Sea Ice dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
LW -> HFLX Reasoning trace
(" ['explanation': "Net longwave flux at surface directly affects the sensible flux. As the longwave flux increases, the surface warns, leading to increased evaporation (latent heat flux) and conduction (sensible heat flux). However, it is elso plausible that surface. net longwave flux.
heat
changes in sensible and latent heat An increase in evaporation conduction could cool the surface, reducing Therefore, Uhere is a high probabilily of keeping Lhis relalionshio and a smaller probabilily ot flipping it. possibility neteorology that these variables intarconnected. Factors such surface characteristies, atnospheric hunidity, and wind speed could mediate or noderate this relationship.";
could affact the net longwave
removing relationship low, it is well established
Figure 12: The CMA reasoning trace for the relationship between net longwave flux at the surface (L.W and sensible and latent heat flux (HFLX).. The model produces an average probability of 70% causal link from longwave flux the surface sensible and latent heat flux.
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Scatterplot (observational samples) with Regression Line
08
HFLX
0.4
LW
Figure 13: Observational relationship between net longwave flux the surface (LW) and latent heat
flux (HFLX), As can seen, there a statistical association between the variables.
KDE Plots
Orgiral
Counterfactual
06
IFLX
02
0.0
12
LW
Figure 14: Output from Deep SCM (DSCM) trained by the CMA demonstrating experimental intervention on net longwave flux at the surface (LW). The DSCM includes a causal link from LW latent heat flux (HFLX). We perform counterfactual query, which sets the value of LW to be 30% greater. The resultant counterfactual distribution (dashed blue) is shifted with respect to the observational distribution (block red). The model suggests that increasing LW leads an increased measurement of HFLX.
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A.4 BENCHMARK: SANGIOVESE
A.4.1 DATASET
The Sangiovese dataset is causal discovery benchmark derived from the conditional linear Gaus- sian network model built in MMgrini et al. (2017) from data collected about grape quality in San- giovese vineyards in Tuscany. In summary, the original dataset consisted of the measurement of var- ious characteristies of grapes collected during growth and post-harvest, such as their pH. polyphenol content, or the mean grape weight. Data was collected between 2007 and 2009. Additionally, dif- ferent well-cstablished interventions for canopy management were applied to different parts of the vineyard. Three intervention types were considered; defoliation and bunch thinning (not applied or applied at 50%), and harvest time (technological or late). The original DAG consists of 15 nodes and 55 edges; the original DAG and its constituent nodes described in Table g
Table 8: Description of variables in Sangiovese dataset
Variable Name
Description of Variable
Treatment
Treatment Type Applied to Must
SproutN
Mean number of sprouts
BunchN
Mean number of bunches
Grape W
Mean weight of grapes
WoodW
Weight of Wood
SPAD06
Soil-Plant Analysis Development in June
SPADO8
Doil-Plant Analysis Development in August Normalized Difference Vegetation Index in June Normalized Difference Vegetation Index in August Total acidity of each must
NDVI06
NDVI08
Acid
Potass
Potassium content of each must pH of each must
pH
Anthoc
Total anthocyanin content of each must Total polyphenol content of each must Potential alcohol
Polyph
Brix
A.4.2 EXPERIMENTAL SETUP
The experimental setup is the same as in the Arctic sea ice benchmark, described in Appendix A.3.2
A.4.3 ADDITIONAL RESULTS
Results for NOTEARS and DAG-GNN are shown in Table 9 The maximum and minimum number of edges found by the NOTEARS algorithm were 22 and respectively, with the maximum and minimum number of true positives being 13 and I, respectively. No significant trend was observed over the NHD for NOTEARS across different hyperparameter settings. similar performance was observed for DAG-GNN and NOTEARS run at A=0.001. The best performing NOTEARS output (hyperparameters 0=0.001, t=0.3) shown in 15 with results for the DAG-GNN shown in Figure As can be seen, both algorithms show similar patterns in true positives and false negatives.
16
Table 10 shows the results of the LLM benchmarks. Whilst 5-turbo is competitive with the data-driven causal discovery approaches, gpt-4 outperforms them across several temperature settings with an average error rate of 0.25%. Figures 17 and 18 illustrate the predicted adjacency matrices compared to the ground-truth for gpt 3.5-turbe and gpt-4, respectively.
Both data- and metadata-driven approaches struggle with the Sangiovese dataset. This would sug- gest that: 1) Establishing causal structure from asymmetries in the dataset is challenging, AND 2) there a lack of evidence to support making causal claims for many of the variables, which likely re- flects less robust domain knowledge in general. As such, the CMA also struggles here and produces only marginally better results than either approach alone
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Table 9: Results for data-driven benchmarks Sangiovese grapes dataset. Dashes indicate cyclic graph was predicted for the corresponding parameters.
Lambda
Threshold Edges
NHD
Ratio
Prec.
Recall
Fl
0.001
0.001
0.1
22
0.240
0.362
0.662
13
0.419
0.102
0.164
NOTEARS
0.001
0.3
0.255
0.286
0.770
0.455
0.075
0.129
0.01
0
0.01
0.1
0.255
0.286
0.893
5
0.385
0.035
0.064
0.01
0.240
0.281
0.855
0.714
0.035
0.066
0.1
0.25
0.270
0,925
0.013
0.026
0.1
0.3
0.255
0.265
0.961
0.006
0.013
DAG-
GNN
0.398
0.439
0.907
0.183
0,202
0.192
N/A
0.3
27
0.265
0.337
0.788
13
0.317
0,102
0.155
MMHC
N/A
0.214
0.408
0.525
0.610
0.490
0.543
GES
N/A
0.291
0.393
0.741
0.417
0.294
0.345
PC
N/A
N/A
25
0.244
0.347
0.706
14
0.560
0.275
0.368
LINGAM N/A
N/A
23
0.296
0.357
0.823
8
0.348
0.157
0.516
Table 10: Large language model benchmarks for the Sangiovese dataset.
Model
Temp.
Error (%)
Edges
NHD
BHD
Ratio
TP
Prec.
Recall
Fl
gpt-3.5
0.00
0.00
89
0.43
0.51
0.84
0.31
0.55
0.40
gpt-3.5
0.20
0.00
89
0.41
0.56
0.73
30
0.34
0.59
0.43
gpt-3.5
0.40
1.50
84
0.41
0.46
0.89
27
0.32
0.53
0.40
gpt-3.5
0.60
0.00
86
0.37
0.48
0.77
0.37
0.63
0.47
gpt-3.5
0,80
3.00
88
0.39
0.49
0.79
0.35
0.61
0.45
gpt-3.5
1.00
9.10
82
0.41
0.47
0.87
26
0.32
0.51
0.39
gpt-4
0.00
67
0.32
0.49
0.65
0.42
0.55
0.47
gpt-4
0.20
0.00
69
0.34
0.48
0.70
27
0.39
0.53
0.45
gpt-4
0.40
0.00
65
0.33
0.48
0.68
26
0.40
0.51
0.45
4
0.60
0.00
63
0.35
0.47
0.74
23
0.37
0.45
0.40
gpt-4
0.80
1.50
59
0.34
0.40
0.85
0.37
0.43
ept-4
1.00
0.00
61
0.33
0.42
0.78
24
0.39
0.47
0.43
CMA
0.6
0.00
24
0.23
0.36
0.63
0.62
0.29
0.40
Binary Adjecency Matrix
Predicted Binary Adjecency 1=0031
Confusion
Bunch
Crppel
Woodw
ezvezs
Aiid
Ivtass
bix
Xethoc
p4yph
Tundh
NEved
NEVes
ied.
SpricH bundh drpan Wabdii
BUNdNN deppe Wast
noves
HOves
pr sethic bohss
Figure 15: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for
NOTEARS predicted DAG from the Sangiovese grapes dataset (A=0.001, ==0.)). We addition-
ally provide illustration of True Positives (TP), False Positives (FP), True Negatíves (TN), and False Negatives (FN) in the third panel.
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Binary Adjacency
Predicted Adjacency
Confusion Matrix
femuut
sinchu
sepew
4Dv00
Acid
Adtwes
armac
Reyph
narcte
Voods
NDVIC
NDVKS
Woodth
BunDN
NDVKS
Relass
pe
sechos
Ro R0300
Figure 16: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for
the DAG-GNN predicted DAG from the Sangiovese grapes dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) the third panel.
True Dinary Adiacency Metrix
Predicted Dinery Adjacency Matrix g, turbo
Confusion Matrix
Sproue
Bancwu
TN
SPAD09
XDV06
SA4D08
six
pl
Artho
Ryyph
Dardh
Wood
SPA500
novee
Kaid
rolyih
paypl
EURONN Coepen mosti
Polans
ge
sethic Poiyon
Figure 17: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for
the gpt-3.5-turbo predicted graph from the Sangiovese grapes dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
True Binary Adjacency Matrix
Predicted Binery Adjacency Metrix gpt-4
Confusion Matrix
WeodW
SP4D0S
Adid
Potass
Bnx
je-
Pelyh
Banck
1540006
NOVEL
NOVER
(oyjph
sceum
ROVER Acd
NOVEU
ROveR
ADd
pe
xeroc Paysh
Figure 18:
Comparison between ground-truth adjacency matrix and predicted adjacency matrix for
the gpt-4 predicted graph from the Sangiovese grapes dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the
third panel.
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Binary Adjacency
Predicted Binary Adjacency Matrix
Confusion Matrix
Sprouth
BunchN
GrapeW
NDVI6
SPAD08
NDV00
Potass
Brix
pt-
Sprouht
Burchil
Cropeit
SPADO6
NDVIDS
SPAD00
NDVID
kid
Potsss
gtt
Poppa
Sprouhti
Burchil
SPADOS
NDVID
kid
Sprouhh BunchN
Gropeit
SPAD96
NDVIBS
NDVI8
ge
Figure 19: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for the CMA predicted DAG from the Sangiovese grapes dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
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A.5 BENCHMARK: ALZHEIMER'S 'S DISEASE
A.5.1 DATASET
Variable description
The variables considered for this benchmark are described in table 11
Table 11: Description of variables in Alzheimer's Disease dataset
Variable Name
Description of Variable
APOE4
Expression level of APOE4 gene
Sex
Biological Sex of Patient
Age
Age of Patient
Education
Educational attainment (years)
Beta Amyloid protein level measured by Florbetapir F
AV45
18
P-tau
Phosphorylated-tau deposition
Brain Volume
Total Brain Matter Volume of Patient
Ventricular Volume
Total Ventricular Volume of Patient
MOCA Score
Montreal Cognitive Assessment Score
Data generation The ground-truth DAG is created in collaboration with 5 domain experts, with expertise in either clinical and/or academic neurology (with a sub-specialist interest in neurodegen- crative/Alzheimerrs disease) or neuroradiology with a subspecialist interest neurodegenerative disease. To accrue expert causal graphs, we build a front-end application in the JavaScript program- ming language (Figure 20). We include edges the final ground-true DAG which are proposed by at least 2/5 experts our consensus heuristic. Figure 21 shows individual expert-based causal graphs, and Figure 22 shows a stacked graph as well as the final ground-truth DAG.
Node labels (separated by
Add Nodes
Chain Nodes
Add Chain
Reset Zoom
Export JSON
Custom json input
Continuous
Apply nature
Chain01
Figure 20: A front-end application to create causal graphs in collaboration with domain experts. Edges can be added to the graph to create Markovian Directed Acyclic Graph (DAG) can be seen for a + b, for example. Associative relationships can also be represented using chains, for example, Chain01 contains nodes d and e, and itself represents causal parent of node f. The colouring represents applying 'nature" to the node, for example 'discrete' or 'continuous' case, red nodes are discrete and blue nodes continuous. The graph can be exported as a JSON object, which can be used for downstream analyses.
this
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Expert
Expert
SEX
age
sex
age
ventricaiar volume
ventricdiar volume
volun
brain
moca
mod
844
education
education
number brain
number
Expert
Expert
age
ventricoiar volume
ventricdiar volume
APQE4
APQE4
moca
4v45
8v45
education
education
tal
tau
slice number
brain MRI
slice number
brain MRI
Expert
Sex
age
ventricular volume
APOE4
brain brainyalur valun
moda
av45
education
slice number
brain MRI
Figure 21: Illustration of causal graphs encoded Directed Acyelic Graphs (DAGs) for each expert. The purple edges indicate unique graph-specific edges, whilst gray edges indicate that the edge exists in at least one other expert graph.
Weighted graph
Consensus graph
age,
sex
sex
ventricula volume
ventricular volume
APOE4
APOE4
bolun
brain volum
moca
moca
8Y45
av45
education
education
taú
tau
slice number
slice number
brain MRI
brain MRI
Figure 22: Left panel: A weighted graph which represents overlay of all individual expert graphs. Here, the purple edges represent edges for removal as they are proposed by a single expert. Right panel: The final concensus graph. Edges are retained if they are proposed by >2 experts.
A.5.2 EXPERIMENTAL SETUP
The experimental setup is the same in the Arctic sea ice benchmark, described in Appendix A.3.2
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A.5.3 ADDITIONAL RESULTS
Results for the NOTEARS and DAG-GNN algorithm can be found in Table The best NOTEARS output is visualized in Figure 23 whilst the DAG-GNN result is shown in Figure 24 The LLM benchmark results are shown Table 13 The metadata-basee (LLM) methods outperform the data-driven approaches. The CMA output is visualised in Figure 27 As can be seen, the CMA
outperform both data- and metadata-driven approaches alone.
Table 12:
Results for data-driven benchmarks on Alzheimer's Disease dataset. Dashes indicate
eyclic graph predicted for the corresponding parameters.
Lambda
Threshold Edges
NHD
BHD
Ratio
TP
Prec.
Recall
F1
0,001
0.001
0.1
0.222
0.296
0.75
0.263
0.132
0.175
NOTEARS
0.001
0.3
0.222
0.321
0.692
0.250
0.1
0.143
0.01
0.01
0.210
0.284
0.739
0.294
0.132
0.182
0.01
0.3
10
0.222
0.272
0.818
0,250
0.1
0,143
0.1
0.1
0.1
0.235
0.259
0.905
3
0.200
0.071
0.105
0.1
0.3
7
0.210
0.259
0.810
3
0.273
0.071
0.113
DAG-
GNN
N/A
0.1
28
0.370
0.370
1.0
7
0.143
0.206
0.169
N/A
0.3
28
0.370
0.444
0.833
7
0.143
0.206
0.169
MMHC
N/A
N/A
11
0.160
0.284
0.565
7
0.636
0.438
0.519
GES
N/A
N/A
10
0.173
0.272
0.636
0.600
0.375
0,462
PC
N/A
0.198
0.321
0.615
0.500
0.500
0.500
LINGAM
N/A
0.185
0.259
0.714
0.600
0.188
0.286
Table 13: Large language model benchmarks for the Alzheimer's disease dataset.
Model
Temp.
Error (%)
Edges
NHD
BHD
Ratio
TP
Prec.
Recall
F1
gpt-3.5
0.0
21
0.38
0.55
0.48
0.62
0.54
gpt-3.5
0.2
7.6
19
0.16
0.38
0.42
11
0.58
0.69
0.63
gpt-3.5
0.4
6.0
19
0.14
0.33
0.41
12
0,63
0.75
0.69
gpt-3.5
0.6
9.1
18
0.20
0.35
0.57
0.50
0.56
0.53
gpt-3.5
0.8
20
0.20
0.32
0.62
0.50
0.62
0.56
gpt-3.5
1.0
9.1
18
0.17
0.44
10
0.56
0.62
gpt-4
0.0
25
0.14
0.48
0.28
0.94
0.73
gpt-4
0,2
0.0
23
0.11
0.46
0.24
15
0.65
0.94
0.77
gpt-4
0.4
1.5
22
0.12
0.4
0.31
14
0.64
0.88
0.74
gpt-4
0.6
0.0
23
0.11
0.41
0.27
15
0.65
0.94
0.77
gpt-4
0.8
21
0.41
0.27
0.67
0.88
0.76
gpt-4
1.0
21
0.43
0.26
0.67
0.88
0.76
CMA
0.6
16
0.07
0.35
0.21
0.81
0.81
0.81
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Thue Binary Adjacency Matrix
Predicted Binary Adjacency Matrix (1 0.1, 2.33
Confusion Matrix
APOE4
Age
TN
AVDD
Too Levelc
Beain Molan
serbriullr
Lducation
Tou Lorals
Bro volume
NOCA Soore
XPOE4
Tou HAAUPPO
n columo
www. schume MOCA Score
A00
2015
Tax Laseett
volums
sekine
Figure 23: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for the NOTEARS predicted DAG for the Alzheimer's dataset (A=0.1, (=.3). We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
True Binary Adjacency Matrix
Predicted B nary Adjacency Matri DAC-GMN (t=0.3)
Confusion Matrix
APOT4
TN
Tho Leualc
Brats Walumne
udvaaiion
Lerals
son Volune
2015
TOU sordl
sen voknno
wentons stLmm
MOCGA Sor
2011
Vamoeuu setume.
ROCA
Figure 24: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for
the DAG-GNN predicted DAG for the Alzheimer's Disease dataset. We additionally provide illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
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True Linary Adjacency Matrix
Predicted tinary Adjacency Matrix GPT-3.5
Confusion Matrix
APULA
Sex
IN
Tou Levels
17
Brain Voture
webrecuer Mouume
Son lohano
APOZE
AMS
TOU sovil
SPOR schano
Wwntitee WAUT MOGA Soor
APOXI
sean valno
Waums
Vent MOCA
Figure 25: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for the GPT3.5 predicted DAG for the Alzheimer's dataset. We additionally provide an illustration of True Positives (TP), False Positives (FPP, True Negatives (TN), and False Negatives (FN) in the third panel.
Irue blnaar Adjacency Matrix
Predicted tinary Adjacercy Matrix GP114
Confusion Matrix
APUE4
Tou Loves
Bram Volume
Votriculr Malumn
Earation
Leves
AROIC
MODA Soors
Tou Lavee
srai velum
West wamss
MOCA
Westricude
Figure 26: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for the GPT4 predicted DAG for the Alzheimer's dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
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True Binarr Adjacency Matrix
Predicted Birary Adjacency Matrix CMA
Confusion Matris
APDE1
E
Sea
PN
Tou Leveta
Rram Vatume
Verdmesfer
Javds Bon volune youn wenne NOC.
Apoc-
MOCA Soon
2002
AOOA 5000s
Figure 27: Comparison between ground-truth adjacency matrix and predicted adjacency matrix for the CMA predicted DAG for the Alzheimer's dataset. We additionally provide an illustration of True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) in the third panel.
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A.5.4 SENSITIVITY ANALYSIS: DEGRADATION OF CAUSAL LINKS
In this sensitivity analysis we perturb the dataset variables using Gaussian noise with increasingly large standard deviation values, which has the effect of reducing the total variation explained by any given variable on its children the causal graph. The effect of these perturbations can be visualised in Figure 28 We run the NOTEARS, DAG-GNN, and CMA algorithms on each noise level and compare their performances.
L3
1.00
0.75
0.50
Add Reditoonall
0.25
Correlation
L1
-0.25
-0.50
Baseline
MOCA
-0.75
W
BV
Tau
APOE4
Sex
Aga
Education
Education
AV45
AV45
Age
Tau
Sex
BV
APOE4
W
MOCA
Figure 28: The strengths of the causal links between the variables are systematically degraded in the AD benchmark dataset. There are three noise levels which add Gaussian noise with mean and standard deviations of 0.4 for L1, 0.8 for L2, and 1.2 for L3. As can be seen, the relationships between all variables are reduced more noise is introduced.
NOTEARS results Inspecting results for NOTEARS in Tables 14 15 and 16 reveals a number
of patterns. First, causal discovery performance is worse with increasing values of the lambda pa- rameter, and this trend is consistent across all noise levels. Second, performance broadly degrades with higher noise levels. The best performance is observed at the 'L1' noise level (Figure 28 LI) with parameters ==0.001 and 1. The worst results were osberved with at the highest noise level L3, with parameters A and 1=0.3
DAG-GNN results Inspecting results for DAG-GNN in Tables 14
15 and 16 showcases the relative robustness of DAG-GNN across noise levels. The same number of true positives are detected across all levels, with less variability observed across other metrics.
CMA results Similarly to the DAG-GNN algorithm, the CMA remains relatively robust across all noise levels, however outperforms both approaches with lower NHD/BHD ratio. Indeed, is likely this improved performance is partly due to the metadata-based modules of the framework, which can still propose reasonable causal structures despite weak causal relationships in the dataset. A visual summary of these results is illustrated in Figure 29
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Table 14: Results for NOTEARS, DAG-GNN, and CMA benchmarks on the Alzheimer's bench- mark dataset- LI noise level e (0, 0.4). Dashes indicate cyclic graph was predicted for the corresponding parameters.
Lambda Threshold Edges
NHD
BHD
Ratio
TP
Prec.
Recall
Fl
0.001
0.001
0.1
14
0.247
0.346 0.284
0.714
0.217 0.157
0.131
0.164 0.098
NOTEARS
0.001
0.259
0.913
0.071
0.01
0.01
0.1
0.247
0.259
0.714
0.217
0.132
0.164
0.01
0.3
0.259
0.259
0.913
3
0.157
0.071
0.098
0.1
0.1
0.259
0.259
1.000
3
0.157
0.071
0.098
0.1
0,3
9
0.235
0.259
0.905
3
0.200
0.071
0.105
DAG-
GNN
N/A
0.1
20
0.272
0.420
0.647
0.212
0,206
0,209
N/A
0.3
19
0.256
0.333
0.778
7
0.226
0.206
0.215
CMA
N/A
N/A
0.086
0.309
0.280
11
0.523
0.647
0.579
Table 15: Results for NOTEARS, DAG-GNN, and CMA benchmarks on the Alzheimer's bench- mark dataset L2 noise level (0, 0.8). Dashes indicate cyclic graph predicted for the corresponding parameters.
Lambda
Threshold Edges
NHD
BHD
Ratio
TP
Prec.
Recall
F1
0.001
0.001
0.1
15
0.284
0.309
0.920
0.159
0.100
0.121 0.105
NOTEARS
0.001
0.3
0.235
0.259
0.905
3
0.2
0.071
0.01
0.01
0.1
15
0.284
0.284
4
0.154
0.100
0.121
0.01
0.3
0.235
0.284
0.826
0.200
0.071
0.105
0.1
0.1
0.1
0.259
0.309
0.84
0.158
0.071
0.098
0.1
0.270
0.270
0.143
0.045
0.069
DAG-
GNN
N/A
0.1
21
0.284
0.383
0.742
0.200
0.206
0.203
0.3
0.235
0.383
0.613
0.259
0.206
0,230
CMA
0.090
0.280
0.300
0.670
0.920
0.770
Table 16: Results for NOTEARS, DAG-GNN, and CMA benchmarks on the Alzheimer's bench- mark dataset L3 noise level e ~ (0 1.2). Dashes indicate cyclic graph was predicted for the corresponding parameters.
Lambda Threshold Edges
NHD
BHD
Ratio
TP
Prec.
Recall
Fl
0.001
0.001
0.272
0.296
0.916
0.167
0.100
0.125
NOTEARS
0.001
0.3
0.247
0.222
1.1
2
0.143
0.045
0.069
0.01
0.01
0.1
0.259
0.359
0.182
0.100
0.129
0.01
0.3
8
0.247
0.296
0.833
2
0.143
0.045
0.069
0.1
0.1
0.1
11
0.259
0.309
0.84
3
0.158
0.071
0.098
0.1
0.3
7
0.235
0.235
2
0.167
0.045
0.071
DAG-
GNN
N/A
0.1
21
0.284
0.383
0.742
7
0.200
0.206
0.203
N/A
0.3
18
0.247
0.346
0.714
0.241
0.206
0.222
CMA
N/A
N/A
12
0.110
0.310
0.360
8
0.67
0.62
0.64
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NOTEARS
0.8
CMA
0.7
NHD/BHD Railo
0.6
044
0.3
L11
L22
L3
Noise
Figure 29: Summary results the best performing NOTEARS and DAG-GNN configurations for each noise level compared with the CMA. At LI, the additional noise added is (0, 0.4). at L2 this is e ~ (O 0.8), and at L3 this is set to e ~ No 1.2). A lower Normalised Hamming Distance (NHD) to Baseline Hamming Distance (BHD) ratio is better.
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A.6 CASE STUDY: ALZHEIMER'S 's DISEASE NEUROIMAGING EXPERIMENT
A. EXPERIMENTAL SETUP
Image pre-processing Structural T1-weighted MRI scans were collected for all participants and linked to their relevant demographic, disease biomarker, and cognitive assessment variables. The earliest Inversion Recovery Spoiled Gradient echo sequence (SAG IR-SPGR) MRI was accrued for each participant, skull-stripped using the HD-BET brain extraction tool (Isensee et al. 2019), and bias-field corrected with the N4 software package (Tustison et al. 2010). All images
resampled to the size of the MNI ICBM152 brain atlas in the NiLcarn software package (Nil) (197 x 233 x 189) with linear interpolation. The resampled images were subsequently rigidly registered to the atlas using ANTs (Avants et al.) 2009). The middle 10 axial slices of each MRI were extracted and their intensity values were normalized by rescaling the minimum and maximum values of each slice to [0, 255]. Each 2D image slice is then saved as a PNG file for training. During training, the image slices were uniformly dequantised by the addition of Gaussian noise (Theis al. 2013). Images were randomly cropped from their original size to 192 x 192 and downsampled to 64 x 64 during training to prevent overfitting. The slices were centre-cropped during counterfactual image inference.
Imaging mechanisms Mechanisms S were defined in the same way as in the synthetic experi- ments A.2.2 By decomposing the image mechanism as per 2.1, we model the image the invertible function
Hing (Uing; PAiig) [Preprocessing CondittionllAfffic/ (PAiig) ) (1mmg)
where the Preprocessing follows RealNVP (Dinh et al., (0166). and the hyper-network for ConditionalAffinea!! is the non-invertible mechanism gimg (Zimg; PAimg) which is implemented as a decoder that outputs the bias for the InnditionalAffine,, (-) transformation with fixed logarithmic- variance of log o² = -5. We require an encoder function Ging (img: PAumg) to generate the latent Zimg- The images are therefore modelled using a CVAE architecture where both encoder and decoder functions are composed of modules of blocks of (LeakyRel. is convolutional layer, BN is batch normalisation, and (ea)))elLU(o) is leaky rectified linear unit with an angle of negative slope parameter c. For all other non-imaging mechanisms, we use similar setup as in the synthetic experiments (Appendix A.222).
BNo, Convo), where Conv
Binary variables such as biological sex require that we learn the binary probability by sampling from a Bernoulli distribution (female male 0). Discrete variables such as APOE4 status and MRI image slice number are sampled from uniform distributions (APOE4 status in {0. 1. 2, and minimum to maximum number of slices, respectively), as per Reinhold et al. (20TT). All learnable parameters in the flows and the CVAE architecture were optimised by a stochastic variational infer- ence approach to estimate the evidence lower bound (ELBO; estimated using 4 Monte Carlo (MC) samples) using the Adam optimizer (Kingma & Ba 2015) with learning rates of 10-3 and 5 x 10-3 respectively. For counterfactual inference, 32 MC samples were taken and the inference result was their average. All learnt mechanisms f: E S were fixed during inference and the single world in- tervention graph (SWIG) formalism was used to produce counterfactuals (Richardson & Robins. 2013). Experiments were parallelized across two NVIDIA RTX 3090 GPUs and one NVIDIA RTX 4090 GPU.
A.7 ADDITIONAL RESULTS
Figure 30 illustrates the CMA's reasoning trace for the relationship between APOE4 and P-tau. The model proposes that the TREM2 gene may confound this relationship. We subsequently ac- crued data for the soluble form of TREM2 (sTREM2) from the Alzheimer's 's Disease Neurolmaging dataset and trained DSCM with a causal link between sTREM2 and P-tau. We perform coun- terfactual intervention and find a statistically significant relationship between sTREM2 levels and P-tau levels following Welch's 's ANOVA test of unequal variance (F 43.52, p < 0.01). Figure 31 illustrates counterfactual interventions against the null intervention. As can be seen, there is a positive relationship between increasing sTREM2 levels and P-tau deposition.
We additionally conduct an observational analysis in two stages. First we regress sTREM2 on P-tau levels, and then we add APOE4 categorical variable. Results are illustrated in Figure 32 and
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respectively, As be seen, there is positive association between sTREM2 levels and P-tau == 5.15.p p < 0.01). As expected, patients with 2 copies of the APOE4 gene display greater levels of P-tau deposition (t 2.76, p 0.006), with the greatest levels observable in homozygous present patients (i.e., those who have two copies of the gene). Finally, we conduct an observational analysis by regressing biological sex onto P-tau levels. We can be seen in Figure 34 females have a higher level of P-tau burden (t 2.29, 0.023).
Figure 33
APOE4 -> P-tau
("explanation": "AP0E4 is najor genetic risk factor for AD and is known to influence tau pathology. Hovever,
downstream effect of unlikely that tau influences APOE4 potentially confound this the di sease prncess. Other genetir factors, such as TREM?, could relationship. Additionally, environmental factors and lifestyle choices such as diet and exercise could also play role. relationship between APOE4 andulated by other factors such as inflammatio,, oxidative stress, and neuronal lass.")
could be mediated
Figure 30: The CMA reasoning trace for the relationship between the APOE4 gene and tau protein pathology. The model suggests that the TREM2 genotype might confound the causal relationship between APOE4 and tau protein.
tau (years)
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200
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NULL +20%
+40%
Atomic intervention
Figure 31: Interventions on STREM2 level and associated counterfactual distributions for P-tau level. The interventions are set as percentages of the original measurement. Statistical annotations
represent Welch's t-tests with Bonferonni correction. **: p S 10-2 ***; p S 10-3 ****.
p S 10- ns: No statistical significance..
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Published as a conference paper at ICLR 2024
800
Tau pratein levels (ngll)
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sTREM2 (pg/ml)
Figure 32: Regression analysis of soluble TREM2 (sTREM2) and phosphrylated tau levels.
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3000 sTREM2 (pg/ml)
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Figure 33: Regression analysis of soluble TREM2 (sTREM2) and phosphrylated tau levels, adjusted for the categorical variables of APOE4 gene.
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Published as a conference paper at ICLR 2024
0.035
Male
0.030
Female
0.025
Density
0.020
0.015
0.010
0.005
0.000
0
20
40
60
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100
Tau protein(pg/ml)
Figure 34: Kernel density estimate plot of the relationship between biological sex and phosphrylated tau levels.
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Published as a conference paper at ICLR 2024
A.7.1 REVIEW OF THE EFFECTS OF STREM2 AND SEX ON THE TAU PROTEIN
STREM2 Neuropathic tau protein Triggering receptor expressed on myeloid cells (TREM2) is an immune receptor in the central nervous system (CNS). It appears to have a posi- tive role in cellular proliferation and survival (Ewers et al. 2019). It has a soluble form (sTREM2), which can be produced from the shedding of TREM2. There are a number of studies which show elevated levels of sTREM2 in AD (Knapskog et al. 2020; Toannides al., 2021 Wilson et al.) 2020). For example, SSurre-- valvee et al. (2019) demonstrated that tau pathology is associated with an increase in sTREM2 the cerebrospinal fluid (CSF). However, presence of amyloid protein alone was not associated with an increase in CSF sTREM2. There is also recent evidence of ele- vated sTREM2 levels in other pathologies including multiple sclerosis (loannides et al., 202I and Parkinson's 's disease subgroups with increased CSF tau (Wilson al. 2020).
Biological sex + Neuropathic tau protein In our analysis, the CMA (fit to the ADNI dataset) proposed that there may be direct causal link between being biologically female (XX chromosome profile), and neuropathic protein deposition. Recent work by Buckley et al. (2020) aimed to inves- tigate sex differences in tau distribution across multiple brain regions of older adults using Positron Emission Tomography (PET) scanning. They used the Alzheimer's ss Disease Neuroimaging Initia- tive (ADNI) and Harvard Aging Brain Study (HABS) datasets. Neuropathic tau protein levels were measured by use of [18F]flortauciprr (FTP), which is tracer agent that allows in vivo quantifica- tion of paired helical filament tau (Ossenkoppele et al. 2018). Their work suggested that women showed statistically significantly higher signal (greater tau levels) in multiple regions of the cortical mantle (p- < 0.007). They additionally wanted to assess whether composite FTP signals in Regions of Interest (ROIs) across the brain
associated with a more rapid cognitive decline. In their study, women with higher FTP signals had a borderline significant higher rate of cognitive decline than men (p- 0.04). Following this, it was thought that perhaps these sex differences were due to sex hormone profiles. To assess this, Wisch et al. (202T) conducted a cross-sectional neu- roimaging study which compared cortical tau deposition (using PET) between cognitively normal males and females. In addition, they also compared preclinical Alzheimer's ' pathology between fe- males who had and had not used hormone therapy (HT). They observed greater tau deposition in females. However, they also observed decreased tau burden females who were HT users, and high- lighted that this relationship should be investigated longitudinally. It was therefore unclear whether hormonal profiles modulated tau protein deposition, or whether this was more directly mediated by genomic/epigenomic markers. deposition. Tau clearance itself is controlled by methylation (Balmik) & Chinnathambi 2021), then (Esteves al. 2019), and finally ubiquitination (Flach et al. 2014).
Yan et al. (2022) demonstrated that the X-linked gene Ubiquitin Specific Peptidase 11 (USPIT)
removes ubiquitin from tau (the final step required for its clearance), which led to its aggregation in-vivo. USP11 has elevated expression in females, and correlated with tau brain pathology. Whilst
this might not yet be the full story, it an exciting research direction, as the mechanism behind the
sexual dimorphism in tau burden is thought to be foundational event in AD development Yan et al. 2022)
52


Reviews:
[{"review_id": "D9HRP9qJdW", "reviewer": "Reviewer_Jpz4", "summary": "The paper introduces a novel framework that synergizes the metadata-based reasoning capabilities of LLMs with the data-driven modeling of Deep Structural Causal Models for causal discovery. The authors evaluated the performance on benchmarks and real-world tasks. Real-world tasks were related to modeling the clinical and radiological phenotype of Alzheimer’s Disease. The experimental results indicate that the CMA can outperform previous approaches to causal discovery and derive new insights regarding causal relationships.", "questions": "1. While the authors do a good job regarding the related work, we consider this could be further enhanced by citing surveys that provide an overview of the relevant topics and domains. E.g., the authors may be interested on the following works: (a) for causal deep modelling: Li, Zongyu, and Zhenfeng Zhu. \"A survey of deep causal model.\" arXiv preprint arXiv:2209.08860 (2022); (b) for Alzheimer disease neuroimaging: Varghese, Tinu, et al. \"A review of neuroimaging biomarkers of Alzheimer’s disease.\" Neurology Asia 18.3 (2013): 239. and Márquez, Freddie, and Michael A. Yassa. \"Neuroimaging biomarkers for Alzheimer’s disease.\" Molecular neurodegeneration 14 (2019): 1-14; and (c) Huang, Yiyi, et al. \"Benchmarking of data-driven causality discovery approaches in the interactions of arctic sea ice and atmosphere.\" Frontiers in big Data 4 (2021): 642182, and Kretschmer, Marlene, et al. \"Using causal effect networks to analyze different Arctic drivers of midlatitude winter circulation.\" Journal of climate 29.11 (2016): 4069-4081.\n2. In the related work section, the authors may consider weighting the views and findings regarding LLMs and causality expressed in the following paper: Zečević, Matej, et al. \"Causal parrots: Large language models may talk causality but are not causal.\" arXiv preprint arXiv:2308.13067 (2023).\n3. When reporting results in Section 4.1, the authors measure average data likelihood and the deviation. It would be helpful to have some reference value to understand whether the reported values are good or not and why.\n4. How is the threshold for DAG-GNN selected?\n5. Table 1: align results to the right so that differences in magnitude are quickly visualized.\n6. Table 2: add up/down arrows near the reported metrics, indicating greater/lower is better.\n7. Table 2: for some algorithms (TCDF, NOTEARS (Temporal), NOTEARS (Temporal)), the authors report results only for the Arctic sea ice dataset, but no clarification is provided as to why no results are reported for the Alzheimer’s disease and Sangiovese datasets.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- the paper proposes an original approach to causal modeling\n - the paper has a good quality: benchmark and real-world tasks are considered, showing promising results in both cases\n - the paper is well structured and written, making it easy to follow\n - the topic is a relevant topic on which much research is being invested, given the new capabilities and opportunities LLMs provide to causal modeling", "weaknesses": "- we have not found strong weaknesses in the paper", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "g7aZUo2QeG", "reviewer": "Reviewer_GSYx", "summary": "This paper proposes to integrate large language models into causal discovery algorithms for multi-modal data and shows superiority of this model is shown in a number of examples.", "questions": "It could be more convincing to analyze the sensitivity of the proposed model in accordance to perturbation of the input parameters, in particular in the presence of weak causal link.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The model architecture is convincing and the extensive numerical experiments show strong promise of the proposed method.", "weaknesses": "The generalization performance/robustness of the proposed method is not completely clear. One challenge in causal discovery is the sensitiveness of the learned causal graph towards perturbation of the distributions, in the presence of weak causal link.\n\nPost-rebuttal: I thank the authors for their response and additional experiments for the case of weak causal link. I am increasing my score to 8.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "UHRG0YtLyO", "reviewer": "Reviewer_CmY7", "summary": "In this paper, the authors devised a causal discovery algorithm that utilizes LLM’s ability on causal reasoning using meta-data. In particular, they proposed Causal Modeling Agent (CMA), which iteratively updates a causal graph through: i) asking LLM for updates on current prediction of edges with previous graph update information; and 2) fitting a model constrained over the intermediately constructed causal graph (using deep learning to model causal mechanism for each variable). Through experiments on benchmark datasets (e.g., Kıcıman et al. (2023)) and a case study of Alzheimer's disease, they empirically demonstrated a potential of their framework outperforming some of causal discovery algorithms and LLMs.", "questions": "The word “metadata” is somewhat used in a mixed manner between domain knowledge already encoded in LLM and memory passed through JSON format. It should be more formally defined. \n\nResults\nGiven that cases with no edges outnumber those with edges, not predicting edges may lead to an increase in accuracy. Thus, a qualitative analysis is necessary since not predicting edges might lead to an increase in the score. Other metrics such as TPR or FDR can be reported.\n\nNovelty\nGiven the abundance of similar papers (Long, S., Piché, A., Zantedeschi, V., Schuster, T., & Drouin, A. (2023). Causal discovery with language models as imperfect experts. arXiv preprint arXiv:2307.02390., Ban, T., Chen, L., Wang, X., & Chen, H. (2023). From query tools to causal architects: Harnessing large language models for advanced causal discovery from data. arXiv preprint arXiv:2306.16902.) in the field, the contribution is not clear. \n\nI noticed discrepancies between what was mentioned and the results such as the performance of gpt-4 in the table 7 in Kıcıman et al. (2023). For example, NHD of GPT 4 in Kıcıman et al. (2023) was reported as 0.22 but you reported 0.35 for GPT 4 in the table 2 in your paper.", "limitations": "", "rating": 3, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 2, "strengths": "Paper is overall written concisely due to multiple modules involved in the framework. The idea of encoding intermediate results in a JSON format and feeding them into an LLM seems clever.", "weaknesses": "- The use of LLM to tweak intermediate results seems clever but it is hard to assess its technical contribution. \n- It is unknown how LLM is doing with respect to its memory. Does LLM always try to update edges in order to maximize data fitting? If the data fitting is based on the currently predicted causal graphs, how can it improve its causal graph? It does not work like an EM algorithm. Does LLM ‘regret’ its previous decision if fitting becomes worse? Considering developing a causal discovery algorithm that is based on local search (incrementally updating causal graph based on its likelihood), how would you compare their learning trajectories?\n- Use of data only to fit the intermediate graph seems not using available dataset in full. Such as conditional independence and other information is all unused.\n- LLM’s stochastic nature is ignored. LLM may answer differently for the same question.\n- It is essential to thoroughly examine the behavior of LLM. How does it adjust the result based on its belief (GPT-4 etc) and intermediate results passed. There are more questions remained than answered.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WfhW7dcOWF", "reviewer": "Reviewer_pSg2", "summary": "The paper combines the meta-data driven Large Language Models (LLMs) and data-driven Deep Structural Causal Models (DSCMs) to construct a novel framework called Causal Modeling Agent for causal discovery. The framework leverages the LLMs' state-of-the-art capability to capture domain knowledge to discover the causal relationship in DSCMs. The framework is tested against a number of benchmarks on the real-world task of modeling the clinical and radiological phenotype of Alzheimer's Disease (AD), which has a ground-truth causal relationship between the vertices. The experimental results suggest that the CMA outperforms purely data-driven and metadata-driven benchmarks. New insights into the causal relationship among biomarkers of AD have also been obtained by CMA.", "questions": "Can the authors provide further evidence from experimental or observational data for the discovered causal relationships with the CMA?", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 4, "contribution": 3, "strengths": "1. The idea to combine LLM and SCM is interesting and novel.\n2. The experimental results are encouraging.\n3. New insights on the causal relationship between biomarkers have been obtained.", "weaknesses": "The contribution would be stronger if further evidence from experimental or observational data can be provided for the discovered causal relationships with the CMA.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (poster)