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How Causality Bridges the Semantic Gap

Shuhao Zhang, Xuran Zhou, Han Guo, Pengtao Xie, Yujia Zheng

arXiv:2610.02594v1cs.LGcs.AIcs.CL

TL;DR

Measured variables may lack names, and latent variables may never be measured, leaving semantics unavailable from the variables’ labels alone. CAUSALBRIDGE learns causal structure from measurements and uses it to align unnamed-variable embeddings with a few named anchors before translating them into names. It more accurately recovers observed and latent semantics than association-based methods, with a wider lead as names become scarcer.

  • Problem

    Systems can contain measured-but-unnamed variables and latent variables that appear only through their effects, while existing naming strategies rely on human knowledge.

    Source: §1 · p. 2

  • Method

    CAUSALBRIDGE discovers causal structure from measurements, optimizes embeddings for unnamed observed and latent variables under graph-implied relations with named variables as anchors, then translates them into names.

    Source: §1 · p. 2, §2 · p. 4, §3.2 · p. 5

  • Results

    CAUSALBRIDGE recovers observed and latent semantics more accurately than association-based methods, and at 50% masking beats the strongest baseline by 2.2 times on undocumented vehicle CAN channels.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    The discovered graph names variables as accurately as the documented graph, and the lead is largest where the least has been written down.

    Source: §5 · p. 9

    Evaluation is limited to systems whose true names are known but masked; the motivating settings, including neural-network features and latent variables in scientific data, have not yet been tested.

    Source: §5 · p. 9

Abstract

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Numerical measurements capture how a system behaves, but often leave the meanings of its variables unspecified. Some variables are measured but never labeled, and others are never measured at all. Existing methods assign semantics to such variables by consulting general human knowledge, but this inherits its biases where that knowledge exists and offers nothing where it does not. We bridge this gap between measurements and their meanings with causal structure instead, reading a variable's semantics from how it acts on other variables. We formalize this as structure-constrained semantic alignment, in which the embedding of each unnamed variable is solved under the dependence relations implied by the causal graph, with the embeddings of a few known names as anchors. Accordingly, we build CausalBridge, a framework that discovers the causal graph from the measurements, latent variables included, solves for the embeddings under those relations, and expresses them as names through a language model. The causal structure reflects the mechanism that generated the measurements and is recovered from the measurements alone, which may make it the one source of information free of bias from human knowledge. We evaluate CausalBridge on five questionnaires and three robotics scenarios, with 20 to 90% of the variable names masked. It recovers the semantics of observed and latent variables more accurately than existing methods that rely on association, and its lead widens as less of the system is documented. The graph it discovers names variables as accurately as the documented one, and a new system is named in minutes and at a fraction of the cost of sampling methods. Once the semantic gap is bridged faithfully, machines can understand the world and take actions causally.

1 INTRODUCTION

The semantic gap includes both measured variables without names and latent variables that appear only through their effects on measurements. CAUSALBRIDGE uses causal structure to align unnamed variables with a few named anchors, and its advantage grows as fewer names are documented.

1 INTRODUCTION

  • Measured-but-unnamed channels and unmeasured latent variables both leave systems without semantics for some of their variables.Latent variables can appear only through their effects on measured variables; naming one makes it a hypothesis people can test or a concept a system can communicate.Source: §1 · p. 2
  • Existing methods can assign the same incorrect name to distinct CAN channels, reflecting language priors more than their different measured signals.AutoInterp labels longitudinal acceleration as pitch rate, while an SASC-style method labels it front left wheel speed.Source: §1 · p. 2
  • CAUSALBRIDGE learns structure from measurements, constrains unnamed-variable embeddings by their relations to named anchors, then has a frozen language model write names.For the CAN example, it names the channel between vertical acceleration and pitch rate longitudinal acceleration.Source: §1 · p. 2
  • At 50% masking, CAUSALBRIDGE outperforms the strongest baseline by 2.2 times on channels of an undocumented vehicle CAN; for latent variables, its lead grows from 1.5 to 1.9 times as masking rises from 20% to 75%.The evaluation covers three robotics scenarios and five questionnaires, with 20–90% of names masked.Source: §1 · pp. 2–3
  • The discovered graph matches the documented graph in naming accuracy, and a new system is named in minutes using 2.6–56.6 times fewer tokens per latent variable than sampled-case methods.The paper reports the largest lead where the least has been documented.Source: §1 · p. 3

2 PROBLEM FORMULATION

The task is to recover names for unnamed observed and latent variables from measurements and a few known names. The paper formalizes this as finding embeddings that satisfy the causal graph’s dependence relations while keeping known-name vectors fixed.

2 PROBLEM FORMULATION

  • Given measurements and a few known variable names, the task is to recover semantics for the remaining observed variables and latent variables that are never measured.Variables are generated by a structural causal model; only a subset of observed variables have known names.Source: §2 · p. 3
  • A causal graph implies conditional independence relations through d-separation, which the paper uses to find semantics without semantic information in the graph.Under Markov and faithfulness assumptions, d-separation corresponds to conditional independence.Source: §2 · pp. 3–4
  • Structure-constrained semantic alignment assigns each variable an embedding, fixes named-variable embeddings as anchors, and seeks an assignment satisfying graph-implied dependence relations.Dependence strength is estimated from measurements, and the unnamed variables’ vectors are translated into names.Source: §2 · p. 4
  • Graph edges and d-separations constrain relations among embeddings, while anchors remove some freedom and existing knowledge regularizes the remaining choices.The regularizer favors English phrases and the language model translates the resulting vector.Source: §2 · p. 4

3 METHOD

CAUSALBRIDGE learns a causal graph, optimizes embeddings for unnamed variables under graph-implied relations and named-variable anchors, then translates the embeddings into names. Its structure-learning, embedding, and translation stages are presented as modular.

3 METHOD

  • CAUSALBRIDGE learns structure from measurements, optimizes unnamed-variable vectors against that structure with named variables as anchors, then translates each vector into a name with a frozen language model.A supplied graph can replace the discovered graph at the embedding step.Source: §3 · p. 4

3.1 CAUSAL STRUCTURE LEARNING

  • For observed robot and vehicle measurements, BOSS searches variable orderings across two consecutive time steps; for questionnaires with latent variables, RLCD recovers structure over observed and latent variables.The discovery algorithms are applied under their assumptions and output a graph without semantics attached.Source: §3.1 · p. 5
  • The learned CAN graph places the unnamed channel on a causal path from vertical acceleration to pitch rate and separates it from steering wheel angle.These relations are learned from measurements without reading names, unlike the baselines, which do not learn structure.Source: §3.1 · p. 5

3.2 CONSTRAINED EMBEDDING OPTIMIZATION

  • Embedding constraints make a variable’s vector reflect its causes, relate causally dependent variables, distinguish unrelated variables, and preserve residual meaning not explained by causes.A learned operator generates vectors from parent vectors; a regularizer favors unit-norm vectors near English phrases and relevant named variables.Source: §3.2 · pp. 5–6

3.3 TRANSLATION VIA PREFIX ADAPTATION

  • The optimized vector is passed directly as a prefix to a frozen language model, preserving its relations while the model writes the name.Decoding the vector into nearest phrases first can discard the relations that distinguish it from other channels; in the example, the prefix yields “longitudinal acceleration.”Source: §3.3 · p. 6
  • The graph and embeddings are computed before the language model is called, so other vector-input models can receive the system’s causal structure.The framework applies to systems with measurements and a few named variables under the conditions of causal discovery.Source: §3.3 · p. 6

4 EXPERIMENTS

CAUSALBRIDGE names observed and latent variables more accurately than baselines, including without a supplied graph. Its gains are strongest where language priors provide little information, and its discovered graph performs comparably to the supplied graph.

RQ1: Does causal structure recover the semantics of the variables a language prior cannot?

  • CAUSALBRIDGE gives the most accurate names on every variable set, including unmeasured latent variables, and needs no supplied structure.Its lead is largest on robotics tasks, where channel labels provide little semantic information, and narrows at 90% masking as few anchors remain.Source: §4 · p. 7
  • On observed variables, its discovered graph reaches NRR@1 at least as high as any baseline given the supplied graph.On latent variables and vehicle CAN, its lead is wider than under the supplied graph at every masking level; on vehicle CAN it closely tracks its supplied-graph run and outperforms it.Source: §4 · p. 8
  • Figure 4 contrasts runs without supplied structure, where CAUSALBRIDGE uses its discovered graph, with runs using the supplied graph.Its rows report NRR@1 and identification, with bands showing one standard error.Source: Figure 4 · p. 8

RQ2: Where does the gain come from?

  • The prefix carries structure into the language model, while the evidence line supplies naming vocabulary; their relative value shifts with masking.The prefix sets channel type and the evidence line resolves coordinates on vehicle CAN and real logs; the low-rank update adds little alone.Source: §4 · pp. 8–9

RQ3: Does one recipe carry across backbones?

  • One adapter recipe generalizes across the 3B, 4B, and 8B backbones, with the largest improvements on channels and convergence at 90% masking.The graph and embeddings do not depend on the backbone; only the adapter is trained per backbone.Source: §4 · p. 9

RQ4: What does naming a new system cost?

  • 2.6 to 56.6 times fewer tokens are used per latent variable than by sampling methods, while latent variables are named more accurately.Ten drives are named from raw traces in about seven minutes; CAUSALBRIDGE writes each name from one prompt rather than describing many examples.Source: §4 · p. 9

5 CONCLUSION AND DISCUSSION

The paper argues that a few named variables let causal relations supply meanings for other variables, with its discovered graph naming variables as accurately as the documented graph. Its evaluation is limited to systems whose true names are known but masked.

5 CONCLUSION AND DISCUSSION

  • Once a few variables are named, another variable’s meaning can be read from its causal relations, and the discovered graph names variables as accurately as the documented one.The paper says the lead is largest where the least has been written down.Source: §5 · p. 9
  • Evaluation covers only systems with known but masked true names, leaving neural-network features, scientific-data latents, and recursively self-improving systems unevaluated.The names must be known to score the names the method writes.Source: §5 · p. 9
  • A system can repeat a loop of measuring, recovering structure, and naming variables, with each recovered name available as an anchor for the next.The paper frames this as expanding the system’s model of the world and vocabulary from its own data.Source: §5 · pp. 9–10

APPENDIX CONTENTS

  • The appendix contents map sections on notation, case studies, theory, datasets, structure learning, embedding optimization, translation, evaluation, and results.The notation table lists symbols in pipeline order, including dependence terms, relation weights, adapter inputs, and naming outputs.Source: APPENDIX CONTENTS · p. 20, Table 1 · p. 20, A NOTATION · p. 21

B CASE STUDIES

The case studies show the context used to name masked variables, while the theoretical background connects causal-graph relations to statistical dependence and rank patterns that can reveal latent variables.

B CASE STUDIES

  • The case studies present model inputs and written names alongside reference names, including a DASS latent-variable example at 50% masking.The example table shows caused-item neighbor lines, an evidence line with named items and factor-score correlations, and the predicted and true names.Source: B CASE STUDIES · p. 21, Table 2 · p. 22

C THEORETICAL BACKGROUND

  • The theoretical background defines a DAG and distribution, and introduces Markov and faithfulness conditions for relating graph separation to independence.The supplied material also states the Gaussian rank and partial-correlation characterization.Source: C THEORETICAL BACKGROUND · p. 24, §C.1 · pp. 24–25

C.1 THE GRAPH AND THE DISTRIBUTION

  • Under Markov and faithfulness assumptions, graph d-separations correspond to distributional independences, expressed in the Gaussian case through vanishing partial correlations.The objective imposes these conditions for the empty set and for parent sets.Source: §C.1 · pp. 24–25

C.2 LATENT VARIABLES AND RANK

  • In a linear structural equation model with independent noise, covariance rank is linked to the size of a t-separating pair, with equality for generic parameters.A rank deficiency among observed variables can reveal latent variables through which every trek between them passes.Source: §C.2 · p. 25

C.3 WHAT STRUCTURE LEARNING RETURNS

The structure-learning results identify conditions under which latent structure can be recovered, while the embedding objective translates graph-implied dependence relations into constraints on variable vectors. The paper also gives a conditional-mutual-information formulation and describes the evaluation data used.

C.3 WHAT STRUCTURE LEARNING RETURNS

  • Under rank faithfulness and Condition 1, RLCD asymptotically identifies the Markov equivalence class of Omin(Os(G)); latent variables are identified through their atomic covers.A discovered latent variable is identified through the items it reaches, which is also how evaluation maps it to a published latent variable.Source: §C.3 · p. 26
  • BOSS supplies an initial graph over questionnaire items for RLCD, while on robot and vehicle channels it searches consecutive time steps and retains forward-time edges.Under its hypotheses, BOSS guarantees the skeleton and unshielded colliders over the two time-step columns.Source: §C.3 · p. 26

C.4 FROM THE GRAPH TO THE OBJECTIVE

  • The embedding objective is a constrained multidimensional scaling of dependence relations onto the unit sphere, with named-variable anchors fixed.With finite data, the loss balances squared violations of the graph-implied relations against the regularizer; the feasible set may be empty unless conditional residual dependences vanish and marginal constraints agree with anchors.Source: §C.4 · p. 27

C.5 THE CONDITIONAL MUTUAL INFORMATION OF THE VECTORS

  • In the Gaussian model of the vectors, residual-vector cosine similarity encodes conditional mutual information through Linfoot’s informational coefficient, allowing dependence constraints to use CMI.The measurement-side dependence is estimated from the data; the Gaussian model applies to the vectors and does not constrain the measurements’ distribution or relation form.Source: §C.5 · pp. 27–28

D DATASETS AND PREPROCESSING

  • The evaluation covers robot and vehicle time series plus questionnaire items, with some true names masked and the remaining names serving as anchors.Each trained part of CAUSALBRIDGE is fitted on data of its own.Source: D DATASETS AND PREPROCESSING · p. 28

D.1 ROBOTICS SCENARIOS

The evaluation spans robot, vehicle, and questionnaire systems with different documentation and variable structures. The sections describe the datasets, masking protocol, training data, and graph-discovery procedures used to prepare them.

D.1 ROBOTICS SCENARIOS

  • The robotics scenarios include simulated arms, real robot logs, and vehicle drives, spanning documented simulator physics, declared platform layouts, and an undocumented vehicle CAN.Simulated-arm and real-log evaluations use 1,200 steps; logs contain 13–28 channels, while each vehicle drive has 12 channels and 2,400 samples, except one shorter drive.Source: §D.1 · pp. 28–29, Table 3 · p. 29

D.2 QUESTIONNAIRES

  • The questionnaire corpus combines public scales, values-survey variables, cognitive tests, and multitasking measures, using published scoring keys as latent-variable references.The five evaluation scales span clinical, personality, mindfulness, vocational-interest, and values measures, with 21–240 items and 3–30 latent variables, including nested keys.Source: §D.2 · pp. 29, 31

D.3 MASKING

  • Names are masked at 20%, 50%, 75%, or 90%, while measured data remain available and every latent variable is a target at each level.Questionnaire scales use five masking folds per level; each robot, log, or drive uses one set, and the same set is reused across methods and structure sources.Source: §D.3 · pp. 31, 33

D.4 TRAINING DATA

  • CAUSALBRIDGE’s trained optimization components and adapters use training data that do not overlap the evaluation scales, arms, logs, or drives.Training includes public questionnaire scales, generated arms and vehicles, synthetic questionnaire scales, and generated robot-adapter examples.Source: §D.4 · pp. 33–34

E CAUSAL STRUCTURE LEARNING

  • Graph discovery learns the graph from measurements alone; BOSS handles robot and vehicle time series, while RLCD with BOSS as its first stage handles questionnaires.The discovered graph is passed to the constrained embedding optimization.Source: E CAUSAL STRUCTURE LEARNING · p. 35

E.1 ROBOT AND VEHICLE CHANNELS

  • For robot and vehicle channels, discovery uses lag-one designs, differences near-integrated channels, regresses retained edges, and refines the summary graph with partial-correlation passes.A physical-law prior may optionally supply within-time-step edge orientations without supplying channel names.Source: §E.1 · p. 35

E.2 QUESTIONNAIRES

  • For questionnaires, BOSS replaces RLCD’s first-stage search, and rank tests introduce latent variables as atomic covers within item groups.The tests use covariance-submatrix rank and canonical correlations with a Wilks statistic; a collider check guards against mistaking a collider for a latent parent.Source: §E.2 · p. 35

E.3 RECOVERED STRUCTURES

The discovered graphs recover meaningful temporal structure in simulated arms but share few edges with supplied graphs on real logs and vehicle drives. The method then aligns unnamed-variable embeddings to graph-implied relations and translates vectors into names using a prefix-adapted language model.

E.3 RECOVERED STRUCTURES

  • On simulated arms, the discovered graph recovers action-to-joint and successive-joint edges, but misses velocity-to-position edges and adds edges between correlated channels.It shares about a quarter of true edges; first-difference inputs make velocity-to-position edges the most frequently missed.Source: §E.3 · p. 36
  • On real logs and vehicle drives, discovered graphs share few supplied edges, and vehicle searches miss all four speed-to-wheel-speed edges.Each vehicle drive yields one to six edges, mostly among accelerations and rates.Source: §E.3 · p. 36
  • Questionnaire graphs are finer than published keys on four scales and cut across the World Values Survey key, whose discovered groups have .53 purity versus .80 to .95 elsewhere.DASS splits three published latent variables into nine groups, with three mixing anxiety and stress; HEXACO discovers 25 latent variables over 64 of 240 items.Source: §E.3 · pp. 36–37

F CONSTRAINED EMBEDDING OPTIMIZATION

  • The embedding optimization takes the discovered graph, measurements, and named-variable names, then returns a vector for every unnamed variable.Named-variable encodings are fixed anchors; the optimized unnamed vectors are passed to the adapter for naming.Source: F CONSTRAINED EMBEDDING OPTIMIZATION · p. 37, §F.1 · p. 37

F.2 THE GENERATING OPERATOR

  • The generating operator adds contributions from a variable’s parents, using edge sign and type, while a sign-consistency penalty keeps learned contributions aligned with positive or negative relations.Negative relations use a learned semantic-negation map, motivated by reverse-keyed questionnaire items whose meaning is opposite the latent parent.Source: §F.2 · p. 37

F.3 THE OBJECTIVE

  • The objective aligns graph-implied independence and dependence with measurement relations, using residual comparisons for conditional dependence and a regularizer to select among feasible embeddings.Its conditional-dependence target compares regression residuals; the CMI variant instead compares conditional mutual information given a pair-specific parent set, without assuming additive noise.Source: §F.3 · p. 37, §F.4 · p. 38, §F.8 · p. 39

F.5 OPTIMIZATION

  • Optimization starts from ridge-regularized least squares and unrolled Adam, then refines each unnamed vector against nearby concept-bank phrases while named vectors remain fixed.The CMI alternative estimates dependence with CCMI and uses ranks and latent codes; estimates use at most 2,000 rows and surrogate tests at the 5% level.Source: §F.5 · p. 38, §F.8 · pp. 39–40

G TRANSLATION VIA PREFIX ADAPTATION

  • The adapter prefixes the frozen language model with vectors and graph-neighborhood evidence, including parent, child, sibling, and named-neighbor information tailored to observed or latent variables.The five input vectors and scalar neighborhood features are mapped to a prefix; the prompt describes incident edges and signed correlations, with unnamed neighbors indicated as hidden labels.Source: G TRANSLATION VIA PREFIX ADAPTATION · p. 40, §G.1 · p. 40, §G.3 · p. 41

G.4 TRAINING

Training adapts the prefix and low-rank updates to predict masked names while encouraging semantic similarity and distinction from sibling names. The same recipe is used across questionnaire and robot domains, and inference greedily decodes a short name.

G.4 TRAINING

  • The frozen language model is trained through a low-rank update and adapter using name likelihood, semantic similarity, contrastive, and sibling-separation losses.The contrastive and sibling losses push outputs toward the true name and away from sibling names.Source: §G.4 · p. 41
  • Training uses one questionnaire adapter and one robot adapter, exposing both to masked names and degraded graphs that resemble discovery errors.Examples also remove prompt or evidence text, or substitute unrelated or sibling vectors, so the adapter learns to use both prefix and text.Source: §G.4 · p. 41

G.5 DECODING

  • At inference, the adapter maps the input to a prefix and the model greedily decodes a short output, taking its first line as the variable name.Source: §G.5 · p. 42

H EVALUATION PROTOCOL

The evaluation scores names against targets using shared embedding and judging protocols, with separate references for discovered latent variables. It compares baselines under controlled masking and graph conditions, while reported fold variation widens as names are removed.

H.1 REFERENCE FOR DISCOVERED LATENT VARIABLES

  • Discovered latent variables are scored against the published latent variable containing the majority of the observed items beneath each discovered group.On the World Values Survey, where groups cross sub-indices, the reference owns about half of the items beneath a discovered latent variable.Source: §H.1 · p. 42

H.2 SCORING SPACE, CANDIDATE POOL AND METRICS

  • NRR@1 and MRR rank names in a shared embedding space, while identification and latent-variable naming quality use a language-model judge and additional validity checks.Naming quality rejects non-name outputs and verbatim item-text copies; NRR@1 requires the true name to rank first.Source: §H.2 · p. 42, §H.3 · p. 43

H.3 THE JUDGE

  • The judge selects the target from five candidates in both candidate orders, and identification credits only cases correct in both orders.Distractors include sibling items or latent variables, or similar robot-channel names, so a broad topic or channel type alone may not distinguish the target.Source: §H.3 · p. 43

H.4 VARIATION ACROSS FOLDS

  • Fold variation grows as masking increases; DASS latent-variable identification falls from .689 ± .183 at 20% masking to .222 ± .208 at 90%.The paper therefore compares methods within the same fold and target; figure bands show standard error across folds or robot systems.Source: §H.4 · p. 43

I BASELINES AND FIDELITY TO THE PUBLISHED DESIGNS

  • Baselines cover concept-bank matching, language-model interpretation, embedding regression, and graph-consuming methods when structure is supplied.The common protocol gives methods the same masked cases, known names, scoring encoder, and judge; structure-free baselines receive no graph or key grouping.Source: I BASELINES AND FIDELITY TO THE… · pp. 43–44
  • Baseline ports preserve selected design elements but make documented adaptations for these data, including replacing visual inputs or changing encoders and training signals.For example, SASC-style uses correlation-based candidates and sign agreement, while DeViSE-style replaces its visual encoder and hinge loss with ridge regression on correlation features.Source: I BASELINES AND FIDELITY TO THE… · pp. 44–45

J.1 RQ1: THE LEAD WITH NO STRUCTURE SUPPLIED

Without supplied structure, CAUSALBRIDGE leads the reported robot and questionnaire comparisons, with stronger gains where names provide little prior information. It also improves channel-type and coordinate recovery and names latent variables from their relations to items.

J.1 RQ1: THE LEAD WITH NO STRUCTURE SUPPLIED

  • At 50% and 75% masking on robot columns, CAUSALBRIDGE’s NRR@1 is 2.2 to 5.1 times the strongest baseline, while identification is 1.6 to 2.9 times and MRR 1.4 to 2.1 times.Across reported columns it leads on every metric, with the largest lead where names provide the least language prior.Source: §J.1 · p. 45
  • At 90% masking on simulated arms, coordinate accuracy is .145 for CAUSALBRIDGE versus .014 for the strongest baseline.On real logs the method scores .579 versus .395 at 90%; on vehicle CAN it scores .367 versus .200 at 50%.Source: §J.1 · p. 46
  • On simulated arms, type accuracy is .537 versus .281 at 50% masking and .577 versus .275 at 75%, measured as what channel type is named.Type accuracy captures what is measured; coordinate accuracy captures which instance of that type is named.Source: §J.1 · pp. 45–46, Figure 18 · p. 48
  • For latent variables, CAUSALBRIDGE’s NRR@1 falls from .540 to .240 as masking rises, while SASC-style falls from .369 to .115; their ratio for CAUSALBRIDGE grows from 1.5 to 1.9 through 75% masking.At 75% masking, latent naming quality is .390 versus .319 for SASC-style; on questionnaire items, CAUSALBRIDGE’s NRR@1 is 1.5, 1.6, 1.7, and 1.3 times SASC-style across the four levels.Source: §J.1 · p. 46
  • Two comparisons qualify the pattern: simulated-arm NRR@1 peaks at 50% masking, and questionnaire-item identification overlaps SASC-style although CAUSALBRIDGE leads NRR@1.Sibling-name copying explains the arm peak; identification accepts a name that reaches the correct latent variable, whereas NRR@1 requires the exact item.Source: §J.1 · pp. 46, 48

J.2 RQ1: THE DISCOVERED GRAPH AND THE SUPPLIED GRAPH

CAUSALBRIDGE performs similarly with its discovered and supplied graph, and its advantage over baselines is wider without supplied structure. A physical prior improves its NRR@1 on simulated arms and vehicle CAN at 20% and 50% masking.

J.2 RQ1: THE DISCOVERED GRAPH AND THE SUPPLIED GRAPH

  • On vehicle CAN, the discovered graph leads the supplied graph at every masking level, with NRR@1 of .900 versus .750 at 20% and .683 versus .650 at 50%.On simulated arms and real logs the supplied graph leads; on latent variables the two graphs define different targets and perform at a similar level.Source: §J.2 · p. 48
  • Graph-using baselines gain from supplied structure, but their gains vary by task and can disappear at 90% masking when few named neighbors remain.On real logs, the base model reaches NRR@1 of .185 and .146 at 50% and 75% masking, versus .111 and .073 for the strongest baseline without a graph; on vehicle CAN, it falls below that baseline from 50% masking onward.Source: §J.2 · pp. 48–49
  • Without supplied structure, CAUSALBRIDGE’s NRR@1 lead over the strongest baseline on vehicle CAN is .800, .200, .167, and .100 across the four masking levels.Under supplied structure, the corresponding leads are .750, .133, .078, and .027; many anchors help constrain unnamed variables, while the difference closes as names are masked.Source: §J.2 · p. 50
  • A physical prior raises NRR@1 on simulated arms from .083 to .104 at 20% masking and from .140 to .157 at 50%.On vehicle CAN, it raises NRR@1 from .800 to .950 at 20% masking and from .367 to .483 at 50%.Source: §J.2 · pp. 50–51, Table 15 · p. 51

J.3 RQ1: THE LEAD REPLICATE BY REPLICATE, AGAINST COPIED NAMES AND SCALE BY SCALE

Replicate-level comparisons show that CAUSALBRIDGE usually leads the strongest baseline, especially on retrieval metrics, while copied-name references can score well on identification without naming accurately. Results also vary across scales and robot instances.

J.3 RQ1: THE LEAD REPLICATE BY REPLICATE, AGAINST COPIED NAMES AND SCALE BY SCALE

  • Without supplied structure, CAUSALBRIDGE leads in 58 of 64 cells and by more than two standard errors in 38, with no cell behind by more than two standard errors.On retrieval metrics, the difference is positive in 38 of 40 cells and exceeds two standard errors in 29; on judged metrics it exceeds two standard errors in 9 of 24 cells and is within two standard errors in 14.Source: §J.3 · p. 52, Table 16 · p. 52
  • Under supplied structure, CAUSALBRIDGE leads in 60 of 64 cells and is behind by more than two standard errors in one cell: item identification at 50% masking.That cell favors SASC-style, whose neighboring-sentence names can identify the latent variable credited by the judge.Source: §J.3 · p. 52, Table 17 · p. 53
  • On latent variables, EFA loading naming reaches NRR@1 of .500 versus .540 at 20% masking and .254 versus .240 at 90%, but its copied item texts receive naming quality of .000.Feature Propagation’s naming quality is .408 versus .446 for CAUSALBRIDGE at 20% masking.Source: §J.3 · p. 52, Table 19 · p. 53
  • CAUSALBRIDGE is first in 15 of 20 item scale-and-masking combinations, and HEXACO is the easiest scale for identification.The main tables pool scales and robot instances by sample; the scale-level tables report results separately, and the theory-based key and response-based measurements define different latent-variable sets.Source: §J.3 · pp. 52–53, Table 18 · p. 53

J.4 RQ2: COMPONENT BY COMPONENT

The component analysis finds that the prefix contributes most, while the evidence line supplies naming vocabulary and can help identify coordinates. One recipe improves all three tested backbones, though their starting levels differ.

J.4 RQ2: COMPONENT BY COMPONENT

  • Removing the prefix costs the most: NRR@1 falls by 50–88% on vehicle CAN, 68% on simulated arms at 75% masking, and 83% on real logs at 90%.On questionnaire items it falls by 25–50%, while identification does not fall; the prefix variant still receives decoded phrases, which retain rough meaning but lose relations to named variables.Source: §J.4 · p. 54
  • The evidence line provides name vocabulary; on real logs at 20% masking, removing it yields NRR@1 of .167, versus .333 without the prefix and .417 for the full method.On item identification, removing the evidence line is 30–38% below the full method at every masking level.Source: §J.4 · pp. 55–56
  • At 90% masking, the variant without the evidence line meets or exceeds the full method on simulated arms, real logs, and latent variables, where few named neighbors remain.On latent variables, identification curves stay within .07 and naming-quality values within .10 across the four variants.Source: §J.4 · p. 57
  • The prefix determines robot-channel type, while the evidence line can determine coordinates: at 50% masking, removing the prefix lowers type accuracy from .683 to .300 on vehicle CAN.Removing the evidence line lowers vehicle CAN coordinate accuracy from .800 to .000 at 20% masking; on simulated arms, the prefix contributes to coordinate accuracy from 50% masking onward.Source: §J.4 · p. 57
  • Without the prefix, the low-rank update coincides with the base model on both questionnaire columns, retaining some gain only in selected robot settings.With a supplied graph, the prefix contributes .13 and .10 NRR@1 on latent variables at 20% and 50% masking under the discovered graph, but almost nothing under the supplied graph.Source: §J.4 · p. 57

J.5 RQ3: ACROSS BACKBONES

  • At 20% masking on vehicle CAN, CAUSALBRIDGE raises NRR@1 from .100, .150, and .050 alone to .800, .550, and .700 across Qwen3-4B, Qwen3-8B, and Llama-3.2-3B.On real logs, the same backbones rise from .000 alone to .417, .417, and .083; type accuracy also improves across the tested backbones.Source: §J.5 · p. 57
  • The backbones differ in starting level and strongest columns, but at 90% masking their NRR@1 values are within .10 on every column.The 3B Llama starts lowest on real logs but reaches the other two by 75% masking; on latent variables it is best at 50% and 75%.Source: §J.5 · pp. 57–58

J.6 TABLES

  • The component and backbone tables report results under both structure sources, with best and second-best values marked within each variable set.Figure 19 presents component removals above and the three-backbone recipe below, with NRR@1 and identification as rows.Source: Figure 19 · p. 56, §J.6 · p. 58

K RESULTS WITH CONDITIONAL MUTUAL INFORMATION AS THE DEPENDENCE MEASURE

The CMI version of CAUSALBRIDGE produces close results to the distance-correlation version across questionnaires and robot channels, with neither consistently ahead. This supports distance correlation as the practical choice in the main method.

K RESULTS WITH CONDITIONAL MUTUAL INFORMATION AS THE DEPENDENCE MEASURE

  • Replacing distance correlation with CMI leaves names essentially unchanged, with close results across questionnaires and robot channels, both graph sources, and masking levels.Neither version is consistently ahead; CMI makes the dependence and independence constraints exact conditional-mutual-information equalities.Source: K RESULTS WITH CONDITIONAL MUTUAL… · p. 61, Table 33 · p. 62, Table 34 · p. 62

L SETTINGS, COMPUTE AND LICENSES

The appendix reports compute and deployment costs, then compares supplied and discovered structures for vehicle, robot, and questionnaire examples. It also specifies how those structures are visualized and notes the data and software licenses.

L SETTINGS, COMPUTE AND LICENSES

  • A100-class GPU runs covered the reported stages; questionnaire and robot adapters were each trained once, taking about one hour and eight and a half hours, respectively.The 150-variable synthetic scale took 107 seconds, while exact rank searches remained practical to about 150 active variables at k ≤3.Source: L SETTINGS, COMPUTE AND LICENSES · p. 64
  • 0.054 seconds per name for CausalBridge, versus 0.32–2.4 seconds for the three sampling methods in timed runs.Sampling methods used 2.6–56.6 times the tokens per latent variable; CausalBridge stayed within a few hundred tokens per column, and the cheapest method named latent variables most accurately.Source: L SETTINGS, COMPUTE AND LICENSES · p. 64
  • Ten vehicle drives go from raw traces to named channels in about seven minutes; naming a new system takes minutes on one GPU.For those drives, structure learning took five seconds, optimization 334 seconds, and name generation 67 seconds.Source: L SETTINGS, COMPUTE AND LICENSES · p. 64
  • The appendix lists public questionnaire datasets and licenses for the datasets, robot logs, and semantic resources used.The listed licenses include MIT, Apache 2.0, and CC BY-SA variants.Source: L SETTINGS, COMPUTE AND LICENSES · p. 65

M RECOVERED STRUCTURES

  • Vehicle, robot, and drive figures place supplied graphs on the left and discovered graphs on the right; questionnaire figures place published scoring keys above recovered structures below.Questionnaire items are marked by identifier and colored by the latent variable governing them.Source: M RECOVERED STRUCTURES · p. 65
  • In the vehicle drive, the discovered graph includes a path from vertical through longitudinal acceleration to pitch rate, plus an edge from pitch rate to front-right wheel speed.The caption also reports two additional edges among rates and lateral acceleration.Source: Figure 21 · p. 65
  • For DASS, six of nine discovered latent variables lie inside one published latent variable; in the World Values Survey, discovered groups cut across Welzel sub-indices more than on any other scale.The DASS structures are shown published above and discovered below.Source: Figure 24 · p. 67, Figure 25 · p. 67

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