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Formalizing Neurath's Ship: Approximate Algorithms for Online Causal Learning

Neil R. Bramley, Peter Dayan, Thomas L. Griffiths, David A. Lagnado

arXiv:1609.04212v3cs.LG

TL;DR

Causal structure learning is computationally demanding, yet people must identify causal models despite limited resources. The paper models learners as locally updating a single hypothesis and finds support for this account in four experiments, alongside evidence of occasional departures from strong sequential dependence.

  • Problem

    Identifying causal models is computationally demanding, and the paper examines how people manage this task despite limited resources.

    Source: General Discussion · p. 47

  • Method

    The paper models learners as maintaining a single hypothesis and updating it locally as they gather evidence.

    Source: General Discussion · p. 48

  • Results

    Across four experiments, participants generated plausible causal models and showed strong sequential dependence in their judgments.

    Source: Summary of Experiments · p. 38

  • Takeaways & Limitations

    The findings support the proposal that people normally maintain a single hypothesis and update it piece by piece.

    Source: Summary of Experiments · p. 38

    Participants showed high sequential dependence but occasionally had to remember and re-report edges they had judged and not yet reconsidered.

    Source: Measurement effects · p. 49, Antifoundationalism · p. 50

Abstract

from arXiv · show

Higher-level cognition depends on the ability to learn models of the world. We can characterize this at the computational level as a structure-learning problem with the goal of best identifying the prevailing causal relationships among a set of relata. However, the computational cost of performing exact Bayesian inference over causal models grows rapidly as the number of relata increases. This implies that the cognitive processes underlying causal learning must be substantially approximate. A powerful class of approximations that focuses on the sequential absorption of successive inputs is captured by the Neurath's ship metaphor in philosophy of science, where theory change is cast as a stochastic and gradual process shaped as much by people's limited willingness to abandon their current theory when considering alternatives as by the ground truth they hope to approach. Inspired by this metaphor and by algorithms for approximating Bayesian inference in machine learning, we propose an algorithmic-level model of causal structure learning under which learners represent only a single global hypothesis that they update locally as they gather evidence. We propose a related scheme for understanding how, under these limitations, learners choose informative interventions that manipulate the causal system to help elucidate its workings. We find support for our approach in the analysis of four experiments.

A computational-level framework for active structure learning

The framework separates causal structure learning into representing models, inferring which model fits observations and interventions, and selecting interventions to reduce uncertainty. It specifies a causal-network representation and Bayesian inference, then values interventions by their expected information gain.

A computational-level framework for active structure learning

  • Active structure learning comprises causal-model representation, inference from observations and interventions, and intervention selection to gather evidence.The framework treats these as three interrelated elements of the learning problem.Source: A computational-level framework for… · p. 5

Representation

  • Causal models are directed acyclic graphs with binary variables and generative connections; causes combine through a noisy-OR with shared strength and background parameters.The representation assumes each cause independently contributes to its effect, and the parameters are shared across components.Source: Representation · pp. 5–6

Inference

  • Interventions fix selected variables while leaving others free, allowing learners to probe causal influences that observations alone may not reveal.The framework also restricts its inference treatment to cases without latent variables.Source: Inference · p. 6
  • Bayesian inference updates model probabilities using the likelihood of observed data under each model, with uncertainty in causal parameters marginalized when needed.Sequential data can be incorporated by using the previous posterior as the next prior.Source: Inference · pp. 6–7

Choosing interventions

  • An intervention’s value is its expected reduction in uncertainty about the true model, averaged over possible outcomes under the current belief.The procedure is greedy: planning multiple steps ahead could select a different intervention.Source: Choosing interventions · pp. 7–8

Behavioral patterns and existing explanations

This section surveys departures of human causal judgments and interventions from idealized Bayesian and active-learning predictions, along with existing accounts based on sequential dependence, local inference, and limited model updating. It frames the question of whether these accounts provide a general-purpose, scalable explanation of learning complex causal world-models and introduces an incremental structure-change algorithm inspired by approximate inference in machine learning.

Behavioral patterns and existing explanations

  • Possible graphs grow rapidly with variables: 3- and 4-variable problems have 25 and 543 graphs; 3-, 4-, and 5-variable problems permit 27, 81, and 243 intervention patterns, and model, intervention, and outcome combinations should be averaged.Source: Behavioral patterns and existing… · p. 8
  • In sequential active learning, judgments show stickiness toward prior hypotheses and recency toward new evidence, which Bramley, Lagnado, and Speekenbrink (2015) modeled with forgetting and conservatism parameters.The forgetting parameter leaks posterior information as it becomes the next prior; the conservatism parameter assigns unusually high probability to the latest causal hypothesis.Source: Behavioral patterns and existing… · p. 9
  • Bonawitz et al. (2014) proposed “win-stay, lose-sample”: learners keep one structural hypothesis until surprised, then resample from the posterior with probability increasing as the observation becomes less probable.The scheme preserves a posterior sample without resampling each trial, but requires storing all past evidence and does not specify how to draw samples.Source: Behavioral patterns and existing… · p. 9
  • Waldmann, Cheng, Hagmayer, and Blaisdell (2008) proposed inferring each variable pair separately, yielding good local inferences but possible Markov violations in complex judgments.The minimal model captured participants’ judgments in two case studies.Source: Behavioral patterns and existing… · p. 9
  • Fernbach and Sloman (2009) attributed spurious direct links in chain judgments to local inference, while Bramley, Lagnado, and Speekenbrink (2015) proposed an endorsement heuristic that updates links from intervention outcomes.The heuristic adds edges from intervened-on variables to activated variables and removes edges to variables that do not activate, producing recency and unnecessary direct links in chains.Source: Behavioral patterns and existing… · pp. 9–10
  • Simple endorsement fit participants’ patterns but was outperformed by Bayesian learning with forgetting and conservatism, and performed poorly with many chains or high true wB.Its success is conditional on matching the situation.Source: Behavioral patterns and existing… · p. 10
  • Bramley, Dayan, and Lagnado (2015) proposed a single-hypothesis model that minimizes edits and unexplained latest states, but gave no procedure for searching for a structure satisfying those constraints.The current model acts as an anchor, and learners try to explain new evidence with minimal changes.Source: Behavioral patterns and existing… · p. 10
  • Existing proposals capture patterns in human causal judgments, but it is unclear whether any provides a general-purpose, scalable explanation of learning complex causal world-models.The authors distinguish behavioral fit from explaining human success on harder problems.Source: Behavioral patterns and existing… · p. 10
  • Win-stay, lose-sample captures behavioral patterns within the normative framework but does not provide a scalable algorithm.Source: Behavioral patterns and existing… · p. 10
  • Simple endorsement offers a scalable heuristic but may not generalize beyond its designed tasks or explain success on harder problems.Source: Behavioral patterns and existing… · p. 10
  • The authors propose an incremental structure-change algorithm inspired by approximate inference in machine learning to meet both desiderata.Source: Behavioral patterns and existing… · p. 10

Algorithms for causal learning with limited resources

The section introduces resource-limited approximations to Bayesian learning and develops local search as a way to update causal hypotheses without enumerating all possibilities. These approaches motivate predictions compared with participants’ behavior in experiments.

Algorithms for causal learning with limited resources

  • The paper draws on approximate-learning algorithms and experiment-design heuristics to formalize Neurath’s ship intuitions and generate predictions for experiments.The predictions are compared with participants’ behavior in several experiments.Source: Algorithms for causal learning with… · p. 11

Approximating with a few hypotheses

  • Particle filtering approximates a posterior with a manageable ensemble of weighted hypotheses, which are reweighted and resampled as evidence arrives.A single-particle account instead treats an individual’s beliefs as behaving most like one hypothesis.Source: Approximating with a few hypotheses · p. 11

Sequential local search

  • MCMC transitions can move to less probable hypotheses and approximate a posterior, whereas hill climbing favors more probable neighbors but can become trapped in a local optimum.Both are presented as possible mechanisms for sequentially changing causal hypotheses.Source: Sequential local search · pp. 11–12
  • Local search changes at most one causal edge at a time, allowing a learner to update a current hypothesis through a short chain without enumerating alternatives.Longer chains make the resulting hypothesis less dependent on its starting point; local search can also help retain information when some evidence is forgotten.Source: Sequential local search · p. 12

Neurath’s ship: An algorithmic-level model of sequential belief

The Neurath’s ship model represents a learner with one current causal hypothesis, updated through local reconsideration of edges as evidence arrives. Search stochasticity and length govern how strongly each new belief remains tied to the previous one.

change

  • After new evidence arrives, the learner locally reconsiders edges in a single current causal model and adopts the resulting model as the new belief.When the belief changes, the evidence is forgotten and collection begins again.Source: change · p. 13

Resampling, hill climbing or random change

  • Edge updates can resample probabilistically, favor the most likely local state through hill climbing, or vary between those tendencies.The parameter ω controls preference for the most likely edge state; setting it to zero produces random movement rather than learning.Source: Resampling, hill climbing or random… · pp. 13, 69

Search length

  • The learner’s search length k is drawn from a Poisson distribution with average λ, which controls how far beliefs move to accommodate recent evidence.Small λ captures conservatism; large λ allows greater movement at the expense of older data represented by the starting belief.Source: Search length · p. 14

Putting these together

  • Averaging transition probabilities over search lengths yields predictions for a new belief given the previous belief and λ, although learners need only sample edges and length.The equation describes where a search process may terminate, not a computation the learner must explicitly perform.Source: Putting these together · p. 14

Selecting interventions on Neurath’s ship: A local uncertainty

This section describes an intervention-selection scheme that uses local uncertainty over a limited set of possibilities, selecting a local focus and an informative intervention relative to it. This approach supports trial-by-trial analysis and allows learner differences to reflect what they focus on.

schema

  • Local and global uncertainty can favor different interventions, allowing the model to explain apparently inefficient choices and diagnose individual or trial-by-trial focus preferences.The section illustrates this contrast with experimental trials and proposes that a learner may choose a test informative for a modest set of alternatives, such as models differing on one edge.Source: schema · p. 16

Three varieties of local focus

  • The section considers edge, effects, and confirmation focus; edge focus compares models differing on one edge, while effects focus asks which variables descend from a chosen variable.In deterministic causal-model identification, the best split is obtained by querying the effects of a randomly chosen variable; edge-focused preferences can favor fixing one endpoint of the target edge on while leaving the other free.Source: schema · p. 16, Edges · p. 17, Effects · p. 18

The two stages of the schema

  • The two-stage scheme first selects the least-certain local focus, then chooses an intervention expected to inform that focus using only the potential next datum.For Stage 2, the model ignores pre-existing evidence and uses what amounts to a uniform prior; both focus and intervention choices can be soft rather than strict maximization to capture decision noise.Source: The two stages of the schema · pp. 16–17

Three varieties of local focus

  • Effect-focused testing fixes the target node “on” while leaving other variables free, whereas confirmatory testing contrasts the current hypothesis with a no-connections null and favors its root nodes.For confirmatory focus, the learner has a single local focus.Source: Effects · p. 18, Confirmation · p. 19
  • Confirmatory testing’s usefulness depends on noise and prior probability, and increases once the tested model has sufficiently high prior probability.A confirmation-focused learner has just one local focus.Source: Confirmation · p. 19

Implications of the schema

  • Intervention choice has two separable stages: choosing what to focus on and selecting an informative intervention relative to that focus.This allows a learner to be poor at choosing a focus yet good at choosing an informative intervention once the focus is set.Source: Implications of the schema · p. 19
  • Learners may differ in the local focuses they favor; those aware of limits in incorporating evidence may ask narrow, single-edge questions, while overly broad focus can impede learning.The schema also allows intervention-choice differences to arise from the types of local focus learners are inclined or able to use.Source: Implications of the schema · p. 19
  • The authors fit behavior by modeling which local focuses learners include in their options, which they choose on each test, and how those choices relate to final performance.This approach uses learners’ choice patterns to diagnose their focus options and selections.Source: Implications of the schema · p. 19

Comparing model predictions to experiments

This section compares the framework with participants’ causal judgments and intervention choices, including whether their interventions align with its predictions. It also describes two new experiments designed to distinguish the framework more clearly from alternatives.

Comparing model predictions to experiments

  • The framework predicts sequential dependence from local edits to a single hypothesis, while local resampling can range from probability matching to hill climbing, which may gain short-term but risk local optima.Source: Comparing model predictions to… · p. 19
  • Participants’ final accuracy was closest to the Neurath’s-ship simulation, which underperformed participants given a full record of prior tests.Neurath’s ship uses only recent data, whereas participants in the information-plus-summary condition could consult all past tests.Source: Comparing judgment patterns · pp. 21–22
  • Participants’ online judgments showed sequential dependence: their edit-distance pattern matched Neurath’s ship, win-stay-lose-sample, and simple-endorser simulations, unlike larger shifts predicted by random or posterior sampling.Source: Comparing judgment patterns · p. 22
  • Participants’ intervention choices were more informative than random selection but less informative than ideal active learning, and fell within the range of simulations using three local foci.Source: Comparing intervention patterns · p. 22
  • Ideal active learning favored one-on interventions for most tests, sometimes switching later to controlled tests when uncertainty concerned direct versus indirect pathways.Source: Comparing intervention patterns · p. 22
  • Edge-focused testing produced a spread of intervention types biased toward controlled one-on, one-off tests, while effects-focused testing favored one-on tests.Source: Comparing intervention patterns · p. 23
  • The confirmation-focused tester predicts a mix of “one-on” and “two-on” interventions, fixing components with children on and leaving components with parents free.Source: Comparing intervention patterns · p. 23
  • In Bramley, Lagnado, and Speekenbrink (2015), participants favored “one-on” tests and more often fixed components with children on (60% vs. 56%; t(24568) = 3.2, p = .001).Components with at least one child in the latest hypothesis were fixed on more often than those believed to have no children.Source: Comparing intervention patterns · p. 23
  • Participants’ intervention choices were less dynamic than the efficient learner’s: single interventions fell only from 78% on the first test to 73% on the last.Source: Comparing intervention patterns · p. 23

Motivating the new experiments

  • The earlier study offered limited discrimination: competing approximations predicted similar accuracy and sequential dependence, while “one-on” tests were also globally informative, especially early in learning.Thus, the strong preference for “one-on” interventions could not establish whether participants used local effect-focused testing.Source: Motivating the new experiments · p. 23
  • The earlier paradigm also allowed unspecified edges and cyclic models, long problems that could produce ceiling performance, and potentially unreliable intermediate judgments and apparent sequential dependence.Participants did not have to update every edge after each test, so lazy responding could be mistaken for genuine belief dependence.Source: Motivating the new experiments · p. 23
  • Experiment 1 used larger, noisier problems with fewer interventions to increase the need for approximations and distinguish the framework from alternatives.The broader range of structures and intervention choices was intended to make competing models more distinguishable.Source: Motivating the new experiments · pp. 24–25
  • The new experiments incentivized accuracy at random learning points and required participants to mark every edge anew after each test.These changes aimed to elicit participants’ best current beliefs and prevent lazy responding from biasing results.Source: Motivating the new experiments · p. 25
  • Experiment 2 compared elicitation procedures, removed pretraining on causal strength and background noise, and probed local problem representations with confidence judgments and explanations.Participants reported confidence about model edges and explained what their interventions were intended to test.Source: Motivating the new experiments · pp. 25–26

Experiment 1: Learning larger causal models

Experiment 1 examines participants’ learning of larger, noisier causal models and their intervention choices. It considers their accuracy and updating patterns, as well as how intervention preferences varied across conditions and trials.

Methods

  • The experiment tested five three-variable and five four-variable problems across four between-subject conditions varying causal strength (wS = .9 or .75) and background noise (wB = .1 or .25).Source: Design · p. 26

Results and discussion

  • Across all four conditions, participants performed above chance but below a Bayes-optimal observer, and final accuracy was closer to Neurath’s ship simulations than to simple endorser, WSLS, random, or ideal responding.Source: Judgments · p. 28
  • Accuracy declined as background noise increased, but showed no evidence of a relationship with causal strength or whether problems had three versus four variables.Source: Judgments · p. 28
  • Accuracy differed by device type: chains were lowest and fully connected models were second-lowest.Source: Judgments · p. 28, Table 1 · p. 29
  • Participants’ probability of judgments decreased rapidly with larger edit distances, a pattern captured by Neurath’s ship and simple endorsement but unlike random or posterior sampling.Source: Sequential dependence · p. 29
  • Win-stay, lose-sample showed a different edit pattern, peaking at zero changes with a second peak matching posterior sampling, while its average edit distance was close to participants’.Source: Sequential dependence · p. 29
  • Participants scoring 22 or more made smaller edits than lower scorers: 0.85±0.95 versus 1.3 ± 1.12 for three-variable and 1.4 ± 1.5 versus 2.4 ± 1.8 for four-variable problems.Edit distance and score were strongly negatively correlated (F(1, 118) = 34, β = −6.7, η2 = .34, p < .001).Source: Sequential dependence · p. 29
  • Aggregating simulated learners with different capacities to search for updates would produce a heavier-tailed edit-distance distribution that more faithfully resembled participants’ choices.Source: Sequential dependence · p. 29
  • The model’s intervention preferences varied by condition and trial, but participants showed no clear condition differences and were much less reactive over trials.The model favored a mixture of “one-on” and “one-on, one-off” interventions, among others in four-variable problems, and favored fixing more variables off when wS was high; “one-on” choices fell below 50% by its final test.Source: Interventions · pp. 29–30
  • Participants were slightly more likely to select one on interventions on their first test (57%) than on their last test (50%; t(238) = 1.7, p = .01).Source: Interventions · p. 30
  • 49% of participants fixed “on” components with at least one child in their latest hypothesis, versus 30% for other components (t(238) = 5.5, p < .001).The overall pattern was not clearly consistent with any one local focus, but might fit a mixture of all three.Source: Interventions · p. 30

Experiment 2: Unknown strengths

Experiment 2 examined how participants learn causal structures and choose interventions when connection reliability and background-noise parameters are unknown. It assessed their judgments, predictions, explanations, confidence, and intervention choices to characterize their learning under this uncertainty.

Experiment 2: Unknown strengths

Methods

  • Participants were not trained on reliability or background-noise parameters, and reported edge confidence and predicted outcomes for variables left free using sliders.Source: Design and procedure · p. 32
  • On a final chain problem, participants explained why they selected each intervention, allowing their stated reasons to be compared with model predictions.Source: Design and procedure · p. 32

Results and discussion

  • Accuracy was above chance in all eight conditions (all t statistics > 6.1, all p values < 0.001).Source: Judgments · p. 33
  • Across considered priors, participants underperformed Bayes-optimal observers except at wS = 0.75, wB = 0.1 under SU and wS = 0.75, wB = 0.25 in the remain condition.The second exception held under all three considered priors.Source: Judgments · p. 33
  • Mean accuracy was 0.61±0.21 in Experiment 2 versus 0.56±0.21 in matched Experiment 1 problems, a nonsignificant difference, t(229) = 1.9, p = .054.Source: Comparison with known strength… · p. 33
  • Participants’ background-noise judgments suffered bias and variance; they were unaffected by connection reliability (wS t(106)=0.88, p=0.37) but affected by higher background-noise levels (wB t(108)=2.7, p=0.008).Source: Comparison with known strength… · p. 34
  • Removing records of previous judgments between trials increased edit distance from .73 in the remain condition to 1.0 in the disappear condition, yet distances remained significantly below UU, SU, SS posterior or random-sampling predictions (all p’s < .001).Source: Comparison with known strength… · p. 35
  • β = .35 for predictions from the latest structure judgment, compared with β = .29 for predictions marginalized over the full posterior, and the relationship was significantly stronger.Reported outcome predictions tracked the predictive distribution under the participants’ latest structure judgment more closely than the full-posterior prediction (Cox test Z = 10.9, p < .001).Source: Additional measures · p. 36
  • “One-on” interventions made up 39% of selections, while constrained “one-on one-off” tests made up 38%, versus 12% in Experiment 1’s three-variable problems.Selections and intervention-sequence informativeness were not closely consistent with global expected information or any single local focus, but could fit a mixture of local effect-focused, edge-focused, and confirmatory strategies.Source: Interventions · p. 36
  • Explanation type predicted performance: categories 1, 2, 4, 5, and 6 were associated with higher final scores than unprincipled or random intervening.Those categories’ scores were [14.5, 12.9, 13.9, 13.9, 13.9] points versus 10.2 for the reference category; primary-category inter-rater agreement was 0.73, with Cohen’s κ = 0.64 ± 0.04.Source: Free explanations · p. 37

Summary of Experiments

  • Participants identified causal structure effectively without specific parameter knowledge, and varying plausible priors for edge reliability or background noise had little effect on judgments or intervention-choice predictions.The priors compared concerned edge reliability (wS) and background noise (wB).Source: Summary of Experiments · p. 39
  • Remain/disappear reporting did not affect overall judgment accuracy, but it affected sequential dependence, especially among lower performers who may have forgotten their previous judgment.Source: Summary of Experiments · p. 39
  • Additional task measures supported the idea that participants represented one model at a time, with intervention-outcome predictions generated from their current hypothesis rather than averaged across models.Source: Summary of Experiments · p. 39
  • Expectation judgments aligned more with participants’ current hypothesis than with marginal likelihoods, although the measures were noisy and the effects small.Source: Summary of Experiments · p. 39

Modeling individual behavior

This section assesses how well Neurath’s ship and local-focus intervention models account for participants’ judgments and intervention choices across three experiments. It examines individual model fits and variation across participants in model recovery and search behavior.

Modeling individual behavior

  • Across the examined experiments, NS simulations qualitatively corresponded to participants’ judgments, and its local intervention scheme corresponded to their interventions.Simple endorsement and WSLS also captured qualitative judgment patterns.Source: Modeling individual behavior · p. 39

Judgments

  • NS had the lowest aggregate BIC (93381), followed by SE (94326), WSLS (97643), NS-RE (101837), Rational (1207209), and Baseline (149313).Table 3 reports the model-fit results across all experiments.Source: Results and discussion · pp. 41–42, Table 3 · p. 42
  • NS best fit Bramley, Lagnado, and Speekenbrink (2015) and Experiment 1, while SE won Experiment 2; all three heuristics beat exact Bayesian inference.NS outperformed WSLS substantially and SE to a lesser degree.Source: Results and discussion · p. 42
  • The best-fitting model varied across participants: WSLS fit 102 best, NS 85, SE 80, NS-RE 70, Rational 28, and Baseline 4.This spread indicates individual differences in learning behavior.Source: Results and discussion · p. 42
  • Generating models were recovered 74% of the time for Bramley, Lagnado, and Speekenbrink (2015), 82% for Experiment 1, and 75% for Experiment 2, versus 17% chance.Baseline-, WSLS-, and SE-generated data were nearly always correctly recaptured.Source: Results and discussion · pp. 42–43
  • NS almost never captured data generated by other models, but NS-generated data were often captured by NS-RE, especially in Bramley, Lagnado, and Speekenbrink (2015) and Experiment 2.NS-RE makes local edits without preferentially approaching more likely models.Source: Results and discussion · p. 43
  • 24/27 simulations using parameters from participants best described by NS were recovered successfully.The analysis suggests some participants classified as NR-RE may instead have been doing something more like NS; in Experiment 1, NS-RE mean accuracy was commensurate with SE, WSLS, and NS.Source: Results and discussion · p. 43
  • Ten participants in Bramley, Lagnado, and Speekenbrink (2015) and 19 in Experiment 2 were best fit by the Rational model, scoring near ceiling: 14.0 of 15 connections and 14.7/21, versus 15.5 for perfect Bayesian integration.Source: Results and discussion · p. 43
  • Participants’ median fitted search lengths (λ) were between 1 and 2 in all three experiments, consistent with generally short searches.The same participant might sometimes not search at all (k = 0) and at other times search much longer than λ.Source: Results and discussion · p. 43
  • Median fitted ω values of 6, 4, and 9.2 across the three experiments suggest moderate hill-climbing, while some participants showed near-deterministic hill climbing.These parameters were fit only to the easier three-variable problems; the largest ω values might have been tempered if the four-variable problems had also been fit.Source: Results and discussion · p. 43

Interventions

  • No single intervention model closely matched participants’ response patterns; their overall choices were consistent with a mixture of local tests, and model fits showed substantial individual variation.Experiment 2 free responses commonly described targeting specific edges, variable effects, and confirmation of the current hypothesis.Source: Models · p. 44, Results and discussion · p. 45
  • BIC 97757 was the mixed local-focus model’s lowest total BIC across the three experiments; effects, global, edges, confirmation, and baseline followed.Mixed was the best overall fitting model for Experiments 1 and 2.Source: Results and discussion · pp. 45–46, Table 5 · p. 46
  • 121 participants were best described by an effects focus and 77 by a mixed focus, while 58, 43, 36, and 35 were best fit by global, confirmation, edge, and baseline models, respectively.Overall, 277/370 participants were fit by one of the local uncertainty-driven models.Source: Results and discussion · p. 46
  • Effects- and edge-focused intervention choices corresponded strongly to participants’ coded explanations in Experiment 2, but confirmatory explanations were most often classified as effect-focused tests of the true model’s root variable.Confirmatory explanations were not correspondingly identified as confirmatory by the model.Source: Results and discussion · p. 46

General Discussion

The discussion considers how causal judgments and intervention choices relate to the single-hypothesis, local-update Neurath’s ship account, while examining its limits and alternative approximations. It frames locally focused interventions as heuristic tests and the model as approximate Bayesian inference that may extend to how people build and adapt theories.

General Discussion

  • Judgments partly reflected the true posterior but showed sequential dependencies, while intervention choices reflected average expected information yet were insufficiently reactive to observed evidence.These findings were reported across three experiments.Source: General Discussion · pp. 47–48
  • The Neurath’s ship account fit best overall, although all three proposals had merit and simple endorsement won in Experiment 2.The account assumes a single causal model that is locally changed to better explain the latest data; some individuals were better fit by win-stay.Source: General Discussion · p. 48
  • Participants’ intervention choices were modeled as stochastic focus on local aspects of uncertainty, producing behavior comparatively invariant to the prior.Modeling suggested a broad spread of local focuses both between and within participants.Source: General Discussion · p. 48
  • The account links single-hypothesis, Gibbs-esque search to approximate inference and explains intervention selection through local rather than global uncertainty reduction.The discussion connects the model to sequential Monte Carlo particle filtering and MCMC, specifically Gibbs sampling.Source: General Discussion · p. 48

Limitations of Neurath’s ship

  • Repeated judgments required assuming they reflected participants’ latest beliefs and did not fundamentally alter learning; the slight reduction in dependence when edges had to be re-reported is consistent with elicitation prompting reconsideration.The authors incentivized best and latest guesses and varied reporting conditions; allowing untouched edges avoided prompting reconsideration but risked conflating incremental change with response laziness.Source: Measurement effects · pp. 48–49
  • The experiments ruled out cycles, limiting application to real-world reciprocal causes; the authors note Neurath’s ship could support online inference for dynamic Bayesian networks or chain graphs.Exploring these structures would require changing the experiment’s semantics so participants could understand reports involving dynamic interactions.Source: Acyclicity · p. 49
  • The modeling assumed noisy-OR likelihoods, and future work may relax that assumption.Participants were trained on noise sources, but the passage identifies the noisy-OR functional form as a pragmatic limitation.Source: Evaluation of evidence · p. 50
  • The assumption that learners consider only one global hypothesis may be too strong for small, constrained problems, where some could reason over whole models.This possibility is especially relevant to three-variable problems, where a learner might shift between structures in a single step.Source: Antifoundationalism · p. 50
  • Although participants’ judgments showed high sequential dependence, the current models accommodate occasional large jumps through variable search length (k), which can let a learner reach a radically different model in a single update.Source: Antifoundationalism · pp. 50–51
  • The model’s recent-evidence assumption is a simplification, since people may retain older evidence, forget recent evidence, or remember summaries selectively.The authors propose studying what learners remember to better understand the trade-off between retaining evidence and absorbing it into beliefs.Source: Selective memory · p. 51
  • The authors fitted a static search-behavior parameter and found evidence of moderate hill climbing, while suggesting that search may become more exploratory or shorten as pressure to settle increases.They also identify experiments distinguishing single-model local search from variational approaches, in which learners maintain multiple local distributions, as future work.Source: Selective memory · p. 51

Alternative approximations and representations

  • Gibbs sampling fit better than Metropolis–Hastings sampling, whose behavioral predictions were similar, supporting the proposal’s emphasis on local inference.The Metropolis–Hastings approach proposes a wholesale alternative and accepts or rejects it, whereas Gibbs updates one part conditional on the rest.Source: Alternative approximations and… · p. 51

Choosing interventions aboard Neurath’s ship

  • Experiment 2 participants chose constrained interventions more frequently than participants in the other experiment.The authors suggest this difference may reflect narrower attention under greater noise uncertainty or differences between participant pools.Source: Choosing interventions aboard… · p. 52
  • Participants’ explanations typically focused on one local aspect of the problem and drew on a mixture of query types across tests.They discussed causal connections, component effects, parameter uncertainty, and sometimes confirmation of their current hypothesis.Source: Choosing interventions aboard… · p. 52
  • Participants’ explanations shifted toward confirming their current hypothesis on the last test, when they knew no further tests would follow.The authors describe this shift as potentially sensible but caution that subjective explanations are problematic to interpret.Source: Choosing interventions aboard… · pp. 52–53
  • Because rule-based explanations are easier to express than complex weighting, participants may describe only one factor and appear to use a one-reason strategy.Source: Choosing interventions aboard… · p. 53
  • With appropriate caution, participants’ explanations offer a window on their subjective active testing and suggest a mixture of heuristic questions, each focused on one uncertainty at a time.Source: Choosing interventions aboard… · p. 53
  • Intervention models fit choices less tightly than beliefs, with greater individual variation and smaller proportional BIC reductions.The authors suggest intervention choices are harder to predict because many tests can be similarly informative and performed in different orders.Source: Choosing interventions aboard… · p. 53
  • Relative to the local intervention schema, most participants tended to target areas of high current uncertainty, though the model does not explain how they did so.Source: Choosing interventions aboard… · p. 53
  • The model unifies modular, constraint-seeking, and confirmatory testing within one schema, and many learners dynamically switch among them.Source: Choosing interventions aboard… · p. 54
  • Free responses suggest that participants’ initial tests addressed specific parts of uncertainty, making the prior choice of focus potentially more important than the final action.Source: Choosing interventions aboard… · p. 54
  • Tests that seemed uninformative about causal structure often targeted other uncertainties, including internal or background noise.Even interventions that appeared wholly uninformative could test hypotheses outside the possibilities the researchers intended participants to consider.Source: Choosing interventions aboard… · p. 54

The navy of one

  • The account frames a single-particle search as bounded rational approximate Bayesian inference, while acknowledging that very short searches may not calculate a proper posterior.Source: The navy of one · pp. 54–55
  • The account spans fully probabilistic to heuristic processing, with a single global model incrementally updated at the higher end of cognition.Retaining Bayesian machinery can express heuristic behavior without a separate process model.Source: The navy of one · pp. 55–56
  • Formalizing approximate inference offers an account that can generalize more readily across tasks than task-specific heuristic models.The authors characterize heuristic models as difficult to combine or apply in broader circumstances when they lack a formal relationship to approximate rationality.Source: The navy of one · p. 55

Scope of the theory

  • The model uses local, conditional edits to one representation, an approach the authors argue need not be limited to causal Bayes nets.They suggest analogous processes could model the induction and adaptation of rich cognitive representations, and how people generate theories and get stuck.Source: Scope of the theory · p. 56
  • Sequential conditional reevaluation shows how radical antifoundationalism can still support theory building.The model’s process incrementally revises a single representation rather than requiring a fixed foundation.Source: Scope of the theory · p. 56

Conclusions

The paper proposes causal theory change through local updates to a single global hypothesis, inspired by Neurath’s ship. Its accounts fit participants’ judgments and information-gathering behavior across experiments, and suggest how people manage learning limitations.

Conclusions

  • The proposed model draws on the Neurath’s ship idea that learners cannot maintain a distribution over all possible beliefs.Instead, learners make sequential local changes to a single representation as they incorporate evidence.Source: Conclusions · p. 57
  • The model accounts for participants’ sequences of causal judgments in three experiments.The authors describe it as a flexible candidate for explaining how complex representations can be formed in cognition.Source: Conclusions · p. 57
  • The authors present the approach as a candidate account of how people construct rich causal representations through interaction with a complex world.Source: Conclusions · p. 57
  • Participants’ information-gathering behavior is consistent with resolving manageable local uncertainty rather than global uncertainty.The authors interpret this pattern as showing awareness of learning limitations.Source: Conclusions · p. 57

Representation and inference

Causal models assign data likelihoods using the states of variables’ parents, excluding variables directly intervened upon. Posterior beliefs over models are then conditioned on prior beliefs, observations, interventions, and—depending on the experiment—known or marginalized parameters.

Representation and inference

  • A noisy-OR causal model assigns each datum a likelihood from the probabilities of non-intervened variables given their parents.A datum can be a complete observation or an intervention outcome.Source: Representation and inference · p. 67
  • The posterior over causal models is conditioned on a prior and data associated with interventions.The model parameters can be conditioned on when known, as in Experiment 1, or marginalized when unknown, as in Experiment 2.Source: Representation and inference · p. 67

Intervention choice

The intervention-choice scheme values interventions by expected uncertainty reduction, using Shannon entropy to quantify uncertainty over causal models. It selects the most valuable intervention at each step and marginalizes over parameter values when they are unknown.

Intervention choice

  • At each time point, the scheme chooses the intervention with the greatest expected value, using the current posterior as the next-step prior.When parameters are unknown, the calculation uses a marginal distribution over them.Source: Intervention choice · pp. 67–68
  • Shannon entropy quantifies uncertainty over causal models, and an intervention’s reduction compares the prior with the posterior after its outcome.Source: Intervention choice · p. 67
  • Intervention value is the expected reduction in uncertainty after observing its possible outcomes.The expectation averages prospectively over the possible outcomes of an intervention.Source: Intervention choice · p. 67

An algorithmic-level model of sequential belief change

The model represents belief as a causal graph and updates it by resampling individual edge states using evidence. Search can involve repeated edge updates and variable lengths; the authors report that testing the initial-graph assumption from later observations yielded better fits and a stronger win over comparison models.

An algorithmic-level model of sequential belief change

  • A belief is a causal model encoded by edge states: i → j, no edge, or i ← j; edge changes that create cycles have zero probability.Source: An algorithmic-level model of… · p. 68
  • The learner selects an edge and resamples its state from a conditional distribution based on the other edges and accumulated evidence.The account uses chains of dependent samples and allows systematic or random edge selection.Source: An algorithmic-level model of… · p. 68
  • If search leaves the model unchanged, the learner keeps accumulating evidence; if it changes the model, that evidence is used up before collection begins again.After k repetitions, the final edge choices constitute the new belief.Source: An algorithmic-level model of… · p. 68
  • Fitting from t=2 onward, rather than relying on the assumed initial unconnected graph, produced better overall fits and a stronger win for Neurath’s ship over the other models considered.Source: An algorithmic-level model of… · p. 69
  • The authors consider search variants that favor the most likely edge state when ω > 1 and increase randomness when ω < 1.They also consider resampling, hill climbing, and random change.Source: Resampling, hill climbing or random… · p. 69
  • Search length is drawn from a Poisson distribution, and averaging across possible lengths gives the probability of each new belief.Source: A distribution over search lengths · p. 69, Putting these together · p. 69
  • The model represents search over beliefs with a transition matrix built by averaging edge-update distributions weighted by edge-selection probabilities.Applying the matrix for a search length gives the distribution over beliefs at the end of that search.Source: Putting these together · p. 69

A local uncertainty schema

The intervention-value calculations vary the uncertainty focus: an edge, a current-versus-null hypothesis comparison, or a variable’s effects. Each focus uses a corresponding prior and posterior entropy calculation.

A local uncertainty schema

  • For an edge focus, confirmation-focus entropy uses a uniform prior over three possible edge states and posterior entropies across possible outcomes.Intervention values are calculated using expected information.Source: Edge focus · p. 70
  • For a variable’s effects, intervention values use expected information with models partitioned by descendant set and posterior entropy summed over possible outcomes.The prior is uniform over models.Source: Effect focus entropy · p. 70
  • For a current-versus-null hypothesis focus, expected information uses a uniform prior over the two hypotheses to calculate prior entropy and posterior entropies.Source: Confirmation focus entropy · p. 70

B Additional modeling details

The additional modeling details specify fitting and decision-noise procedures, marginalization over unknown noise in Experiment 3, and the baseline used to evaluate model fits.

B Additional modeling details

  • Models were fit by maximum likelihood, with Brent or Nelder–Mead optimization and repeated random starts to check convergence to global optima.Source: B Additional modeling details · p. 71
  • A lapse parameter ϵ models occasional inattention or forgotten outcomes; predicted belief probabilities mix model predictions with uniform noise.The noise mixture limits the influence of random judgments on other model parameters.Source: B Additional modeling details · p. 71

Marginalization

  • For Experiment 3, 1000 paired uniform wS and wB samples were averaged to compute marginal likelihoods and posteriors, then expected information gain.Source: Marginalization · p. 71

Evaluating fits

  • The baseline model serves as the null model for calculating BIC and pseudo-R2 for all other models.Source: Evaluating fits · p. 71
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