Models for the balance scale

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Talk. Summary of empirical resultsSummary of modelsHow rule like are children?Do connectionist networks explain children's behavior?New data: transitionsAn alternative ACT-R modelNewer data: RTHow to proceed?. Balance scale task. Rule I: only weightRule II: also distance when weights equalR

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Models for the balance scale

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1. Models for the balance scale On balance, connectionist networks do not work Han van der Maas University of Amsterdam Brenda Jansen, Hedderik van Rijn, Maartje Raijmakers, Philip Quinlan

2. Talk Summary of empirical results Summary of models How rule like are children? Do connectionist networks explain children’s behavior? New data: transitions An alternative ACT-R model Newer data: RT How to proceed?

3. Balance scale task Rule I: only weight Rule II: also distance when weights equal Rule III:guess on conflict items Rule IV: compare torques Addition: compare sums Buggy: shift weights until non-conflict item QP: all conflicts balance SDD: smallest distance down

4. Short history of balance scale research Piaget & Inhelder: proportional reasoning Siegler: Item types, Rules 1 to 4, role of encoding, overlapping waves Wilkening: Functional measurement (Kerkman & Wright) Ferretti & Butterfield: torque difference effect Others: Addition rule, Buggy rule, QP rule

5. Short history of balance scale modeling Production Rule Models Klahr & Siegler, ’78, representational issues Sage & Langley, ’83, ’87, construction of rules by discrimination analysis, transition mechanism Newell, ’90, SOAR Decision-Tree Models Schmidt & Ling, ’96, incremental decision tree learning Connectionist models McClelland, ’89, ’95, back-propagation Shultz et al, ’91, ’94, ’95, modification of network topology by cascade-correlation New ‘symbolic’ model: ACT-R (van Rijn ‘03)

6. McClelland’s PDP model & Shultz’s CC model Rule like behavior of children can be explained with implicit, gradual, patterns of activation: non-symbolic Main points of evidence: Simulate rule-like data Stages Torque difference effect Addition rule

7. Critical evaluation Why? Connectionist account is dominant PDP and CC are applied to a large number of cognitive tasks PDP and CC are extremely simple models Dynamic system account and connectionist claim to model higher order cognition (i.e., replace information processing accounts) Progress requires criticism Note This is not a dynamical critique ! In the past I criticized the dynamical approach for partly the same reasons (higher order cognition)

8. So let’s see.. Do connectionist models really simulate rule-like data? Analysis of rules Discontinuities How important is the torque difference effect? Can PDP/CC models do the torque rule?

9. Rules Criteria of Reese ‘89: regular, consistent, discontinuous, transferable, evidenced from different sources, conscious Data of Siegler suggest consistency Analyzed with Rule Assessment Methodology (RAM) RAM also used for analysis of data simulated with networks

10. Rule Assessment Methodology 6 item types Compare answer patterns with answer patterns according to the rules Allow certain misfit (20%) Additional criteria for some rules If two rules fit equally well then ‘unclassified’ If overall fit >80% then acceptable This type of procedure is used a lot in research in which subjects have to be classified

11. Problems of RAM Informal procedure (no statistical underpinning) Arbitrary criteria that can not be generalized Rules are pre-specified What to do with ties? Rule III cannot be detected The necessity of individual rules cannot be assessed There are no formal fit measures

12. Latent class analysis Advanced rule assessment Established statistical technique Many books, courses and papers Commercial and free programs Hundreds of applications in all sciences in last years Especially useful in this case Categorical latent structure model Can be used explorative (to find new rules) and/or confirmative (to test rules) Solves problem of ties Criteria based on statistical theory

13. Example of LCA: conflict balance

14. Note: LCA is not without problems Requires lots of data Baysian methods allow analysis of all items at once Model selection criteria AIC or BIC, bootstrap statistics Rule change during test Hidden Markov models Nevertheless LCA (and related statistical techniques) are a welcomed step forward

15. Evidence for rule use by children LCA models give rise to good fits; at least 80 % of the classes can be ascribed to known rules Verbal justifications: awareness From different sources: justifications, RT Transferable: Siegler ‘81 Discontinuous change from R1 to R2 Hysteresis (Jansen &vdMaas, 2001) bimodality Many children and adults master rule IV

16. But... Rule III: guessing Mostly addition Not perfect fit of rule model Given strictness of LCA Rule switching Is variability in rule use Measurement error Variability near transitions Torque difference effect

17. Rule classification depends on torque difference Ferretti & Butterfield investigated 4 levels: On non-conflict items: 1, 3, 12 and 24-30 units On conflict items: 1, 3, 5, and 18-24 units They found a significant effect of torque difference Torque difference effect

18. Re-analysis Jansen & vdMaas ‘97

19. Analysis of PDP model No fitting latent class model (Jansen & vdMaas, ‘97) Only 19 % of classes fit a known rule Stages are artifact of scoring method (Raijmakers, ‘96): only continuous change no bimodality No Rule IV

20. Analysis of the CC model Quinlan, Rendall, Jansen, Booij, vdMaas (in revision) report on two independent replications of CC model: Latent class models fit (multigroup LCA) 9 class multigroup model LR=61.24 p(bootstrap) = .54, best BIC

21. LCA model interpretations Very clear: Rule 1 Rule II Mixture of additive rules No pure addition Two ‘odd’ rules No Rule IV Developmental order partly incorrect

22. Rue IV: torque Jansen & vdMaas ‘02: 11% use Rule IV (LCA), note: no training, no feedback! Rule IV is easily taught (but hard to discover) LCA of CC and PDP show (no signs of) Rule IV Only 10 % of the torque items (which are failed with the addition rule) are solved correctly Long training, on selective sets of items does not help

23. Can PDP and CC multiply? Additive activation function At best: mimic multiplication with moderate success with careful choice of weights in a limited training and test set

24. Evaluation CC Works better than PDP Rule I, II and weighted addition (Wilkening) LCA models fit But Rule I, II (i.e. weight preference) by bias in input Weighted addition is in the activation rule LCA shows no weighted addition by children on balance scale No Rule IV

25. Transition model

26. New data We found small but significant evidence for hysteresis in the transition from Rule I to Rule II

27. Act-R model of balance scale

28. Model evaluation: Rules, rule construction Developmental order Product difference effect for extreme PD differences Rule I to II transition (possibly hysteresis as function of saliency) Learning without feedback Disadvantages elements of rules transferred from other domains Constraints build in Rules a bit ‘too good’ Open question: can we model this Rule I-II transition in a neural network?

29. Newer data: RT’s We found support for rules with response times Computer test (10* 7 items), 147 children 44 undergrads. Rule classification by cluster analysis (agreement with rule assessment method high) Fit of RT models Regression models (using R packages lm, nls and nlme) Linear part (rules + item characteristics) non-linear (exponential learning function) mixed effects (individual intercepts & slopes) Estimates of duration of processing stages, learning and age parameters, and inconsistency parameters + fit measures

30. Mean RT’s rules * types

31. Compensation: Addition or buggy rule Addition rule: sum weight and distance on each side and compare the sums. Buggy rule: shift the pile with the largest number of weights until either distances or weights are equal. Problem: response patterns are the same !

32. Addition or buggy

33. Model 2 fit (example compensation rule)

34. Weight-distance items

35. RT: new challenge We think the RT data provide a new challenge for computational modeling of the balance scale task Lots of new effects to explain Both connectionist and Act-R models are able to predict response times Usher & McClelland model

36. On balance, they do not work… Connectionist networks may fail to mimic rules but since children’s behavior is not perfectly rule-like, there is always room for discussion We think behavior is more rule-like than connectionist networks can explain, based on: Torque difference effect not relevant LCA results Humans use Rule IV Transitions between rules McClelland ‘95: PDP models leave out explicit rules (which humans can and do use) and are therefore missing an important aspect of human cognition

37. Explicit cognition Brain activity is largely implicit, graded, continuous and parallel But in the end: explicit cognition is what makes us different from our cats and dogs

38. Is there a future for connectionist modeling of balance scale? Yes! Use more complex models PDP and CC are more than 10 years old Allow multiplicative activation rules, or allow torque by different input representations Use networks that show phase transitions in their learning behavior Use networks with interesting dynamical properties ART networks provide interesting possibilities for the future Many more new models Combine neural with symbolic architectures Rule extraction from neural networks Focus on new data

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