{"title": "A Quantitative Model of Counterfactual Reasoning", "book": "Advances in Neural Information Processing Systems", "page_first": 123, "page_last": 130, "abstract": null, "full_text": "A Quantitative Model of Counterfactual\n\nReasoning\n\nDaniel Yarlett\n\nDivision of Informatics\nUniversity of Edinburgh\n\nEdinburgh, Scotland\ndany@cogsci.ed.ac.uk\n\nMichael Ramscar\n\nDivision of Informatics\nUniversity of Edinburgh\n\nEdinburgh, Scotland\nmichael@dai.ed.ac.uk\n\nAbstract\n\nIn this paper we explore two quantitative approaches to the modelling of\ncounterfactual reasoning \u2013 a linear and a noisy-OR model \u2013 based on in-\nformation contained in conceptual dependency networks. Empirical data\nis acquired in a study and the \ufb01t of the models compared to it. We con-\nclude by considering the appropriateness of non-parametric approaches\nto counterfactual reasoning, and examining the prospects for other para-\nmetric approaches in the future.\n\n1 Introduction\n\nIf robins didn\u2019t have wings would they still be able to \ufb02y, eat worms or build nests? Pre-\nvious work on counterfactual reasoning has tended to characterise the processes by which\nquestions such as these are answered in purely qualitative terms, either focusing on the\nfactors determining their onset and consequences (see Roese, 1997, for a review); the qual-\nitative outline of their psychological characteristics (Kahneman and Miller, 1986; Byrne\nand Tasso, 1999); or else their logical or schematic properties (Lewis, 1973; Goodman,\n1983). And although Pearl (2000) has described a formalism addressing quantitative as-\npects of counterfactual reasoning, this model has yet to be tested empirically. Furthermore,\nthe non-parametric framework in which it is proposed means certain problems attach to it\nas a cognitive model, as discussed in  6.\n\nTo date then, the quantitative processes underlying human counterfactual reasoning have\nproven surprisingly recalcitrant to philosophical, psychological and linguistic analysis. In\nthis paper we propose two parametric models of counterfactual reasoning for a speci\ufb01c\nclass of counterfactuals: those involving modi\ufb01cations to our conceptual knowledge. The\nmodels we present are intended to capture the constraints operative on this form of infer-\nence at the computational level. Having outlined the models, we present a study which\ncompares their predictions with the judgements of participants about corresponding coun-\nterfactuals. Finally, we conclude by raising logistical and methodological doubts about a\nnon-parametric approach to the problem, and considering future work to extend the current\nmodels.\n\n\f2 Counterfactuals and Causal Dependencies\n\nOne of the main dif\ufb01culties in analysing counterfactuals is that they refer to alternative\nways that things could be, but it\u2019s dif\ufb01cult to specify exactly which alternatives they pick\nout. For example, to answer the counterfactual question we began this paper with we\nclearly need to examine the possible states of affairs in which robins don\u2019t have wings in\norder to see whether they will still be able to \ufb02y, eat worms and build nests in them. But the\nproblem is that we can imagine many possible ways in which robins can be without wings\n\u2013 for instance, at an extreme we can imagine a situation in which the robin genotype failed\nto evolve beyond the plankton stage \u2013 not all of which will be relevant when it comes to\nreasoning counterfactually.\n\nIn the alternatives envisaged by a counterfactual some things are clearly going to differ\nfrom the way they are in the actual world, while others are going to remain unchanged.\nAnd specifying which things will be affected, and which things will be unaffected, by\na counterfactual supposition is the crux of the issue. Counterfactual reasoning seems to\nrevolve around causal dependencies: if something depends on a counterfactual supposition\nthen it should differ from the way it is in the actual world, otherwise it should remain\njust as it is. The challenge is to specify exactly what depends on what in the world \u2013 and\ncrucially to what degree, if we are interested in the quantitative aspects of counterfactual\nreasoning \u2013 in order that we can arrive at appropriate counterfactual inferences. Clearly\nsome information about our representation of dependency relations is required.\n\n3 Dependency Information\n\nFortunately, data is available about people\u2019s representations of dependencies, albeit in a\nlimited domain. As part of an investigation into feature centrality, Sloman, Love and Ahn\n(1998) explored the idea that a feature is central to a concept to the degree that other features\ndepend on it. To test this idea empirically they derived dependency networks for four\nconcepts \u2013 robin, apple, chair and guitar \u2013 by asking people to rate on a scale of 0 to 3 how\nstrongly they thought the features of the four concepts depended on one another. One of\nthe dependency structures derived from this process is depicted in Figure 1.\n\n4 Parametric Models\n\nThe models we present here simulate counterfactual reasoning about a concept by oper-\nating on conceptual networks such as the one in Figure 1. A counterfactual supposition\nis entertained by setting the activation of the counterfactually manipulated feature to an\nappropriate level. Inference then proceeds via an iterative algorithm which propagates the\neffect of manipulating the selected feature throughout the network.\n\nIn order to do this we make two main assumptions about cause-effect interactions. First we\nassume that a node representing an effect,\n, will be expected to change as a function of (i)\n, has itself changed, and (ii) the degree\nthe degree to which a node representing its cause,\nto which\n, will\naffect a target node,\n, independently of one another and in a cumulative fashion. This\nmeans that the proposed models do not attempt to deal with interactions between causes.\n\n. Second, we also assume that multiple cause nodes,\n\ndepends on\n\n\u0001\u0003\u0002\u0005\u0004\u0007\u0006\n\nThe \ufb01rst assumption seems warranted by recent empirical work (Yarlett & Ramscar, in\npreparation). And while the second assumption is certainly not true in all instances \u2013\ninteraction effects are certainly possible \u2013 there do seem to be multiple schemas that can be\nadopted in causal reasoning (Kelley, 1967), and it may be that the parametric assumptions\nof the two models correspond to a form of reasoning that predominates.\n\n\n\u0001\n\n\u0001\n\n\ffeathers\n\nsmall\n\nbeak\n\nflies\n\neats\n\ntwo legs\n\nliving\n\nmoves\n\nwings\n\nbuilds\nnests\n\nchirps\n\nred\nbreast\n\neats\nworms\n\nlays\neggs\n\nFigure 1: Dependency network for the concept robin. An arrow drawn from feature A to\nfeature B means that A depends on B. Note (i) that only the strongest dependency links\nare shown, but that all dependency information was used in the simulations; (ii) there\nis a numeric strength associated with every dependency connection, although this is not\nshown in the diagram; and (iii) the proposed models propagate information in the opposite\ndirection to the dependency connections.\n\n4.1 Causal Dependency Networks\n\nThe dependency networks obtained by Sloman, Love and Ahn (1998) were collected by\nasking people to consider features in a pairwise fashion, independently of all other fea-\ntures. However, causal inference requires that the causal impact of multiple features on a\ntarget node be combined. Therefore some preprocessing needs to be done to the raw depen-\ndency networks to de\ufb01ne a causal dependency network suitable for using in counterfactual\n, in\n,\n, are\n\ninference. The original dependency networks can each be represented as a matrix \nwhich\u0001\nin concept\u0007\n\nas judged by the original participants. The modi\ufb01ed causal dependency networks,\nde\ufb01ned as follows:\n\nrepresents the strength with which feature\u0005 depends on feature\u0006\n\n\u0002\u0004\u0003\n\n(1)\n\n(2)\n\nwhere\n\n\u0002\u0004\u0003\n\u000b\r\f\u000f\u000e\u0011\u0010\u0011\u0012\nwhere\u0015\u001c\u001b\u001e\u001d ;\n\notherwise.\n\n\u0003\u0014\u0013\n\n\u0002\u0004\u0003\n\t\n\t\u0018\u0017\u001a\u0019\n\n\u0010\u0011\u0012\u0016\u0015\n\nThis transformation achieves two things. Firstly it normalises the weights to be in the range\n0 to 1, instead of the range 0 to 3 that the original ratings occupied. Secondly it normalises\nthe strength with which each input node is connected to a target node with respect to the\nsum of all other inputs to the target. This means that multiple inputs to a target node cannot\nactivate the target any more than a single input.\n\n4.2 Parametric Propagation Schemes\n\nWe can now de\ufb01ne how inference proceeds in the two parametric models: the linear and\ndenote the feature being counterfactually manipulated (\u2018has\nrepresents\nto have changed as a result of the counterfactual\n\nbe a matrix in which each component !\n\nthe noisy-OR models. Let\u001f\nwings\u2019 in our example), and let \nthe amount the model predicts feature\u0006\n\n\u0002#\"\n\n\u0006\n\u0006\n\u0001\n\u0006\n\b\n\u0006\n\u0001\n\u0006\n\u0001\n\u0006\n\u000e\n\u0013\n\u001d\n$\n\fiterations. To initialise both models all predicted levels of change\n\n, are initialised to 0:\n\n, after\n\nmodi\ufb01cation to\u001f\nfor features other than the manipulated feature,\u001f\n\u0007\t\b\n\u0001\u0006\u0005\n\n4.2.1 Linear Model\n\n!\u0002\u0001\u0004\u0003\n\nThe update rules for each iteration of the linear model are de\ufb01ned as follows. The manipu-\nis set to an initial activation level of 1, indicating it has been counterfactually\n\nmodi\ufb01ed1. All other features have their activations set as speci\ufb01ed below:\n\nlated feature\u001f\n\nThis condition states that a feature is expected to change in proportion to the degree to\nwhich the features that in\ufb02uence it have changed, given the initial alteration made to the\n, and the degree to which they affect it. The general robustness of\nlinear models of human judgements (Dawes, 1979) provides grounds for expecting a good\ncorrelation between the linear model and human counterfactual judgements.\n\nmanipulated feature\u001f\n\n4.2.2 Noisy-OR Model\n\nThe second model uses the noisy-OR gate (Pearl, 1988) to describe the propagation of\ninformation in causal inference. The noisy-OR gate assumes that each cause has an inde-\npendent probability of failing to produce the effect, and that the effect will only be absent\nif all its associated causes fail to produce it. In the counterfactual model noisy-OR propa-\ngation is therefore formalised as follows:\n\n$\f\u000b\n\n\t\u000e\r\u0004\u000f\n\n(3)\n\n(4)\n\n(5)\n\n$\f\u000b\n\n!\u0010\u0001\n\n\u0019\u0012\u0011\u0014\u0013\n\n\u0019\u0015\u0011\n\nThe questions people were asked to validate the two models measured how strongly they\nwould believe in different features of a concept, if a speci\ufb01c feature was subtracted. This\ncan be interpreted as the degree to which their belief in the target feature would vary given\nthe presence and the absence of the manipulated feature. Accordingly, the output of the\nnoisy-OR model was the difference in activation of each node when the manipulated node\n\n\u001f was set to 1 and 0 respectively2.\n\n4.2.3 Clamping\n\nBecause of the existence of loops in the dependency networks, if the counterfactually ma-\nnipulated node is not clamped to its initial value activation can feed back through the net-\nwork and change this value. This is likely to be undesirable, because it will mean the\nnetwork will converge to a state in which the required counterfactual manipulation has not\nbeen successfully maintained, and that therefore its consequences have not been properly\nassimilated. The empirical performance of the two models was therefore considered when\n\nhence will not affect the correlational results.\n\nat convergence are simply multiples of the initial value selected and\n\n1Note that the performance of the linear model does not depend crucially on the activation of\u0016\nbeing set to 1, as solutions for\u0017\ndegree of change (expressed in Pearl\u2019s causal calculus as\u0018\u001a\u0019\u001c\u001b\u001e\u001d \u001f\n\n2This highlights an interesting difference in the output of the two models: the linear model outputs\nthe degree to which a feature is expected to change as a result of a counterfactual manipulation di-\nrectly, whereas the noisy-OR model outputs probabilities which need to be converted into an expected\n\n%'&)(+*-,\t./\u0018\u001a\u0019\u001c\u001b0\u001d1\u001f\n\n%'&\u001423*-, ).\n\n!\u0004\"$#\n\n!\u0004\"$#\n\n\"\n\n\t\n\u001d\n!\n\u0001\n\"\n\u0002\n\b\n\u000f\n\u0001\n!\n\u000f\n\"\n$\n\"\n\u0002\n\t\n\u000f\n\u0012\n\b\n\u000f\n\u0001\n!\n\u000f\n\"\n$\n\u0013\n\fthe activation of the manipulated node was clamped to its initial value, and not clamped.\n\nThe clamping constraint bears a close similarity to Pearl\u2019s (2000) \u2018\u0001\u0001\n\nwhich prevents causes of a random variable \u0002\noccurred in order to bring \u0002\n\naffecting its value when an intervention has\n\n\u0012\u0003\u0002\n\n\t\u0005\u0004\n\n\u0013 \u2019 operator,\n\n4.2.4 Convergence\n\n\t\u0006\u0004 about.\n\nPropagation continues in both models until the activations for the features converge:\n\n\u0012\b\u0007\n\n\u0012\n\t\n\n!\u0010\u0001\n\n!\u0002\u0001\n\n\t\f\u000b\u000e\n\n(6)\n\nThe models thus offer a spreading activation account of the changes induced in a conceptual\nnetwork as a result of a counterfactual manipulation, their iterative nature allowing the\neffect of non-local in\ufb02uences to be accommodated.\n\n5 Testing the Models\n\nIn order to test the validity of the two models we empirically studied people\u2019s intuitions\nabout how they would expect concepts to change if they no longer possessed characteristic\nfeatures. For example, participants were asked to imagine that robins did not in fact have\nwings. They were then asked to rate how strongly they agreed or disagreed with statements\nsuch as \u2018If robins didn\u2019t have wings, they would still be able to \ufb02y\u2019. The task clearly\nrequires participants to engage in counterfactual reasoning: robins do in fact have wings \u2013\nin normal contexts at least \u2013 so participants are required to modify their standard conceptual\nrepresentation in order to \ufb01nd out how this affects their belief in the other aspects of robins.\n\n5.1 Method\n\nThree features were chosen from each of the four concepts for which dependency informa-\ntion was available. These features were selected as having low, medium and high levels\nof centrality, as reported by Sloman, Love and Ahn (1998, Study 1). This was to ensure\nthat counterfactuals revolving around more and less important features of a concept were\nconsidered in the study.\n\nEach selected feature formed the basis of a counterfactual manipulation. For example, if\nthe concept was robin and the selected feature was \u2018has wings\u2019, then subjects were asked\nto imagine that robins didn\u2019t have wings. Participants were then asked how strongly they\nbelieved that the concept in question would still possess each of its remaining features if\nit no longer possessed the selected feature. For example, they would read \u2018If robins didn\u2019t\nhave wings, they would still be able to \ufb02y\u2019 and be asked to rate how strongly they agreed\nwith it.\n\nRatings were elicited on a 1-7 point scale anchored by \u2018strongly disagree\u2019 at the lower end\nand \u2018strongly agree\u2019 at the upper end. The ratings provided by participants can be regarded\nas estimates of how much people expect the features of a concept to change if the concept\nwere counterfactually modi\ufb01ed in the speci\ufb01ed way. If the models are good ones we would\ntherefore expect there to be a correlation between their predictions and the judgements of\nthe participants.\n\n5.2 Design and Materials\n\nParticipants were randomly presented with 4 of the 12 counterfactual manipulations, and\nwere asked to rate their agreement with counterfactual statements about the remaining,\n\n\u0004\n\u0013\n\"\n$\n\u0011\n\"\n$\n\u0004\n\u0002\n\u0013\n\fCounterfactual Concept\nrobin-wings\nrobin-lays-eggs\nrobin-eats-worms\nchair-back\nchair-arms\nchair-holds-people\nguitar-neck\nguitar-makes-sound\nguitar-used-by-music-groups\napple-grows-on-trees\napple-edible\napple-stem\nMean\n\nn\n13\n13\n13\n8\n8\n8\n8\n8\n8\n8\n8\n8\n\nLinear Model\n\nNoisy-OR Model\n\nClamped Non-Clamped Clamped Non-Clamped\n-0.870**\n-0.521*\n-0.066\n-0.451\n-0.530\n-0.815**\n-0.760*\n-0.889**\n0.235\n-0.748*\n-0.207\n-0.965**\n-0.549\n\n-0.739**\n-0.278\n-0.009\n-0.178\n-0.358\n-0.917**\n-0.381\n-0.939**\n0.290\n-0.905**\n-0.288\n-0.961**\n-0.472\n\n-0.062\n0.121\n-0.017\n0.148\n0.036\n-0.957**\n-0.181\n0.895**\n0.263\n-0.921**\n0.000\n-0.893**\n-0.131\n\n-0.044\n-0.105\n-0.069\n0.191\n0.042\n-0.928**\n-0.242\n-0.920**\n0.225\n-0.838**\n0.361\n-0.948**\n-0.273\n\nTable 1: The correlation between the linear and noisy-OR models, in the clamped and\nnon-clamped conditions, with participants\u2019 empirical judgements about corresponding in-\n\nferences. All comparisons were one-tailed (* \n\n\u001d\u0002\u0001\n\n\u001d\u0004\u0003 ; ** \n\n\u001d\u0005\u0001\n\n\u0019 ).\n\nunmanipulated features of the concept. People read an introductory passage for each infer-\nence in which they were asked to \u2018Imagine that robins didn\u2019t have wings. If this was true,\nhow much would you agree or disagree with the following statements...\u2019 They were then\nasked to rate their agreement with the speci\ufb01c inferences.\n\n5.3 Participants\n\n38 members of the Division of Informatics, University of Edinburgh, took part in the study.\nAll participants were volunteers, and no reward was offered for participation.\n\n5.4 Results\n\nThe correlation of the two models, in the clamped and non-clamped conditions, is shown\nrepeated-measures ANOVA revealed that there was a main effect\n\nin Table 1. A \u0006\b\u0007\t\u0006\n\u001d\u0005\u0001\r\u0003\u000f\u000e , \nof clamping (\n\n\u0019\f\u000b\n\u001d\u0002\u0001\n\u001d , \n\u0012\u0011\u000e ), and no interaction effect (\n\n(\n\n\u0019\u0011\u000b\n\u001d\u0002\u0001\nof both the linear (Wilcoxon Test, Z \t\n\u000e\u0014\u0013 , \nmodel (Wilcoxon Test, Z\t\n\n\u001d\u0005\u0001\n\u001d\f\u0010 ), no main effect of propagation method\n\u0019 ). The correlations\n\u0006 , \n\u0019 , one-tailed) differed signi\ufb01cantly from 0 when\n\n\u0003 , one-tailed) and the noisy-OR\n\nclamping was used.\n\n\u001d\u0005\u0001\n\n\u0019\f\u000b\n\n5.5 Discussion\n\nThe simulation results show that clamping is necessary to the success of the counterfactual\n\nmodels; this thus constitutes an empirical validation of Pearl\u2019s use of the \u2018\u0001\n\noperator in modelling counterfactuals. In addition, both the models capture the empirical\npatterns with some degree of success, so further work is required to tease them apart.\n\n\u0013 \u2019\n\n6 Exploring Non-Parametric Approaches\n\nThe models of counterfactual reasoning we have presented both make parametric assump-\ntions. Although non-parametric models in general offer greater \ufb02exibility, there are two\nmain reasons \u2013 one logistical and one methodological \u2013 why applying them in this context\nmay be problematic.\n\n\u000b\n\u000b\n\u001d\n\u0012\n\u0019\n\u0019\n\u0013\n\t\n\u0019\n\t\n\u001d\n\u0012\n\u0019\n\u0019\n\u0013\n\t\n\u0019\n\u0001\n\u0012\n\t\n\u0019\n\u0012\n\u0019\n\u0019\n\u0013\n\u000b\n\u0006\n\u0001\n\u0010\n\u000b\n\u001d\n\u001d\n\u0006\n\u0006\n\u0001\n\u000b\n\u001d\n\n\u0012\n\u0002\n\t\n\u0004\n\f6.1 A Logistical Reason: Conditional Probability Tables\n\nBayesian Belief Networks (BBNs) de\ufb01ne conditional dependence relations in terms of\ngraph structures like the dependency structures used by the present model. This makes\nthem an obvious choice of normative model for counterfactual inference. However, there\nare certain problems that make the application of a non-parametric BBN to counterfactual\nreasoning problematic.\n\nto a combinatorial explosion in the number of parameters required. If \u0005\n\nFor non-parametric inference a joint conditional probability table needs to be de\ufb01ned for\nall the variables upon which a target node is conditioned. In other words, it\u2019s not suf\ufb01cient\nis required. This leads\n\n\u0013 , . . . , \n\b\u0003\u0002\nis a vector of\u0006\n\u0002 represents the number of discrete classes that the random variable\n\ncan take, then the number of conditional probabilities required to compute the interaction\nbetween\n\n\u0013 alone; instead, \n\nin the general case is:\n\n\u0012\u0001\n\n\u0013 , \nto know \nelements in which\u0015\n\nand\n\n\u0012\u0001\n\n\u0002\u0005\u0004\u0007\u0006\n\n\u0012\u0001\n\n\u0012\u0004\n\n(7)\n\nOn the assumption that features can normally be represented by two classes (present\nor absent), the number of probability judgements required to successfully apply a non-\nparametric BBN to all four of Sloman, Love and Ahn\u2019s (1998) concepts is 3888. Aside\nfrom the obvious logistical dif\ufb01culties in obtaining estimates of this number of parameters\nfrom people, attribution theorists suggest that simplifying assumptions are often made in\ncausal inference (Kelley, 1972). If this is the case then it should be possible to specify a\nparametric model which appropriately captures these patterns, as we have attempted to do\nwith the models in this paper, thus obviating the need for a fully general non-parametric\napproach.\n\n6.2 A Methodological Reason: Patterns of Interaction\n\nParametric models are special cases of non-parametric models:\nthis means that a non-\nparametric model will be able to capture patterns of interaction between causes that a para-\nmetric model may be unable to express. A risk concomitant with the generality of non-\nparametric models is that they can gloss over important limitations in human inference.\nAlthough a non-parametric approach, with exhaustively estimated conditional probability\nparameters, would likely \ufb01t people\u2019s counterfactual judgements satisfactorily, it would not\ninform us about the limitations in our ability to process causal interactions. A parametric\napproach, however, allows one to adopt an incremental approach to modelling in which\nsuch limitations can be made explicit: parametric models can be generalised when there is\nempirical evidence that they fail to capture a particular kind of interaction. Parametric ap-\nproaches go hand-in-hand, then, with an empirical investigation of our treatment of causal\ninteractions. Obtaining a good \ufb01t with data is not of sole importance in cognitive mod-\nelling: it is also important for the model to make explicit the assumptions it is predicated\non, and parametric approaches allow this to be done, hopefully making causal principles\nexplicit which would otherwise lie latent in an exhaustive conditional probability table.\n\n7 Closing Thoughts\n\nGiven the lack of quantitative models of counterfactual reasoning, we believe the models\nwe have presented in this paper constitute a signi\ufb01cant contribution to our understanding\nof this process. Notably, the models achieved a signi\ufb01cant correlation across a sizeable\ndata-set (111 data-points), with no free parameters. However, there are limitations to the\n\n\t\n\b\n\u0002\n\t\n\t\n\b\n\u0006\n\t\n\b\n\u0002\n\u000b\n\u0001\n\u0001\n\u0001\n\b\n\u0006\n\u0013\n\u0001\n\u0002\n\u0001\n\n\u0006\n\u0013\n\u0002\n\u0007\n\u0002\n\u0015\n\u0002\n\fcurrent models. As stated, the models both assume that causal factors contribute indepen-\ndently to a target factor, and this is clearly not always the case. Although a non-parametric\nBayesian model with an exhaustive conditional probability table could accommodate all\npossible interaction effects between causal factors, as argued in the previous section, this\nwould not necessarily be all that enlightening. It is up to further empirical work to unearth\nthe principles underpinning our processing of causal interactions (e.g., Kelley, 1972); these\nprinciples can then be made explicit in future parametric models to yield a fuller under-\nstanding of human inference. In the future we intend to examine our treatment of causal\ninteractions empirically, in order to reach a better understanding of the appropriate way to\nmodel counterfactual reasoning.\n\nAcknowledgements\n\nWe would like to thank Tom Grif\ufb01ths, Brad Love, Steven Sloman and Josh Tenenbaum for\ntheir discussion of the ideas presented in this paper.\n\nReferences\n\n[1] Byrne R.M.J. and Tasso A. (1999). Counterfactual Reasoning with Factual, Possible, and Coun-\nterfactual Conditionals, Memory & Cognition, 27(4), 726-740.\n\n[2] Dawes R.M. (1979). The Robust Beauty of Improper Linear Models in Decision Making, Ameri-\ncan Psychologist, 34, 571-582.\n\n[3] Goodman N. (1983; 4th edition). 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(1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference,\nMorgan Kaufmann, San Mateo, California.\n\n[11] Pearl J. (2000). Causality: Models, Reasoning, and Inference, Cambridge University Press,\nCambridge.\n\n[12] Roese N.J. (1997). Counterfactual Thinking, Psychological Bulletin, 121, 133-148.\n\n[13] Sloman S., Love B.C. and Ahn W.K. (1998). Feature Centrality and Conceptual Coherence,\nCognitive Science, 22(2), 189-228.\n\n[14] Yarlett D.G. and Ramscar M.J.A. (2001). Structural Determinants of Counterfactual Reasoning,\nProceedings of the 23rd Annual Conference of the Cognitive Science Society, 1154-1159.\n\n[15] Yarlett D.G. and Ramscar M.J.A. (in preparation). Uncertainty in Causal and Counterfactual\nInference.\n\n\f", "award": [], "sourceid": 1943, "authors": [{"given_name": "Daniel", "family_name": "Yarlett", "institution": null}, {"given_name": "Michael", "family_name": "Ramscar", "institution": null}]}