Structured Correlation from the Causal Background
Ralf Mayrhofer1 , Noah D. Goodman2 , Michael R. Waldmann1 , and Joshua B. Tenenbaum2
1 {rmayrho, mwaldma}@uni-goettingen.de; Department of Psychology, University of Goettingen, Germany.
2 {ndg, jbt}@mit.edu; Department of Brain and Cognitive Sciences, MIT.

Abstract
Previous research has cast doubt on whether the Markov condition is a default assumption of human causal reasoning—as
causal Bayes net approaches suggest. Human subjects often
seem to violate the Markov condition in common-cause reasoning tasks. While this might be treated as evidence thathumans are inefficient causal reasoners, we propose that theunderlying human intuitions reflect abstract causal knowledge
that is sensitive to a great deal of contextual information—knowledge of the “causal background”. In this paper, we introduce a hierarchical Bayesian model of causal backgroundknowledge which explains Markov violations and makes additional, more fine-grained predictions on the basis of causally
relevant category membership. We confirm these predictions
using an experimental paradigm which extends that used inprevious studies of “Markov violation.”
Keywords: causal reasoning; hierarchical Bayesian models;
categorization; Bayes nets; Markov condition

Introduction
Causal Bayes nets are an increasingly popular approach tohuman causal learning and reasoning (Waldmann & Martignon, 1998; Steyvers, Tenenbaum, Wagenmakers, & Blum,
2003; Griffiths & Tenenbaum, 2005; Sloman, 2005). However, recent research (see Rehder & Burnett, 2005; Waldmann, Cheng, Hagmayer, & Blaisdell, 2008; Walsh & Sloman, 2007) has cast doubt on the Markov condition, a central
assumption of this approach. This condition states that a variable, conditioned on its causes, is independent of all variables
except its effects (Pearl, 2000; Spirtes, Glymour, & Scheines,
2000). A cause thus “screens off” its effects: for example, ina simple common-cause network (Fig. 1a) inferences about
the state of an effect variable, E3 , depend only on the stateof its cause C and not on the states of the other effects (E1and E2 ). However, it is not difficult to create scenarios inwhich our intuitions seem to violate this condition: Suppose
one day while in your favorite cafe you badly need to checkyour email, but your laptop is unable to connect to the wireless network. From the past you know that e-mail does notalways work, but there is a fairly high probability that it does
when the transmitter is on. You see (from the indicator lights)
that the transmitter is on, so you ask your neighbor if his computer is working, but it also isn’t connecting. You ask four
more people and get the same answer. Will you try anotherperson? You probably expect that the seventh computer will
be no different than the first six—also not working. If we takethe causal structure of this situation to be the overt one—thetransmitter independently causes each computer to connect tothe internet—then the Markov condition implies that the failure of any number of computers should not affect the probability that a final computer can connect. Thus, your intuition
that six instances of failure indicate that a failure in the sev-

enth case is likely is a case of a “Markov violation”1 . Such“Markov violations” in common-cause scenarios were experimentally documented by Waldmann, Mayrhofer, and Hagmayer (2007). Earlier work by Rehder and Burnett (2005)
explored analogous “Markov violations” in reasoning aboutcausally related properties of object categories.
One interpretation of this finding is that the competencyof human causal reasoning is well described by causal Bayes
nets, but that humans are simply not efficient causal reasoners. Another interpretation is that the causal Bayes net formalism is not able to capture human causal reasoning, andshould be abandoned. We offer a third interpretation: thatcausal Bayes nets can be used to describe human causal reasoning, but that the overt causal structure must be augmented
to account for knowledge of the causal background structure.
The causal background is the sum of all of the unobservable causal influences potentially affecting a specific causal
system. This goes beyond overt causal structure that is explicitly known in a given situation, summarizing the additional subtleties of the real system. We do not propose thatpeople have specific knowledge of causal background factors (though this may sometimes be the case) but that they
have abstract knowledge about classes of background factorsthat may influence causal relations between various classesof events. This knowledge allows people to account for manypotential background influences implicitly, in aggregate form,
without representing them explicitly or individually. This isrelated to the notion of the alternative cause in situations ofelemental causal induction (Cheng, 1997), however it extends
the alternative cause to non-elemental situations in which thealternatives may affect multiple variables.
If this causal background knowledge is a coherent and useful form of abstract knowledge, rather than a mere reflectionof inefficiencies in causal reasoning, then we should be able
to reveal finer structure within the intuitions of “Markov violation”. Imagine that, in the example above, the seventh computer is of a completely different type than the other six (say,a Mac instead of a PC). In this case you may be much more
likely to try once more to check your email, using this newcomputer. Motivated by this intuition, we consider the effectof categorical knowledge on “Markov violation”. It is wellknown that categories can influence causal inferences andvice versa (Lien & Cheng, 2000; Waldmann & Hagmayer,
2006; Ahn, 1998; Rehder, 2003). The notion that objects
have abstract causal types, which reflect similarities in causal
powers, has been formalized in the causal-schemata model(Kemp, Goodman, & Tenenbaum, 2007) within a hierarchi1 We use the quoted term “Markov violation” to indicate simply
the empirical phenomenon.

cal Bayesian framework. In this paper we attempt to formalize category-based causal background knowledge in a similar
hierarchical Bayesian model. This model then motivates an
empirical demonstration which parallels the opening example, showing fine grained sensitivity of “Markov violation” tocontext.

by3 :P(En =1|C, E1 , ..., En−1 )= ∑ P(En =1|C, PN, E1 , ..., En−1 )P(PN|C, E1 , ..., En−1 )PN= ∑ P(En =1|C, PN)P(PN|C, E1 , ..., En−1 )PN(1)Modeling
In this section we build up a hierarchical Bayesian model
which captures a reasoner’s knowledge about both overt
causal structure, and the structured causal background. We
make the simplifying assumption that the overt causal structure, given by specific world-knowledge or experimental instructions, is that of a simple common-cause network (as inFig. 1a). This structure predicts an influence of the cause
on each of the effects, but predicts that the effects are otherwise independent (a consequence of the Markov condition).
However, there may be many unobserved factors which could
potentially affect any system—preventive causes, missing enabling conditions, etc.—and these factors can cause additional correlations between the effect variables. We will capture these background correlations by augmenting the overt
causal structure with an unobserved common noise sourcethat represents all of the additional influences which potentially affect the different effect variables. Further, unobserved
factors are more likely to be shared by objects within a category than those in different categories. In our model thiskind of sensitivity to category structure is a consequence ofa refined common noise source: we include a separate noise
source for each category, shared by effect variables pertaining
to category members. Finally, we will augment the model tohandle both uncertainty over the causally relevant categories,
and the role of perceptual features in suggesting relevant categories.

In the first step we have introduced the hidden variable PN
and marginalized over its possible values; in the second step
we have used the independence of the effect variables given
all of their parent variables (this is the Markov condition,
which once again holds formally after accounting for the
background influences via PN). The reasoning captured by
Eq. 1 can be considered as a two step process: First the stateof PN is inferred, then, given that state, the state of the target
effect En is inferred.

Structuring the background correlation
Let us say now, that we are given the knowledge that the effect variables partition into distinct categories, and that all the
relevant unobserved causal factors are specific to a single category. How can we capture this additional structure? Insteadof a single hidden variable PN we will have one such variable PNz j for each category z j of the partition z (see Fig. 1c).
Writing Ez j (excluding En ) for the effect variables in the z j thcategory, we now have:P(En =1|C, E1 , ..., En−1 , z)
= ∑ P(En =1|C, PNz j )P(PNz j |C, Ezn )

(2)

PNzn

Notice that the state of En only depends on the states of othervariables within its category.

Uncertainty over relevant categories
Background correlation
We begin with the standard common-cause network shown in
Fig. 1a, which we assume to be parametrized as in a noisy-or
causal structure: the cause C has power wC to bring about
each effect independently (see Cheng, 1997). To capture
unobserved influences common to all of the effect variables
we introduce an additional variable PN which also acts as a
causal parent to all of the effects, preventing C from bringing
about its effects via a noisy-and-not parametrization2 . With
what probability will the final effect En be on, given the statesof the common cause C and the other effects E1 ...En−1 (butnot the unobservable background variable PN)? This is given

2 It is possible to also include a common generative influence, wehave omitted this for reasons of simplicity.

However, we may suspect that there are causally relevant categories, but not know what they are. For instance, if the effects pertain to particular objects which have features suggesting separate categories. We may maintain uncertainty
over the correct categorization, while still gaining the ability to leverage any categories which present themselves, byplacing a prior distribution on categories (this is very similar
to the construction of causal schemata in Kemp, Goodman,
and Tenenbaum (2007)). We use a Chinese Restaurant Process prior P(z) on the partition z (see Pitman, 2002), and we
additionally assume that there are features F associated withthe objects that also depend on the partition through the like3 The learner in our experimental situations doesn’t know the true
value of the causal strengths wC and wPN . We place a Beta priorover these causal strengths to represent this uncertainty, and must
Rthen integrate over the strengths, giving: P(En =1|C, E1 , ..., En−1 ) =P(En =1|C, E1 , ..., En−1 , w)dw. These integrals all have standard
solutions, and we omit them from our exposition for clarity.

C

C

E1

E2

E3

E1

E2

(a)

…

PN

En

(b)

C

PN1

E1

E2

E3

PN2

…

En

(c)

Figure 1: a) An example of a simple causal Bayes net, here a common-cause model. The cause variable C is connected toits effects via causal arrows. The state of each effect node depends only on the state of the cause node. Due to the Markov
condition, the effects are independently influenced by C. b) Modeling the Markov violation by inserting a common preventive
noise node PN. This hidden node interacts with C. If PN is present, the causal influence of C is lowered for all effects. Thus,
if E1 , ..., En−1 were observed absent, even if the cause is present, PN likely, and this lowers the probability of En . c) Adding
structure to the simple PN-model. For each cluster effect variables a common preventive noise node is added. This leads todependent conditional probabilities within clusters but not between clusters.
lihood P(F|z)4 . Hence:

Experiment

P(En =1|C, E1 , ..., En−1 , F)
= ∑ P(En =1|C, E1 , ..., En−1 , z)P(F|z)P(z)(3)

zModel predictions
What inferences do we expect on the basis of this model?
In the case without categories (Fig. 1b) this is quite simple.
When the common cause is present, if observed effects are off
it will provide evidence that PN is on (preventing the activityof C). Hence, the conditional probability of the target effectshould decrease with an increasing number of absent collateral effects. That is, we predict a violation of the Markov
condition—or rather the Markov condition as it would holdfor a simple model lacking PN. In the case that the common
cause is absent, the conditional probability of the target effect will be independent of the states of the collateral effects(since the preventer has nothing to prevent). In this case theprobability of the target effect should only depend on its baserate.
Whenever a causally relevant category is known with certainty (Fig. 1c), the same predictions apply with respect tothose effects belonging to the same category, but not to theeffects of the other category. That is, we predict only within
category “Markov violations”.
When there is uncertainty over the relevant categories theinfluence of the PN nodes is marginalized out over all possibilities. This leads to graded influences, since there will besome weight on partitions that puts any group of effects together. The more probable a partition given the features, thehigher the influence of category members within this category on each other. This leads to the prediction that the moresimilar two effects are, the more dependent their conditional
probabilities should be.

We wished to test whether the “Markov violation” phenomenon in human causal inference is sensitive to category
structure, as predicted by the model. For this purpose we useda version of the “mind sending” task used by Waldmann et al.
(2007), which adapted the mind reading alien task of Steyvers
et al. (2003) and combined it with the reasoning task introduced by Rehder and Burnett (2005). In this task, subjects
were presented with four aliens, Gonz, Murks, Brxxx, andZoohng, who mostly think of nothing but sometimes think of“POR” (“food” in the alien language), further, Gonz is able
to transmit its thoughts of POR to the other aliens. Thus thethoughts of Murks, Brxxx, and Zoohng are statistically dependent on the thoughts of Gonz. In the test phase subjects
rated the conditional probability of a target alien, e.g. Murks,
thinking of POR given the thoughts of the other aliens. The
results showed that participants’ estimate of the target effect’s probability are—in this case—strongly influenced by
the thoughts of the other effect aliens.
The model predicts that the presence/absence of events
which are similar to the target event have more impact on
estimated probability than the presence/absence of events dissimilar to the target event. To investigate the influence of category structure we adapted the task of Waldmann et al. (2007),by increasing the number of effects aliens and introducing a(binary) color feature for the effect aliens. As in the originaltask, subjects had to reason about the thoughts (POR/nothing)of a target effect alien in a common-cause structure with respect to the thoughts of the common cause alien and the othereffect aliens.

Method

4 We take the dependence of features on category to be given as

Participants 60 students from the University of Goettingen(33 female, 27 male; mean age 25.1 years) participated in
exchange for candy.

in the infinite mixture model (see Pitman, 2002)—a model of categorization related to Anderson’s rational model (Anderson, 1991).
Other feature-based categorization models could be substituted.

Procedure and Materials In the instruction phase we presented participants with instructions about six aliens: Gonz,bility judgment.)

Brxxx

MezuktGonz?
KarkasZoohng

Murks

Figure 2: An example of a test panel used in the experiment. Experimental manipulation varried the thoughts of themind sending alien Gonz (POR or nothing) and the numberof aliens of each color thinking POR. As dependent measuresubjects were requested to rate the probability that the target
alien (indicated by a question mark above its head) thinks of
POR in the given situation. (Thus 0, 1, or 2 aliens of the samecolor as the target alien might successfully think of POR, and0, 1, or 2 aliens of the other color.)

Murks, Mezukt, Zoohng, Brxxx, and Karkas, who usually think of nothing (illustrated by an empty bubble) andsometimes think of POR (indicated by a bubble containing“POR”). Participants were told that Gonz is able to transmit its thoughts into the heads of the other aliens (i.e., Gonz
served as cause in a common-cause structure with five effects). It was emphasized that the effect aliens frequently
think of POR if Gonz thinks of POR, and that Gonz isn’t
perfect, which means that he sometimes fails to transmit histhoughts due to inattention; the four possible cases with twoaliens thinking of POR or nothing were shown as illustrations. In addition, it was pointed out that there were twokinds of effect aliens and that these kinds are indicated by
the color of the aliens: green and yellow; but it was not specified what influence this might have (leading to uncertainty
about the relevance of this feature). Identical pictures wereused for all effect aliens—with two different colors—in order
to exclude influences of implicit features. There were always
two green collateral effect aliens and two yellow collateraleffect aliens. The target effect alien’s color was counterbalanced across subjects. In the test phase subjects were confronted with several test panels with all the non-target aliens
thinking of POR or nothing (for an example see Fig. 2). For
each panel, subjects were requested to imagine ten situations
with the given configuration and then to judge, in how manyof these situations the target alien—Murks—would probably
think of POR. (It was pointed out that this is merely a proba-

Design The predictions were tested in a 2x3x3 repeatedmeasurement ANOVA design, with the following factors: thethoughts of the cause alien (nothing vs. POR, i.e., C=0
and C=1); the number of effect aliens of the same category(i.e., same color) as the target alien that are thinking of POR(0, 1, or 2—three levels); the number of effect aliens of the
opposite category (i.e., other color) thinking of POR (0, 1,
or 2—also three levels). Because this would lead to 18 test
cases, subjects were tested on both levels of the C factor ina balanced partial confounded randomized block design (seeKirk, 1995). So, within the 3x3-combinations on each levelof C only three cases were given to each participant, chosensuch that each level of each factor was realized once for eachsubject (so, three subjects constitute a full data set). The orderof the test cases was randomized across subjects.
Predictions Let us clarify the predictions of our model applied to this particular task. First, there should be a strong
influence of the state of the common cause (C=0 vs. C=1),
as was found by Waldmann et al. (2007). When the common
cause is absent (C=0), the conditional probability of the target effect should be nearly independent of the states of othereffects (also found by Waldmann et al.). In the case that thecommon cause is present (C=1), and hence the preventive
noise sources come into play, the model predicts that a strong“Markov violation” should be found with respect to alienswith the same color; a smaller violation should be found withrespect to the differently colored aliens. The latter prediction
follows because, although we have only one feature, there is
uncertainty about the causal relevance of this feature, and thepossible existence of causally relevant categories is not betrayed by obvious features.

Results
The main prediction is about different effect sizes of themain effect (i.e., different amounts of “Markov violation”),
hence the results are displayed as marginalized main effects
in Fig. 3. The left hand side of the figure shows the data, theright hand side the model predictions5 .
In general, the ratings are much higher when the cause
alien thinks of POR, which reveals a strong influence of the
C factor, as expected from causal Bayes net theory. If thecause alien thinks of nothing (the lower two lines), the number of effect aliens thinking of POR has no influence on theratings; the number of same colored effect aliens thinking of
POR has no effect (F = 1.91), and neither does the numberof different colored effect aliens thinking of POR (F < 1).
This is in line with the model predictions. If the cause alien
5 To compute the predictions a Monte Carlo simulation with
50,000 repetitions was performed. Only the background rate of theeffects and the general strength of the preventive noise nodes were
adjusted by hand and set to a Beta(1,3) and a Beta (5,1) distribution.
The other parameters of the model were set to values given by thesimple model applied to the data of Waldmann et al. (2007) and byKemp, Goodman, and Tenenbaum (2007).

Human Data

Model Predictions

9
8

10
same category
opposite categ.

CC: POR
CC: nothing

POR thoughts of target alien

POR thoughts of target alien

10

7
6
5
4

3
2
1
0

9
8

same category
opposite categ.

CC: POR
CC: nothing

7

6
5
4
3

2
1
r=.99

0
0/2

1/1

2/0

Number of effect aliens thinking of POR/nothing

0/2

1/1

2/0

Number of effect aliens thinking of POR/nothing

Figure 3: Mean rating and standard error of the number of times the target aliens thinks of POR plotted against the numberof times collateral aliens think of POR (in the left panel the experimental data, in the right panel the model predictions). The
upper two lines in each panel correspond to the sending alien thinking of POR, the lower two lines to the sending alien thinking
of nothing. Results shown as marginalized main effects: The solid lines average over the number of opposite category aliens
thinking of POR (showing in column 1/1, for instance, the average ratings over conditions in which the number of oppositesame aliens thinking of POR was: 0-1, 1-1, and 2-1.) The dashed line averages over the number of same category aliens. The
columns indicate the number of effect aliens within the relevant group thinking of POR. (Note that the solid and dashed series
thus represent the same data, averaged in two different ways in order to examine the effects within each category separately.
The crossing of the lines thus cannot be interpreted as an interaction.)
thinks of POR and can therefore transmit its thoughts (the upper two lines), both types of effect aliens have an influence.
The number of same category effect aliens thinking of POR
reveals a stronger effect (F2,112 = 22.53, p < 0.0001) than thenumber of opposite category effect aliens thinking of POR(F2,112 = 6.60, p < 0.01). (The interaction of these factors
isn’t significant (F4,112 = 1.44, p = 0.224)6 , as predicted by
the model.)To test the main hypothesis—whether the number of samecolor effect aliens thinking of POR has the greater influenceon the subjective conditional probability of the target effect—a test must be constructed, because no standard procedure isavailable. The interesting question is, whether the same category factor explains more variance than the opposite category
factor, which simply means that it has more influence on thedependent measure. So, an appropriate statistic for testing
this is the ratio of the variances explained by the first andthe second factor. The empirical variance ratio is 3.413. To
test its significance, we constructed a bootstrap distributionof this variance ratio under the null hypothesis that the variances are equal (i.e., the ratio is 1). For this purpose we drew
6 Because Figure 3 only shows marginalized main effects, the interaction cannot be obtained by simple examination. The slopes in
the figure display the different strengths of the factor’s influence on
the ratings (the higher the slope the more variance is explained by
the corresponding factor).

bootstrap samples from the data under the assumption thatthe factors are exchangeable (i.e., that color doesn’t matter)7 .
The p-value of the empirical variance ratio with respect to thebootstrap distribution is 0.026. Hence, the number of samecolor effect aliens thinking of POR has a greater influenceon the subjective conditional probability rating of the target
effect than the number of different color aliens thinking of
POR. This is in line with the main prediction of the model.

Discussion
Our experimental results replicate the finding of “Markov violation”, but also shows that this phenomenon is sensitive tosubtle contextual factors—”Markov violations” are stronger
within categories of effects than between them. This is consistent with the idea that people have abstract intuitions about
influences of the underlying causal background structure.
Other current theories of human causal reasoning suggest that
Markov violations express irrationalities of the human mind,
and offer no compelling account of the sensitivity of these
violations to context. In contrast, we have developed a hierarchical Bayesian model that provides a rational account ofthese findings. The model was designed to capture abstract
causal background structure associated with causal types ofobjects, and does so by extending the causal Bayes nets picture with a form of abstract knowledge.
7 10,000 bootstrap samples were drawn.

Our experiment was designed to demonstrate that similarity structure affects causal reasoning in “Markov violation”
situations. For this purpose we limited the task to a single
relevant feature, the aliens’ color. In this setting the ambiguity of causally relevant categories is minimized. By contrast, consider the earlier example, when no computer willconnect to the wireless network in your favorite cafe—while
you are pushing to finish your CogSci paper. These computers have many features—Macs vs. PCs, laptops with whitecases vs. those with black cases, etc. These features do notweigh equally on your intuitions about whether a final computer will work: it’s brand may matter, but probably not it’ssize. Our model allows that different features might have different causal relevance for the inference task of interest. Indeed, confronted with evidence the model will learn whichfeatures matter with respect to any structured dependencies inthe data and which features do not. However, little is known
about how and whether people acquire such abstract knowledge about the causal background; further experiments areclearly needed.
We believe that the flavor of causal reasoning studied
here—which accounts for causal background structure—isnecessary to capture real-world human reasoning abilities. In
most natural contexts there are a myriad of influences whichcannot each be considered individually, but whose overall influence is significant. Yet it is likely that richer forms of
causal background knowledge than we have considered will
be needed to capture human intuitions. These might, for instance, include notions of agent-patient relationships or domain restrictions on causal influence. In addition, it is important to consider other modeling approaches to these issues,
such as the combination of causality and similarity describedin Kemp, Shafto, Berke, and Tenenbaum (2007).
We have suggested that experimental tests showing“Markov violations” do not reflect fundamental human irrationality, but the invocation of an important form of abstract
knowledge about the causal background structure. If our interpretation is correct, then a great deal of important psychology is hidden within “Markov violations” and similar phenomena.

Acknowledgments
The research was supported by a grant of the Deutsche
Forschungsgemeinschaft (DFG: WA-621/20-1,2) [RM,
MRW] and by the J. S. McDonnell foundation causal learning collaborative initiative [NDG]. In addition, the authors
would like to thank Marie-Theres Kater for her assistance indata collection.

References
Ahn, W. (1998). Why are different features central for thenatural kinds and artifacts? the role of causal status in determining feature centrality. Cognition, 69, 135-178.
Anderson, J. R. (1991). The adaptive nature of human categorization. Psychological Review, 98, 409-429.

Cheng, P. W. (1997). From covariation to causation: A causal
power theory. Psychological Review, 104, 367-405.
Griffiths, T. L., & Tenenbaum, J. B. (2005). Structure andstrength in causal induction. Cognitive Psychology, 51,
354-384.
Kemp, C., Goodman, N. D., & Tenenbaum, J. B. (2007).
Learning causal schemata. In Proceedings of the 29th Annual Meeting of the Cognitive Science Society.
Kemp, C., Shafto, P., Berke, A., & Tenenbaum, J. B. (2007).
Combining causal and similarity-based reasoning. Advances in Neural Information Processing Systems, 19.
Kirk, R. E. (1995). Experimental design: Procedures for the
behavioral sciences (3rd ed.). CA: Brooks/Cole.
Lien, Y., & Cheng, P. W. (2000). Distinguishing genuine
from spurious causes: A coherence hypothesis. Cognitive
Psychology, 40, 87-137.
Pearl, J. (2000). Causality: Models, reasoning, and inference. Cambridge, MA: Cambridge University Press.
Pitman, J. (2002). Combinatorial stochastic processes. Lecture notes for Saint Flour Summer School.
Rehder, B. (2003). Categorization as causal reasoning. Cognitive Science, 27, 709-748.
Rehder, B., & Burnett, R. (2005). Feature inference and thecausal structure of categories. Cognitive Psychology, 50,
264-314.
Sloman, S. (2005). Causal models: How people think about
the world and its alternatives. Oxford: Oxford University
Press.
Spirtes, P., Glymour, C., & Scheines, R. (2000). Causation,
prediction, and search (2nd ed.). New York: Springer.
Steyvers, M., Tenenbaum, J. B., Wagenmakers, E.-J., &Blum, B. (2003). Inferring causal networks from observations and interventions. Cognitive Science, 27, 453-489.
Waldmann, M. R., Cheng, P. W., Hagmayer, Y., & Blaisdell,A. P. (2008). Causal learning in rats and humans: A minimal rational model. In N. Chater & M. Oaksford (Eds.),
The probabilistic mind: Prospects for bayesian cognitive
science. Oxford: Oxford University Press.
Waldmann, M. R., & Hagmayer, Y. (2006). Categories andcausality: The neglected direction. Cognitive Psychology,
53, 27-58.
Waldmann, M. R., & Martignon, L. (1998). A Bayesian
network model of causal learning. In Proceedings of the20th Annual Conference of the Cognitive Science Society.
Waldmann, M. R., Mayrhofer, R., & Hagmayer, Y. (2007).
Mind reading aliens: Causal capacities and the Markov
condition. Unpublished manuscript.
Walsh, C. R., & Sloman, S. A. (2007). Updating beliefs withcausal models: Violations of screening off. In M. A. Gluck,
J. R. Anderson, & S. M. Kosslyn (Eds.), A Festschrift for
Gordon H. Bower. New York: LEA.