Upsetting the contingency table: Causal induction over sequences of point events

Transcription

1 Upsetting the contingency table: Causal inuction over sequences of point events Michael D. Pacer Thomas L. Griffiths (tom Department of Psychology, 321 Tolman Hall Berkeley, CA 9472 USA Abstract Data continuously stream into our mins, guiing our learning an inference with no trial elimiters to parse our experience. These ata can take on a variety of forms, but research on causal learning has emphasize iscrete contingency ata over continuous sequences of events. We present a formal framework for moeling causal inferences about sequences of point events, base on Bayesian inference over nonhomogeneous Poisson processes (NHPPs). We show how to apply this framework to successfully moel ata from an experiment by Lagnao an Speekenbrink (21) which examine human learning from sequences of point events. Keywors: causal inference; continuous time; stochastic processes; Bayesian moels Introuction Only a single bullet is neee to en a life. One momentary event quickly ruptures a slew of causal mechanisms, having effects that persist long after the trigger was pulle. We unerstan this causal inference easily enough, but how o we manage to arrive at that inference? More precisely, what formal tools can characterize events that last an instant but leave profoun consequences in their wake? Learning from instantaneous events is one of the most important forms of causal inuction in the real worl, but is not the case consiere in most moels of human causal reasoning. More commonly, these moels rely on the existence of aggregate statistics often in the form of a contingency table escribing the frequencies with which ifferent combinations of events occur (Griffiths & Tenenbaum, 29). The most traitional case is the 2 2 contingency table that escribes two variables that are either present or not. It is out of this statistical paraigm that many of the moels of human causal inference an learning have arisen (Gopnik et al., 24). The worl offers more than contingency tables an frequency counts. Different kins of ata warrant ifferent kins of inferences. For example, when Greville an Buehner (21) coerce events embee in continuous-time into contingency tables, they are challenge with ientifying an (assume) unerlying trial structure. To fill a 2 2 table of event pairs, one nees to slice the continuum into trials; a variable is counte if an event occurre in the slice an not if it i not. But ifferent slicing regimens can warrant ifferent inferences about the same ata. This arises because of a funamental asymmetry between the times at which events occur an when then o not occur. In a contingency table both X an X are treate as the same kin of entity, but instantaneous events can be counte while the the continuous expanse of nothingness in which events are embee by efinition cannot be counte, only measure. But there is another way to moel sequences of point events occurring in continuous-time. Here, we buil a moel of causal inuction on the framework escribe in Pacer an Griffiths (212), which uses a moel of events embee in continuous-time. It computes probabilities not in terms of the frequency with which events co-occur, but irectly from the temporal istances between cause an effect events. Importantly, we accomplish this without neeing to uniquely match iniviual cause events with iniviual effect events. We rely on probabilistic graphical moels to escribe the structure of causal relations. We use Poisson processes an operations over these processes to escribe the functional relationships between variables, which provie the structure by which we ientify an organize kins of events. Our focus in this paper is on causal inuction from sequential point processes sequences of events that are transient moments in continuous time. This is an important case for moeling causal inuction, escribing a wie range of real-worl settings in which people perform causal inuction such as trying to infer which particular interaction with which particular other person le to coming own with the flu. Bramley, Gerstenberg, an Lagnao (214) raise a concern that these cases present a challenge for the continuoustime causal network approach presente by Pacer an Griffiths (212). The paper will procee as follows. We review Poisson processes, which form the founation for the rest of the paper. We introuce a physical analogue of these formal structures to provie intuition to the unerlying mathematics of nonhomogeneous Poisson processes. These processes form the basis of our framework when combine with probabilistic graphical causal moels. We then successfully apply the framework to Experiment 2 from Lagnao an Speekenbrink (21) a paraigm for human jugments of causal structure from sequences of point events. We then iscuss the implications of this work for statistical an mechanistic theories of human causal reasoning. Formal Founations In this section we escribe formal founations for our framework (cf., Pacer & Griffiths, 212). We explain a class of point-process moels (processes that escribe the occurrence of point-events which have a location, but no measurable uration) calle Poisson processes that are efine in terms of a space an a positive rate function that gives the expecte number of events to be foun in any subspace. This ratefunction will epen upon the ientity an time of events that occur uring the course of the processes activity (cf.,

2 Simma & Joran, 21; Blunell, Beck, & Heller, 212). To accommoate this kin of conitional intensity function we use nonhomogeneous Poisson processes (NHPPs), whose rate functions are not constant. We will first escribe two ways of looking at homogeneous Poisson processes: arrivals an rates. After eveloping an intuition for Poisson processes with constant rate functions, we will then consier the general case of NHPPs an how to moel sequences of point events in terms of a generative moel that iteratively analyzes iniviual arrivals relative to the rates inuce by previous arrivals. Poisson Processes: Arrivals an Rates Poisson processes can be interprete in a number of ways that len themselves more or less easily to ifferent applications. We will escribe Poisson processes in two senses: the arrival process sense an the rate-of-events sense. 1 Sequences of point events can be escribe sequences of successive arrivals, making the arrival perspective convenient for computing point event likelihoos. But it is easiest to conceive of the causal effects of point events in terms of their altering event rates. Both perspectives prove useful. Arrivals The arrival sense can be unerstoo by anyone who has ever waite in a queue. You will have to wait some amount of time before your turn, an we can assign a probability that you will be serve by time t. If you were next in the queue an it were governe by a homogeneous Poisson Process with rate λ, the waiting time istribution of being serve by time t woul be an exponential istribution with mean 1 λ (t Exp( 1 λ ) p(t) = λe λt ). Interestingly, this istribution is memoryless, such that, regarless of how long we have waite, we still expect to wait the exact same amount of time it has no memory of how long it has been since the last event. This memorylessness property oes not hol for the general class of NHPPs. Rates Equivalently, we can count the number of events that occur in a measurable time-perio, rather than looking at the elays between each event. Poisson processes efine a rate of events, which escribes the expecte count of events to occur in any interval. A homogeneous Poisson process has a constant rate, λ, an for a time interval with length τ we can expect to see event-count istribution governe by a Poisson ranom variable, with mean (λ τ ). In the case of nonhomogeneous Poisson processes, we will have a rate-function efine over time λ(t). Integrating this function over some time-interval efines how many events are expecte to occur (i.e., for τ [a,b) the expecte event count is τ[a,b) λ(s)s). 1 Poisson processes can be efine over higher imensional spaces (e.g., R 3 ) than the real line. This complicates the arrival perspective, which implicitly relies on the orer that events arrive. The event-rate perspective is unchange in higher imensions; in that sense it coul be sai to be more funamental than the arrival perspective. In this paper, for simplicity we focus on processes efine over time ([, )). Combining Perspectives The arrival perspective gives us a probability istribution over intervals of time (i.e., intervals efine from now until the next event arrives), while the rateof-events perspective gives us an instantaneous measure of event likelihoo which is comprehensible only in terms its integration over intervals of time. The former is more useful in cases where events are analyze one at a time. For example, when simulating epenent event sequences or calculating the probabilities of event sequences in terms of the likelihoo of each event s occurrence given the previous relevant occurrences. In our moel of Lagnao an Speekenbrink (21), we will use this perspective to efine the likelihoo of each inter-arrival perio conitional on the previous events. The rate-of-events perspective is useful when simulating many events when the rate is inepenent of the particular occurrence of the events. The rate perspective is also useful for calculating event likelihoos, when the interval uring when the events occurre is known, but the exact occurrence times are unknown. Pacer an Griffiths (212) use this technique to analyze the ata given to participants in Greville an Buehner (27) in which ata were presente in a tabular form that escribe the ay uring which bacteria ie but not the exact timing of the events. This property allows us to recover a trial structure from continuous-time by integrating over intervals of time an treating occurrences within those intervals as events that occurre in those trials. Most importantly for our uses, it is most straightforwar to see causes as altering the rate-of-events an then computing an expecte wait-time istribution base on those altere event rates. Describing effects in terms of rate changes will be the key to the causal aspect of our framework. Fortunately, Poisson processes have two closure properties, superposition an thinning, that allow us to create continuous time analogs of noisy-or an noisy-andnot (for more etails, see Pacer & Griffiths, 212). Superposition an Thinning in Poisson Processes This section will evelop a rough physical moel to ai in thinking about NHPPs as forme by functions on homogeneous Poisson processes. Namely, we aim to provie an intuition for the superposition an thinning closure-properties of homogeneous Poisson processes from the rate-of-events perspective. We o so by sketching a mechanistic picture of a particle emission system that exhibits these properties. First, consier a ecaying raioactive material which releases particles at a constant rate, λ. With a particle etector aroun the material you recor the time-stamp at which particles hit the etector. A particle is expecte to hit the etector, on average, every 1 λs. This etector will then be recoring a homogeneous Poisson process with rate λ. Suppose you were to place a barrier to block some of the paths leaing from materials to the etector (call the proportion blocke π π 1, as in the parameter associate with the orange filter in Figure 1). From the etector s perspective, events associate with particles blocke by a filter are events that never occurre. This process is known as filtering

3 particle paths in transit etecte filtere raioactive materials particle etector filter (π) Figure 1: Particle emission etector moel for visualizing superpositioning an filtering Poisson processes. Color istinguishes particle origins prior to etection, with color lost in etection. Filtere events are never etecte. the Poisson process, an if π is inepenent of the generating process, filtering gives a Poisson process with rate (1 π)λ. Suppose you were to place another raioactive material in the etector, of a ifferent kin than the original, but which i not interact with the original raioactive material (see the blue an green materials in Figure 1). From the perspective of the etector, there is no ifference between the particles hitting it from ifferent materials it only knows that a particle hits it an when it hits it. If we suppose the rate of this new material s emitting particles is constant at λ 1, then the total set of events woul be a Poisson process with rate λ + λ 1. This is the superposition property of Poisson processes: if jointly inepenent, the union of the events from two Poisson processes will be a Poisson process whose rate is their sum. Superposition an thinning allow us to see how the rates of Poisson processes can change without altering their unerlying structure. We can apply these transformations at particular times or intervals of time, thereby proucing an increase (via superposition) or a ecrease (via thinning) in the rate of events while at all times maintaining its ientity as a Poisson process. Applying superposition or thinning as timeepenent functions thus allows one way to create NHPPs that nonetheless can be unerstoo in terms of component processes an their transformations. Causal Graphs an Sequential Point Processes With superposition an thinning in our tool belts, we can iscuss causal point processes feeing back into themselves. By applying thinning an superposition epenent on time we can buil NHPPs from component processes. If we apply superposition an thinning functions relative to the occurrence of events at particular times (e.g., by scaling their effects relative to the istance from the events occurrence time), then we can efine NHPPs relative to the occurrence of particular events. If we then efine the set of events capable of evoking changes in the Poisson process as themselves being the outcome of Poisson processes (see also Blunell et al., 212), then we have successfully manage to create a system of causal relations in terms of Poisson processes. It is worth noting that if we want to escribe the causal effects of point events at all, we will nee some form of influence function that lasts beyon the occurrence of the point event. Because point events are instantaneous, if a cause s effects last while cause persists the effects woul have to occur at that same instant. But we are seeking to moel causal relations that are not only embee in a moment in time, but across time. We nee to consier how events that occur at one point an time can influence events or, more precisely, the unerlying rate of events at later time points an intervals. We nee a way to escribe what kins of things exist which things are relate to one another an in what ways they are relate i.e., we nee an ontology, plausible relations, an functional forms (Griffiths & Tenenbaum, 29). Probabilistic graphical moels are formal structures for expressing stochastic epenencies between variables. But are our events vali variables? To efine a set of possible graphs over variables, we usually nee to know what those variables are. Given that we o not know when events will occur events woul make events a ba caniate for graphical noes. Furthermore, there s no obvious way to say whether event at time t counts as the same event (or even kin of event) as that which happens at t 1. Without some other information we woul be left with a proliferation of variables equal to the number of events we observe. Instea, we will consier the case where events have signature ientities that ientify which sequence of events each event belongs to. These sequences will be our variables. Both events an variables feature in our ontology, but the graphs will be efine over potential relations between variables which will be associate with sequences of point events. Events will be the components through which variables actually affect one another. As in Griffiths an Tenenbaum (29), to compute posteriors over graphs, we will nee a prior over the possible relations between variables. Often this will be mae easier by knowing (from our ontology) which things are potential causes of which effects. When all possible binary relations coul be graphs it is a challenge not because it is ifficult to impose a prior per se, but because of how rapily the number of graphs grows in terms of the number of variables. This is especially pernicious because by using continuous-time causal networks, we can represent irecte cycles as easily as any other relationship, which allows even more graphs than in the traitional causal Bayes nets framework. For each of these relationships, we will express a functional form efining how variables relate to one another. In particular, we nee to relate the kin of relationship between variables (e.g., generative, preventative) to the how the influence of particular events associate with those variables changes over time. Our approach is to see an event(t ) in the cause-variable s event set as inucing a NHPP on the variable s chil noes. For generative causes, the NHPP starts with a maximum rate(ψ) immeiately when it is triggere

4 with the rate ecaying exponentially accoring to a ecay rate(ϕ) scale istance from the cause event ((t t )). This prouces a NHPP with rate function λ(t) = ψe ϕ(t t ). We compute likelihoos for events sequences as follows. We compute a likelihoo for an event using the rate function of its associate variable uring its wait-time interval. We then upate the rate functions for variables that are effects of the variable associate with the event, an iterate this process until we exhaust the event sequence. The waiting-time likelihoo for each event can be further broken own into waitingtime likelihoos for the Poisson process composing its variable s rate function at the instant the event occurs. Note that from the superposition perspective an event s waiting-time implies that none of the variable s processes prouce events until the point when the event in question occurs. The general form of the cf (cumulative ensity function) of the waiting time istribution until the first event of a NHPP with time-epenent rate function λ( ), is: F(T t) = F(t) = 1 exp( λ(s)s). This can be converte to a pf by taking the erivative accoring to t, which (assuming that the erivative exists) becomes, p(t) = λ(t)exp( λ(s)s). Moeling Continuous Event Streams: Lagnao & Speekenbrink, 21 In this section we will moel Experiment 2 from Lagnao an Speekenbrink (21) using the formal elements escribe in Pacer an Griffiths (212) an above. As mentione, Bramley et al. (214) raise a concern that the framework escribe in Pacer an Griffiths (212) oes not aress sequences of point events. Here, we apply this framework to moeling ata from real-time event streams, characterizing sequences of point events an using them for computing the same inferences Lagnao an Speekenbrink (21) aske of their participants. Moreover, our moels jugments closely match average human responses, which suggests the framework succees both at characterizing the sequences of point events an at proucing moels capable of causal inference that comparable to that of human beings facing the same problems. Experiment Description Experiment 2 of Lagnao an Speekenbrink (21) has the form of participants observing a continuous sequence of events (as a vieo) that represent the time-course of various kins of seismic activity, specifically three kins of seismic waves (which we shall refer to as A, B, C) an earthquakes (E). The goal of the participants was to infer which (if any) of the seismic waves were the cause of the earthquakes. Earlier work suggests people will lessen their jugments of causal attribution between two variables if there is a longer(or more variable) elay between the occurrence of two events. t t However, this coul either be because there is something specific about long elays between causes an effects, or that longer elays allow more opportunities uring which other events coul occur that are not causally relate, thus weakening the connection between the original two variables of interest. 2 Accoring to the esign of the experiment unknown to the participant only one of the types of wave(a) was a cause of earthquakes, but sometimes non-causal waves woul occur in the interval of time between the cause an its effects. This allows us to isentangle the two explanations for reuce causal strength ue to longer elays. The length of time between the cause an its effect an the commonness of mi-interval events sometimes were the primary ifferences between the experimental conitions. The experiment ha a 2 2 structure. DELAY-LENGTH coul be LONG (mean elay between cause an effect = 6s) or SHORT (mean elay between cause an effect = 3s) in both the stanar-eviation is.1s. The probability a non-cause event occurre between the cause an the effect was LOW ( 35% of the time a non-cause event woul occur between a cause an its effect) or HIGH ( 65%). These probabilities are approximate because the event sequences were ranomly sample an so cannot be expecte to exactly match expecte percentages. The authors chose elay istributions between the occurrence of cause an lure events to prouce these probabilities in aggregate across samples. They generate sixty atasets per conition that represente the time-stamps an ientities of events that occurre in the movie. Of these, the first twenty atasets were use, with each of twenty participants participating in all four conitions exactly once. They were tol that each animation woul last no more than 1 minutes. 3 After each vieo participants were aske to provie jugments about the seismic waves that they ha just observe. Participants were first aske to rate the extent to which each wave was a cause of earthquakes on a scale of (oes not cause the effect) to 1 (completely causes the effect). This provies an absolute jugment of each wave s causal properties since the rating provie for one of the waves i not constrain the rating provie for the other waves. Participants were then aske for comparative ratings, in which they ivie 1 points amongst the three types of cause. Builing the Moel We treate the problem as one of structure inuction. That is, given the knowlege that there are three possible cause variables ({A,B,C}) of the effect in question (E) an the ata D, we want to infer a posterior over the possible graphs linking 2 We shoul note that more opportunities is actually somewhat misleaing as opportunities in plural form suggests that there woul be a countable number of opportunities uring which these events coul intervene. It is more accurate to say that long elays allow for a larger, continuous amount of opportunity (a mass noun). 3 Though participants i see multiple conitions, we treat each trial inepenently rather than attempting to etect orer effects. While we acknowlege its potential usefulness, aressing this is outsie the scope of our analysis.

5 the causes to the effects. Then we will use this posterior to compute measures analogous to those given by participants. Graphs an Parameterization We consiere graphs with any subset of three potential inepenent causes {A, B,C}. All causal links were generative an non-interacting with the other causal links. Thus the total rate of effects uner a graph woul be the superposition of all Poisson processes inuce by the activity of cause-events accoring to the graph. As escribe in Pacer an Griffiths (212) this is the continuoustime analog to the Noisy-OR parameterization of a causal graph. Because causes were inepenent accoring to all graphs, the likelihoo of their occurrences can be remove from our likelihoo calculations. In aition to a base-rate process PP λ, which we assume was a homogeneous Poisson process with rate λ, we allowe each cause-event (t [X] ) to initiate a NHPP with maximum rate (ψ X ) that ecays exponentially(ϕ X ) relative to the the istance from the cause event( t t [X] ). We sample these parameters in a similar manner to Pacer an Griffiths (212). We use uniform ranom variables (u U( 1.5,1.5 )) uner a transformation (λ = e u ) to etermine our initial timescale (in secons), which acts as our base-rate PP. This creates a approximate scale-free baseline parameter(λ 1 λ ) from which other parameters can be sample. We sample ψ X Γ(λ,1) (the maximum rate inuce by a single event of type X occurring) an ϕ X Γ(λ,1) (the rate at which the intensity ecays accoring to the istance in time from that instance) associate with each potential cause X {A,B,C}. Each cause instance(t [X] ) prouces a NHPP with rate function ψexp( ϕ(t t [X] )). Because the baseline istribution is scale-free an efines other scales, these parameters are not fit to the ata. A misfit baseline scale prouces overflow, unerflow or other numerical an computational issues that result in moel failure. But, any success can only stem from the moel s structural commitments an the relation to the moele ata. Data The ata use to generate stimuli in Lagnao an Speekenbrink (21) are organize by the time-step (in millisecons) that an event occurre an the ientity of the kin of event (i.e., A,B,C or E). Though there were sixty generate sequences consistent with the esign principles of their experiment, we use only the first twenty which correspone with the conitions that they ran in their stuy. Structure Inference For each graph (G α G) an ataset (D) we take our sample parameters({θ} m {1,...,M} ; for us, M = 1) an compute: L(D G α ) 1 M M m=1 exp(l(d G α,θ m )) We compute log-likelihoos (l( )) uner G α an Θ m as follows. For computational efficiency, we eliminate events o not alter other events (e.g., uner graph B E, we consier Human: Absolute Moel: Absolute Cause A Cause B Cause C r.9466, p <1 5 Figure 2: Top: Mean absolute jugments, Experiment 2 of Lagnao an Speekenbrink (21). Bottom: m abs moel. likelihoos of E- an B-events but eliminate A- an C-events). Using the reuce event set ({,t 1,...,t i,...,t n }), we can partition the observation interval an consiering each interval event-ientity pair τ t j,t j+1 (X j,x j+1 ) τ j conitione on previous events associate with cause X (t [X] j) we can calculate the log-likelihoo. The total log-likelihoo: l(d G α,θ) = Λ (t,t n ) +λ {t} n, where Λ (,tn ) is the log-likelihoo component of the intervals, Λ (,tn )=λ (t n t )+ [ [ ψ X τ j D X, X E G α ϕ X (e ϕ Xt j e ϕ Xt j+1 ) t [X] t j [e ϕ Xt [X] ]]], an λ {t} n is the log-likelihoo component of the point events, λ {t} n = [log(λ + t j t [E] X, j X E G α [ψ X (e ϕ Xt j ) [e ϕxt [X] t [X] t j ]])]. Using the likelihoo estimate(l) plus the prior for each graph (in our case, uniform over graphs p(g α ) 1, α), we can then compute the posteriors for all graphs. L(D G α ) p(g α ) p(g α D) = Gα G (L(D G α ) p(g α )) Comparison to Human Responses People juge causes, not graphs; we nee a way to map posterior probabilities p(g α D) to causal jugments. Lagnao an Speekenbrink (21) aske participants for absolute measures (assign each potential cause a value on scale from to 11) an comparative measures (assign a total of 1 points to the three causes). We will moel jugments as statements about structure inferences (not strength estimations). We interpret the absolute score in terms of a probability that a particular variable is thought to be present by marginalizing over the probabilities given to the graphs that inclue that variable is a cause. I.e., m abs (X N; p(g D)) = p(g α D) G α (X E) G α

6 Note if the complete graph were to receive all the probability measure, then m abs (A), m abs (B), an m abs (C) woul each equal 1, an so their sum woul equal 3. Thus, this is not a probability measure in the usual sense because we have not efine probabilities over causes, but over graphs. However, by normalizing by the sum of these measures over all noes, we can aapt the absolute measure to saying something about the comparative importance of the ifferent noes in proucing the effect. This will sum to 1, but still shoul not be interprete as anything like a irect probability of the cause being present. Gα (X E) G m comp (X N; p(g D)) = α p(g α D) x N Gα (x E) G α p(g α D) Finally, we coul consier the comparative prompt as implying that there is only one cause, an so we shoul only consier those graphs which attribute a single cause for proucing the effect in question. In fact we can say that uner the restriction that only one cause may exist the graph incluing X as its sole cause is the measure of the comparative importance of X (since it is the only graph with that cause). Results m unique (X N; p(g D)) = p(x E D) X {A,B,C} p(x E D) We fin an excellent fit between our moels preicte values an average human jugments for both absolute (r.94, p < 1 5, see Figure 2) an comparative jugments(m comp r.98, p < 1 7 ; m unique r.98, p < 1 7, see Figure 3) of the ifferent kins of waves causal importance. Discussion Learning from streams of events unfoling in continuous time is an important case to consier in stuying human causal inuction. We have shown how this case can be accommoate within the framework of Pacer an Griffiths (212), successfully preicting mean human jugments of real-time event streams use in Lagnao an Speekenbrink (21). We know of no other framework that has been brought to bear on realtime event streams. Often, statistical frameworks of theories of human causal inference are contraste with mechanistic theories. The kins of arguments use against statistics are that they neglect to inclue information about temporal an spatial characteristics of the moele processes. We have seen success in moeling causal inuction that relies explicitly on the temporal aspects of events. We see this as a step towars reconciling mechanism an statistics. More sophisticate graphical moels can be built that articulate complex causal mechanisms that are sensitive to particular temporal an spatial properties. This gives a hope for issolving statistical-mechanistic ivie an the birth of a new synthesis with statistical inferences compute over mechanistic theories. Human: Comparative Moel: Comparative Moel: Unique Cause A Cause B Cause C r.9777, p <1 7 r.9766, p <1 7 Figure 3: Top: Mean human comparative jugments from Lagnao an Speekenbrink (21). Mile: m comp moel. Bottom: m unique moel. Acknowlegments. We thank Maarten Speekenbrink an Davi Lagnao for proviing their stimuli an results. Support for this work was provie by the National Defense Science & Engineering Grauate Fellowship (NDSEG) Program aware to MP an grant FA from the Air Force Office of Scientific Research aware to TG. References Blunell, C., Beck, J., & Heller, K. A. (212). Moelling reciprocating relationships with Hawkes processes. In Avances in Neural Information Processing Systems (pp ). Bramley, N. R., Gerstenberg, T., & Lagnao, D. A. (214). The orer of things: Inferring causal structure from temporal patterns. In Proceeings of the 36 th Annual Conf. of the Cognitive Science Society. Gopnik, A., Glymour, C., Sobel, D., Schulz, L., Kushnir, T., & Danks, D. (24). A theory of causal learning in chilren: Causal maps an Bayes nets. Psychological Review, 111, Greville, W., & Buehner, M. (27). The influence of temporal istributions on causal inuction from tabular ata. Memory & Cognition, 35, Greville, W., & Buehner, M. (21). Temporal preictability facilitates causal learning. Journal of Experimental Psychology: General, 139(4), 756. Griffiths, T. L., & Tenenbaum, J. B. (29). Theory-base causal inuction. Psychological review, 116(4), 661. Lagnao, D. A., & Speekenbrink, M. (21). The influence of elays in real-time causal learning. The Open Psychology Journal, 3(2), Pacer, M., & Griffiths, T. (212). Elements of a rational framework for continuous-time causal inuction. In Proc. of the 34th Conf. of the CogSci Society. Simma, A., & Joran, M. (21). Moeling events with cascaes of poisson processes. In International conference on uncertainty in artificial intelligence.