VIOLENCE IN THE SECOND INTIFADA: A DEMONSTRATION …

VIOLENCE IN THE SECONDINTIFADA: A DEMONSTRATIONOF BAYESIAN GENERATIVECOGNITIVE MODELING

Percy K. Mistry and Michael D. LeeUniversity of California Irvine, United States

ABSTRACTJeliazkov and Poirier (2008) analyze the daily incidence of violence duringthe Second Intifada in a statistical way using an analytical Bayesian imple-mentation of a second-order discrete Markov process. We tackle the samedata and modeling problem from our perspective as cognitive scientists. First,we propose a psychological model of violence, based on a latent psychologicalconstruct we call “build up” that controls the retaliatory and repetitive violentbehavior by both sides in the conflict. Build up is based on a social memory ofrecent violence and generates the probability and intensity of current violence.Our psychological model is implemented as a generative probabilistic graphi-cal model, which allows for fully Bayesian inference using computationalmethods. We show that our model is both descriptively adequate, based onposterior predictive checks, and has good predictive performance. We thenpresent a series of results that show how inferences based on the model canprovide insight into the nature of the conflict. These inferences consider thebase rates of violence in different periods of the Second Intifada, the natureof the social memory for recent violence, and the way repetitive versus retal-iatory violent behavior affects each side in the conflict. Finally, we discusspossible extensions of our model and draw conclusions about the potential

Topics in Identification, Limited Dependent Variables, Partial Observability,

Experimentation, and Flexible Modeling

Advances in Econometrics, Volume 40A, 65�90

Copyright r 2019 by Emerald Publishing Limited

All rights of reproduction in any form reserved

ISSN: 0731-9053/doi:10.1108/S0731-90532019000040A005

65

http://dx.doi.org/10.1108/S0731-90532019000040A005

theoretical and methodological advantages of treating societal conflict as acognitive modeling problem.

Keywords: Second Intifada; Bayesian methods; cognitive models;vector autogressive models; modeling violence; hierarchical models

1. INTRODUCTIONFrom the outside perspective of cognitive scientists � sharing with economistsan interest in understanding human decision-making, choice, judgment, andpreference, but working with different problems and data � Dale Poirer’s focuson Bayesian methods has always been a natural point of contact. As early as1988, in an article titled “The Subjective Perspective of a ‘Spiritual Bayesian’,”Dale Poirer was viewed as having “preached a rousing sermon for the Bayesiancause” (Rust, 1988, p. 145). With one or two prescient exceptions (e.g.,Edwards, Lindman, & Savage, 1963), it took cognitive scientists longer to beginadvocating for Bayesian methods as an effective way to relate psychologicalmodels to behavioral data. The initial focus was on the ability of Bayesian meth-ods to account for model complexity in comparing different models (Myung &Pitt, 1997), but the general coherence, completeness, and consistency ofBayesian statistics progressively has come to be appreciated throughout thecognitive sciences. Bayesian methods are now widely used and accepted in areasincluding perception, learning, memory, and decision-making, and Bayesianmethods are standard components of coursebooks (Farrell & Lewandowsky,2018; Kruschke, 2011; Lee & Wagenmakers, 2013) and handbooks (Lee, 2018;Myung, Pitt, & Kim, 2005) in the field.

In this chapter, we revisit one of Poirer’s most recent Bayesian applications.Jeliazkov and Poirier (2008) consider a problem and dataset, involving thepattern of violence during the Second Intifada between Israel and Palestine,that has received considerable attention in political science and statistics (Asali,Abu-Qarn, & Beenstock, 2017; Frisch, 2006; Golan & Rosenblatt, 2011;Haushofer, Biletzki, & Kanwisher, 2010; Jaeger & Paserman, 2006, 2008, 2009).The intifada violence dataset measures the numbers of fatalities for both theIsraeli and Palestinian sides over the course of more than 2,400 days, anddivides the days into 10 meaningful periods delineated by significant political ormilitary events.

Jeliazkov and Poirier (2008) model the daily incidence of violence using ananalytical Bayesian implementation of a second-order discrete Markov process.They find that the data are “characterized by weak dynamics and strong insta-bility across sub-periods, showing distinct violence patterns within each politicalregime.” Many other authors have used vector autoregression (VAR) statisticalmethods to model the data. Jaeger and Paserman (2008) use a VAR frameworkto test the relationships between fatalities on either side, and show evidencefor a single directional Granger causality, with Palestinian fatalities beingcausally dependent on previous Israeli fatalities, but not the other way round.The evidence seems to suggest that Palestinian violence is random and cannot

66 PERCY K. MISTRY AND MICHAEL D. LEE

be predicted based on previous Israeli violence. However, Asali et al. (2017) usenon-linear VAR models and find evidence for a bidirectional effect of fatalitieson either side being influenced by previous violence, but with a faster and moreaggressive reaction from Israel. Haushofer et al. (2010) use VAR methods andfind evidence for retaliation by Israelis. They also argue that taking into accountnon-lethal violence reveals retaliation by Palestine, especially when consideringthe firing of Palestinian rockets. They show a retaliatory effect in the probabilityof violence rather than the number of killings of Israelis by Palestinians, andconclude that Israeli violence leads to escalation rather than incapacitation. Thisconclusion contrasts with that reached by Frisch (2006), who argue that Israelicounterterrorism violence leads to increasing Palestinian incapacitation. Golanand Rosenblatt (2011), however, reply that the retaliatory effects shown byHaushofer et al. (2010) are not significant. They argue that many of the assump-tions required by the statistical analyses employed by both Haushofer et al.(2010) and Jaeger and Paserman (2008) are violated. Golan and Rosenblatt(2011) find that the nature of the conflict varies significantly over time, shiftingbetween regimes of retaliatory and non-retaliatory violence.

As this summary of previous research makes clear, how to analyze the intifadadata remains an open statistical question, and the conclusions that can be drawnremains an open political and behavioral science question. Accordingly, in thischapter, we address the same broad research goal of trying to model the sequenceof fatalities over days and time periods in the data, aiming to understand the pat-terns of violence. Our approach, however, is to treat the problem as a psychologicaland cognitive modeling challenge. Our aim is to develop a descriptively adequatemodel, cast in terms of psychologically interpretable processes and variables thatprovide insight into the underlying dynamics of the conflict. This cognitive scienceperspective leads to some theoretical and methodological differences from previousstudies of the data. Theoretically, we aim to develop generative probabilisticmodels of the observed that involve psychological theory, rather than rely on stan-dard statistical models. Methodologically, we do not seek models that are analyti-cally tractable. Instead, we adopt the formalism provided by graphical models(Jordan, 2004; Koller, Friedman, Getoor, & Taskar, 2007; Lee & Wagenmakers,2013) and use standard JAGS software that applies MCMC computationalmethods to make approximate Bayesian inferences (Plummer, 2003).

The structure of this chapter is as follows. In the next section, we describe thedata in a little more detail. We then develop a cognitively inspired model of thegeneration of day-by-day violence by both sides. We describe the implementationof the model as a graphical model, and use this implementation to applyBayesian computational methods to make inferences. Our results’ section focuseson four model-based inferences, which provide insights into how the two sides inthe conflict remember and are affected by previous violence, and how they reactin terms of producing new violent behavior. Finally, we describe some possibleextensions to our model, and discuss the theoretical and methodological differ-ences arising from our cognitive modeling approach to the problem.

67Violence in the Second Intifada

2. DATAThe data take the form of counts of fatalities, vs;t and v0s;t, for violence on the tthday committed, respectively, by the sth side and against the s0th side. Becausethere are two sides, the violence committed by the first side is the violencereceived by the second side, so that v1;t ¼ v02;t. Similarly, the violence committedby the second side is the violence received by the first side, so that v2;t ¼ v01;t. Thedata consist of fatalities on both sides over a period of 2,434 days. This includes240 violent days (9.9%) with 465 Israeli fatalities by Palestinians, and 1,247 vio-lent days (51%) with 4,088 Palestinian fatalities by Israelis. There are 178 days(7.3%) with violence by both sides, and 1,125 days (46%) with no violence byeither side. The highest numbers of fatalities on a single day are 14 and 62,respectively, for the two sides. Across the 10 periods, the average daily numberof fatalities ranges from a low of 0.005 to a highest of 1.15 for violence byPalestine, and from a low of 0.64 to a high of 7.79 for violence by Israel.

Fig. 1 presents a visual summary of the sequences of counts of violence thatcomprise the data. The top panel shows violence by Israel against Palestine, and

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Fig. 1. The Pattern of Change in Weekly Violence over Time. Notes: The top panelshows violence by Israel against Palestine, and the bottom panel shows violence byPalestine against Israel. Bars show the range in violence each week, and the solidline shows the average violence within the week. The broken lines show the division

of weeks into the 10 periods.


the bottom panel shows violence by Palestine against Israel. In both panels thex-axis ranges over all of the days in the data. As mentioned earlier, the data alsoinclude a partition of each day into one of 10 periods, corresponding to differentinterpretable social and political contexts, with the period for the tth daydenoted by pt. Fig. 1 shows these periods using dividing lines: the presence ofEhud Barak, then Ariel Sharon, Operation Defensive Shield, its ceasefire, thedeath of Arafat and then the incapacitation of Sharon, three periods surround-ing the Lebanon war, and the final truce. The violence counts are summarizedvisually at the level of weeks, with bars showing the minimum and maximumcounts in each week, and a solid line showing the average violence in each week.

It is clear from Fig. 1 that the counts of violence differ significantly acrossdifferent periods, and that the violence by the two sides is often asymmetric. Forexample, at the beginning of the Sharon Stroke period, there is significant vio-lence by Israel, but very little by Palestine. It is also clear from Fig. 1 that manydays and weeks have no observed violence. Thus, it seems likely that any ade-quate model needs to allow for changes across periods, must model the violencegenerated by each side as dependent on each other but different from oneanother, and needs to have the capability to explain a mixture of violent andnon-violent days.

3. BUILDING A PSYCHOLOGICAL MODELThe cornerstone of our model is a latent psychological construct, which we callbuild up, that measures the strength of the tendency toward violence for eachside over time. It can be conceived as something like a societal disposition orattitude toward violence. Fig. 2 provides an overview of how our model assumesthat build up mediates the transition from previous violence to future violencefor a side in the conflict. Previous violence, committed by either side, acts asstimulus that is remembered. Violence produced by the same side contributes toa repetitive memory, while violence committed by the other side contributes to aretaliatory memory. Fig. 2 then shows how build up is assumed to play a role inthe decision-making processes that determine the probability that the side willgenerate violence in response to build up, and the intensity of this possible vio-lence. Together, the decisions about whether to produce violence and, if so, howintense it should be, result in the action of new violent behavior.

3.1. Memory for Violence

We represent the repetitive and retaliatory memories as, respectively, αs;t and βs;tfor the sth side at the tth time. These memories are recency-weighted tallies ofthe previous violence by the same side and by the other side:

αs;t ¼ γαs;pt vs;t þ 1� γαs;pt

� �αs;t�1; αs;1 ¼ vs;1

βs;t ¼ γβs;pt v0s;t þ 1� γβs;pt

� �βs;t�1; βs;1 ¼ v0s;1:

ð1Þ


The weights γαs;pt and γβs;pt quantify how persistent the influence of previousviolence is, in societal memory. If a recency weight is close to one, only the mostrecent events will be significant. As a recency weight decreases toward zero, theinfluence of distant events increases. These assumptions give societal memorysimilar properties to standard assumptions in cognitive science about the natureof human episodic memory (Norman, Detre, & Polyn, 2008).

3.2. Build Up

Build up depends on these memories of previous violence. Formally, the psycho-logical build up ψ s;t for the sth side at the tth time is given by a weighted combi-nation of the retaliatory αs;t and repetitive βs;t components:

ψ s;t ¼ ωαs;ptαs;t þ ωβ

s;ptβs;t: ð2Þ

The magnitude of the weights ωαs;pt and ωβ

s;pt correspond to the relative impor-tance of the repetitive and retaliatory components for the sth side during thetime period pt.

The sign of these weights correspond to the valence of the influence. If aweight is positive, the influence of previous violence is to aggregate furtherviolence, while if a weight is negative, the influence of previous violence is todeter further violence. Further violence aggregated by previous violence by thesame side is naturally interpreted as repetitive violence, while further violenceaggregated by previous violence by the other side is naturally interpreted asretaliatory violence. When the influence of previous violence from the same side

Violence by other(Stimuli)

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Build up(Attitude)

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Fig. 2. Overview of a Psychological Model of How One Side in a ConflictGenerates Violent Behavior on the Basis of Build Up Caused by Previous Violence.


is negative, one interpretation is that the side is satisfied with the impact of theirprevious actions, and another interpretation is that the capacity for additionalviolence by that side has been exhausted. When the influence of previousviolence from the other side is negative, one interpretation is that the capacity ofthe side subjected to the previous violence has been significantly diminished, andanother possible interpretation is that the previous violence has acted as aneffective deterrent.

3.3. Probability of Violent Attacks

The probability of violent attacks πt at any time t is based on the latent build upψ t�1 and is modeled with a logistic function:

πs;t ¼ νs;pt

�1þ exp

� ψ s;t�1 � τs;ptÞ� �

λs;pt

� �� ð3Þ

that includes a threshold value τs;pt , an upper bound νs;pt , and a scale λs;pt . As thebuild up ψ s;t�1 crosses the threshold τs;pt , the probability of violence will be halfthe maximum possible probability νs;pt . How quickly the probability increases ordecreases as the build up increases or decreases is controlled by the scale λs;pt .The non-linear relationship between build up and response probability is onecommonly made in cognitive modeling. For example, at the most basic level ofindividual decision processes, the sigmoidal relationship is often used to maplatent strengths to behavioral probabilities in psychophysics (Wichmann & Hill,2001). At a higher level, similar functional forms have been used to describe theprobability of a person engaging in violence, depending on a social distance met-ric (Bhavnani, Donnay, Miodownik, Mor, & Helbing, 2014).

The probability πs;t is used to generate a latent indicator δs;t ∼Bernoulliðπs;tÞ,with δs;t ¼ 1 indicating the sth side perpetrated violence on the tth day, andδs;t ¼ 0 indicating that they did not.

3.4. Intensity of Violent Attacks

The intensity of observed violence is also modeled as depending on the combina-tion of the latent build up ψ s;t�1, and a base rate ϕpt of violence in the period pthat is independent of the latent build up. The intensity is calculated as:

θs;t ¼ ψ s;t�1 þ ϕs;pt�1; ð4Þ

and is the basis of modeling the observed violence as:

vs;t ∼Poisson ðθs;tÞ if δs;t ¼ 1 and θs;t > 0

0 otherwise:

(ð5Þ

Together, the modeling approach of considering the probability of violenceand the intensity of violence as separate but interrelated processes means that


observed non-violent days can occur through two meaningfully different routes.One route is when non-violence arises structurally through a failure to reach athreshold needed for the possibility of violence. The other route is that non-violence eventuates through random variation in sampling, when a potentiallyviolent day just happens to produce no violent acts. The latent binary indicatorδs;t distinguishes these two routes.

3.5. Interaction between Sides

Fig. 3 shows how the basic decision-making process for each side in the conflictinteract in the complete model. Each side processes previous violence and pro-duces new violence following the overview presented in Fig. 2. The sides arelinked by their encountering the new violence. As Fig. 3 shows, the violence pro-duced by one side is perceived and remembered as its own repetitive violence,and by the other side as retaliatory violence.

3.6. Graphical Model

Fig. 4 shows a graphical model representation, which also incorporates theassumptions we make about prior distributions on the model parameters.Graphical models provide a convenient formalism for expressing generativeprobabilistic models in psychology, and the approach has the advantage ofbeing especially well suited to the application of computational methods forBayesian inference. Graphical models were developed and remain widely used inartificial intelligence and machine learning (e.g., Jordan, 2004; Koller et al.,2007; Pearl, 1998), and are progressively being adopted in the cognitive sciences(Lee & Wagenmakers, 2013).

In a graphical model, nodes in a graph represent parameters and data, andthe graph structure indicates how the parameters generate the data, with

Violence by self Violence by other Violence by self Violence by other

Repetitive memory Retaliatory memory

Build up

Prob of violence Intensity of violence

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Retaliatory memory Repetitive memory

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Prob of violence Intensity of violence

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Fig. 3. Overview of How the Two Sides Interact in Our Model. Notes: Both sidesremember and produce violence following the basic model in Fig. 2, with violence

produced by each side being perceived as repetitive and retaliatory acts.


children depending on their parents. Observed variables, usually in the form ofdata, are shown by shaded nodes, while latent variables, usually in the form ofparameters, are shown as unshaded nodes. Discrete-valued variables are shownby square nodes, while continuous-valued variables are shown by circular nodes.Stochastic variables, defined in terms of their distributions, are shown by single-bordered nodes, while deterministic variables, defined as functions of othervariables, are shown by double-bordered nodes. Encompassing plates show inde-pendent replications of the graph structure.

The graphical model in Fig. 4 shows visually how the latent build up ψ isresponsible for generating the observed counts of violence v by influencing boththe probability of violence π and the intensity of violence θ. The graphical modelalso shows, via the broken arcs, the “feedback loop” by which the counts ofviolence update the cumulative totals α and β that determine the build up forthe next day. The graphical model representation captures visually the assump-tions about period-by-period change in the parameters that guide the cumulativeviolence totals, build up, and the probability and intensity of violence. Theseparameters lie in the outer plate corresponding to the different periods, and areassumed to be independent across periods.

Fig. 4. Graphical Model Representation of the Model of Intifada Violence. Notes:The graph on the left expresses the relationship between model parameters and data,showing how psychological variables and processes are assumed to generateobserved counts of violence across days and periods. The equations on the right

provide the formal statistical assumptions about the parameters and data.


Besides presenting the graphical model, Fig. 4 presents the definition of deter-ministic nodes, and the distributional assumptions for stochastic nodes. Thesedistributions include prior distributions over model parameters, which arerequired for Bayesian inference. The priors are selected so that the recencyweights γαs;pt and γβs;pt are always between zero and one. The relative weights on

repetition and retaliation, given by ωαs;pt and ωβ

s;pt , are given a zero-centeredGaussian prior. This means that they place equal prior probability on both posi-tive and negative influences of repetitive and retaliatory build up, but that thelevel of possible build up is unbounded. The priors for the scale and thresholdparameters of the logistic function driving the probability of violence arecentered at one and zero respectively.

4. MODELING RESULTSWe applied the graphical model in Fig. 4 to the data, using the default Markov-chain Monte-Carlo (MCMC) methods implemented by JAGS. Our results arebased on 5,000 samples from each of three independent chains, after discarding5,000 burn-in samples in each chain. The convergence of the chains was assessedusing the standard R̂ statistic (Brooks & Gelman, 1997), which measures theratio of within-to-between-chain variance, with ratios close to one being indica-tors of reasonable levels of convergence.

We begin by assessing the descriptive adequacy of the model, through a pos-terior predictive check. We then test the predictive capabilities of the model,examining its ability to forecast the level of violence for future days, based onknowledge of previous violence. Finally, we examine inferences about modelparameters to understand the dynamics of the observed violence. This involvesinferences about the level of build up for retaliatory and repetitive violence forboth sides during different periods, the relationship between build up and theprobability and intensity of violence, and the relative impact violence has onbuild up as time progresses.

4.1. Descriptive Adequacy

A minimum requirement for a model to be useful is that it is able to describe thedata it is designed to understand and explain. Without some level of descriptiveadequacy, it is not clear that inferences about parameters are meaningful, andunlikely that the model will have any predictive capability. We follow the stan-dard Bayesian approach of comparing the posterior predictive distribution ofthe model to the observed data (Gelman, Carlin, Stern, & Rubin, 2004; Shiffrin,Lee, Kim, & Wagenmakers, 2008). The posterior predictive distribution is thedistribution over data generated by the model, based on the posterior distribu-tion of parameters. This provides a basic test of the ability of the model tore-describe data it has already seen.

Fig. 5 summarizes both the data and the posterior posterior distribution,using the same approach used to show the data in Fig. 1. The top panel showsviolence by Israel against Palestine, and the bottom panel shows violence by


Palestine against Israel. Within each panel, the data are again summarized bybars showing the range in violence over each week and a solid line showing thepattern of change in the average weekly violence. The posterior predictive distri-bution is summarized in exactly the same way, but is shown in mirror-imagereflection about the x-axis, increasing downward in the lower half of the axis.This means that agreement between the data and the posterior predictive distri-bution can be assessed visually by the extent of symmetry in the reflection.

The results in Fig. 5 show that the model is able to describe (or “fit”) theaverage weekly violence very well. The descriptive adequacy for the daily rangesis not quite as impressive. In particular, the model seems often to underestimatethe maximum extent of violence. Nonetheless, overall, we regard the posteriorpredictive check as providing evidence that the model is capable of capturingmuch of the variability in the data, and provides grounds for believing thatinferences about model parameters can usefully be interpreted.

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Fig. 5. Descriptive Adequacy of the Model. Notes: The top panel shows violenceby Israel against Palestine, and the bottom panel shows violence by Palestine againstIsrael. The broken lines show the division of weeks into the 10 periods. Within eachpanel, the data are summarized in the upper half of the axis, and the posteriorpredictive distribution is summarized in the lower half of the axis. For both the dataand posterior predictive distribution, bars show the range in violence over each

week, and the solid line shows the average violence within the week.


4.2. Predictive Performance

We now move from testing the adequacy of our model’s descriptions towardtesting its predictions. To enable the model to make predictions with a fixed lag,we iterate the model on a daily basis, providing the model only the last 30 daysof data at a time, and making predictions for the next 1; 2;…7 days.

The top two panels of Fig. 6 show the change in an error measure as descrip-tion, based on data already observed, and then consider genuine predictions ofthe violence count one day ahead, two days ahead, and so on, up to one weekahead. The error measure is the mean absolute difference between an integerprediction, formed by rounding the appropriate part of the predictive distribu-tion, and the actual observed violence count for that day. In this way, the pre-dictions can be conceived as simple whole-number forecasts that might be usedin real-world application.

In the top panels of Fig. 6, the “describe” value corresponds to this errormeasure for the posterior predictive distribution, while the “predict days ahead”correspond to the predictions. The “repeat yesterday” value corresponds to aheuristic prediction that simply predicts the same violence as observed the previ-ous day. The left panel measures the average error over all of the days in thedata, while the right panels only considers those days in which violence was

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Fig. 6. Descriptive and Predictive Adequacy of the Model, and an AlternativeHeuristic “Repeat Yesterday” Model. Notes: The top two panels show the meanabsolute difference between model posterior predictive distributions expressed asinteger counts. The bottom panels show the pattern of modeled and observed valuesfor the posterior predictive description, the one-day and one-week prediction, and

the repeat yesterday heuristic.


observed. In both cases, not surprisingly, the error increases as descriptionmoves to prediction, and as predictions are made further into the future. The rel-atively slow increase in error, however, and the absolute value of the errors, areencouraging. Even predicting one week ahead, the model is on average onlyabout one violence count away from the truth. The model predicting one dayahead also clearly outperforms the simple heuristic of repeating yesterday’sviolence count, under both measures.

The bottom panels of Fig. 6 provide results showing how modelpredictions � for the posterior predictive description, one-day and one-weekpredictions, and the repeat yesterday heuristic � relate to the observed data. Ineach panel, the x-axis corresponds to the predicted violence, the y-axis to theobserved count, and the area of the square is proportional to how often thiscombination occurred over all days. Correct predictions on the diagonal havedarker shading, and the hit rate (HR) and false-alarm rate (FAR) for detectingnon-violent days with zero counts of violence are also listed.

This set of results makes clear that the superior accuracy of the descriptiveaccount is based on its knowledge of the violent versus non-violent days. It hasperfect signal detection performance with respect to this distinction. That is, ithas a 100% accuracy in classifying zero (non-violent) versus non-zero violentdays. The predictive models, however, have lower hit rates (correctly predictinga violent day) and much higher FARs (incorrectly predicting what turns out tobe a non-violent day, as a violent day). However, the additional greater error ofthe repeat yesterday heuristic is due to a significantly lower hit rate. This heuris-tic thus incorrectly predicts many more violent days as non-violent. In the real-world setting of conflict, this is arguably the most serious type of error, and the54% hit rate of the heuristic is problematically low. The 80% and 81% hit ratesof the one-day and one-week predictions made by the model, in contrast, arerelatively impressive.

4.3. Levels of Build Up in Different Periods

Fig. 7 shows the distribution of the level of build up, aggregated over eachperiod. The bars show the 95% range of values of ψ for both the the retaliatoryand repetitive components for both sides, and the broken lines show the medianvalue. Recall that build up can be positive or negative, and has different inter-pretations for the retaliatory and repetitive components. A positive retaliatorybuild up corresponds psychologically to an escalation of conflict, with greaterlikelihood of violence causing further violence, while negative values are bestinterpreted as some sort of capacity reduction that prevents retaliation.Consistent with these interpretations, the results in Fig. 7 show that negativeretaliatory build up is only associated with Palestine, and that positive retalia-tory build up is at its greatest for both sides during the Operation DefensiveShield period.

Meanwhile, repetitive build up is best understood in terms of the execution ofstrategies or plans that span a number of days. Positive repetitive build up corre-sponds to the ongoing execution of action that leads to observations of violence,


whereas negative repetitive build up corresponds to the cessation of action.Fig. 7 shows that Palestine has relative low repetitive build up, either positive ornegative, in any period. Israel, in contrast, shows large levels of build up inseveral periods, including Operation Defensive Shield and around the Lebanonwar, consistent with the execution of planned action.

4.4. Build Up and the Probability of Violence

Fig. 8 shows the relationship between the probability of violence and latentbuild up during each period. This corresponds to the form of the logistic func-tion we use to model π, based on the expectation of the inferred threshold,bound, and scale parameters for each period. The changes in the functional rela-tionship from period to period are generally interpretable. For example, there isa significant flattening of both functions to lower probabilities of violencebetween the Operation Defensive Shield and ceasefire period, and there is anincrease then decrease for both sides through the pre-, during, and post-Lebanon war periods. Fig. 8 also shows that Israel almost always is inferred tohave a great probability of violence for the same level of build up, and that thisprobability increases more quickly as build up increases. We also note that thefunctions relating build up to the probability of violence are most similarbetween the two periods in the first period of the intifada, and in the final periodwhen a truce had been established.

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Fig. 7. The Distribution of Build Up of the Repetitive and RetaliatoryComponents of Build Up for Each Period. Notes: The bars show the 95% range ofbuild up over all days in a period, and the broken line shows the median level of

build up during a period.


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Fig. 8. The Probability of Violence as a Function of Latent Build Up on Either Side.

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4.5. Build Up and the Intensity of Violence

Fig. 9 shows the results of an analysis that explores the usefulness of the day-to-day monitoring of violence for modeling the intensity of violence. This analysisis based on the definition of the intensity θ as the additive combination of thebuild up ψ , which varies day to day, and a base rate ϕ for the overall period.The dynamics ratio ψ

=ð ψ þ ϕ Þ quantifies the importance of build up beyond

the base rate in determining the modeled intensity of violence. Ratios near zerocorrespond to the situation in which the base rate provides most of the explana-tion, and changes in observed violence from day to day do not affectthe intensity of subsequent violence. One interpretation of this situation is thatthe psychological factors that determine the intensity of violence have beeninternalized, and are not sensitive to external events. The base rate quantifiesthis internalized intensity for each period. Larger values of the dynamics ratio,however, correspond to situations where observed violence from day to daydoes affect the intensity of subsequent violence. One interpretation is that the sit-uation is more dynamic and reactive, and so the daily pattern of change in buildup is needed to model observed violence.

The results in Fig. 9 show that the dynamics ratio is generally near zero, inall periods, for the repetitive component of violence for Palestine and retaliatoryviolence for Israel. The dynamics ratios are often higher, and show more period-to-period variability, for the retaliatory component for Palestine and the repeti-tive component for Israel. This suggests that daily changes in violence affect theintensity with which the Palestine side retaliates to violence, and the Israel siderepeats current violence. The impact of specific periods are also generally inter-pretable. For example, there is a large increase in the repetitive dynamics forIsrael around the Lebanon war and the during Operation Defensive Shield, buta sudden fall during the truce period.

4.6. Impact of Recent Violence on Build Up

Fig. 10 shows how the repetitive and retaliatory components impact latent buildup over time. Within each period, the curves show the relative impact of vio-lence on the current level of build up, for violence that occurred between oneday and one week prior to the day being considered. One way to think of thesecurves is as impulse functions, showing the impact that one unit of violencefrom a previous day has on the latent build up ψ for the current day. This meansthat the y-values on the curves are not inherently meaningful, but their relativevalues are. In particular, the rate of decay in the impact on build up as the num-ber of days increases quantifies how previous violence “washes out” of thesystem.

For most periods, the results in Fig. 10 show that the influence of previousviolence decreases to a negligible level within a week. There are, however, somenotable exceptions. For example, in the case of Israel during the period whenArafat dies, and during the pre-Lebanon war, the retaliatory impact remainshigh after seven days. Fig. 10 also shows how retaliatory influence can be


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Fig. 9. Distribution of the Ratio of Latent Build Up to the Base Rate in Determining the Intensity of Violence. Notes: The panelscorrespond to the retaliatory and repetitive components for Palestine and Israel, and the distribution of the ratio is shown for each

period.

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repeat by Palestineretaliate by Palestinerepeat by Israelretaliate by Israel

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Fig. 10. Recency Impacts of the Repetitive and Retaliatory Components of Latent Build Up. Notes: Each panel corresponds to one ofthe 10 named periods, and the horizontal line denotes no impact. Within each panel, the x-axis corresponds to the number of days fromthe current day, ranging from one to seven, and the y-axis corresponds to the effect on build up. The different lines correspond to the

repetitive and retaliatory impact by Israel and Palestine.

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positive or negative in different periods, and thus switch in the qualitative natureof its impact as the broader social and political context changes. Interestingly,these qualitative changes are only inferred to occur for Palestine. For example,the influence of retaliation by Palestine is positive during the Sharon period, butbecomes significantly negative during the Operation Defensive Shield period.One interpretation of this change is that the Operation Defensive Shield periodwas one with a goal of capacity destruction by Israel. Conversely, there is a largeincrease in the retaliatory component in the period in which Arafat died.Finally, it is interesting to note the large difference in the dilution of dynamicrepetitive and retaliatory impacts between the ceasefire period and the truceperiod.

5. EXTENSION TO A SPATIAL MODELAn interesting extension of our model, which we leave largely as future work,involves incorporating spatial and structural information relevant to the patternsof intifada violence using network models. Jaeger and Paserman (2006) highlightthat there are at least three different factions within Palestine, and that the pat-terns of violence perpetrated by these factions, as well as the response of Israelto each of these factions, may vary significantly. Bhavnani et al. (2014) suggestthat both spatial proximity and social distance can drive the variability in con-flict between groups. For example, it is plausible that violence in different citiesgenerated by either side may not influence future violence in the same way. Ourmodel can be extended to incorporate the influence of such factions, and the dif-ferent influence of violence across different geographical locations. This requirestreating the violence perpetrated by each faction, or within each city, as a sepa-rate time series, with separate repetitive and retaliatory weight coefficients.

As a proof of concept, we present illustrative results from a simple implemen-tation of such a model. We implemented this model using weekly rather thandaily data, based on 19 cities for which fatalities data was separately available.The extended network model infers the repetitive and retaliatory violence com-ponents between each pair of 19 cities. Fig. 11 summarizes the results just forthe city of Nablus, and only in relation to retaliatory violence by Palestine. Thegeographical location of Nablus and the other cities is shown on the right-handside. The panels on the left correspond to the different periods, with lines linkingNablus to the other cities representing the expected retaliatory weight for theother city in the period. Solid lighter lines correspond to positive weights, whilebroken lighter lines correspond to negative weights. The thickness of each linecorresponds to the absolute magnitude of the weight. Fig. 11 shows geographi-cal differences in which cities have positive and negative relationships to Nablus.It also shows a clear pattern of change in these retaliatory violence weights overthe periods, with the magnitude of positive weights increasing to some nearbycities, and the magnitude of negative weights increasing to a cluster of more dis-tant cities. We note that there was significant violent activity in Nablus duringthe Battle of Nablus during Operation Defensive Shield, resulting in an Israeli


Barak Sharon Defensive Shield Ceasefire Arafat dies

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Fig. 11. Illustration of Inferences from a Network Model, for the Case of Retaliatory Violence by Palestine. Note: Repetition based onviolence in a single city.

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victory, and a number of the negative retaliatory weights increase in magnitudeduring that period.

6. DISCUSSIONOverall, our basic model provides suitable descriptive and predictive capabilityand allows us to make inferences about the psychology of the underlying pro-cesses that generated violence during the intifada. We first discuss the theoreticalconclusions coming from our results, before comparing our approach to previ-ous statistical models of the intifada data, and finally considering some possibleextensions of our model.

6.1. Theoretical Conclusions

Our model of the pattern of violence in the intifada is predicated on the assump-tion that there is a latent psychological variable, which we call build up, thatunderlies the repetitive and retaliatory forces that generate observed violence.Build up arises from a social memory of previous violence, which we modeled asa decaying trace of previous repetitive and retaliatory violent acts. Our modelalso conceives of build up as being a common cause for both the probabilitythat a violence will occur on a given day and, given that it does occur, the inten-sity of that violence. These two components are modeled in different ways. Theprobability of violence is a non-linear function of build up, controlled by anupper bound, a threshold, and a scale. The intensity of violence combines buildup with a base rate that together determines the overall magnitude of violencefor a given day. To the extent that our model demonstrated a basic level ofdescriptive adequacy, and a reasonable level of predictive capability, these fun-damental theoretical assumptions are supported. We think conceiving the cycleof violence in terms of changing patterns of latent build up is a useful psycholog-ical perspective. We also think the evidence for the common cause approachprovides an interesting insight into the theoretical foundations of violence. Itwould be straightforward to propose a theoretically less-constrained model thattreated the probability of violence and the intensity of violence as two indepen-dent processes. The success of our model suggests that these two processes canbe linked, which makes for a simpler and more constrained model, and for amodel with greater explanatory power (Lee, 2018).

Besides supporting a general theoretical position that the latent construct ofbuild up is a useful way to model the patterns of violence, our model allows formore detailed inferences about specific aspects of the data. Most obviously, byallowing key model parameters to vary across the periods, it is possible to makeinferences about underlying changes as the social and political context changes.Our results are certainly consistent with the broad theoretical conclusions drawnby Jeliazkov and Poirier (2008, p. 18), who found the data to be “characterizedby weak dynamics and strong instability across sub-periods, showing distinctviolence patterns within each political regime.” We hope we have demonstratedhow our assumptions about detailed psychological variables and mechanisms


allow for potentially more fine-grained insight. We highlighted a number ofspecific inferences in our results, showing how periods like Operation DefensiveShield and surrounding the Lebanon war had very different violence dynamicsfrom periods of truce or ceasefire. We also identified more nuanced and insight-ful inferences, concerning the asymmetry in the dynamics of repetitive and retal-iatory components for the Israeli and Palestinian sides, and the different recencyimpacts recent violence had on build up in different periods for each both repeti-tion and retaliation for both sides.

We are not political scientists or military historians, and so do not want todraw broader interpretations from our results. But, we are sure the results couldbe interpreted more generally by those with the right domain knowledge. Themodel-based inferences we have reported provide insight into the differencesbetween the two sides in the conflict, and the differences in the processes theyused that lead to observed violence through repetitive and retaliatory behavior.

6.2. Methodological Conclusions

An interesting consequence of approaching the current problem as a psychologi-cal modeling problem is that our methods deviate in a number of ways fromprevious approaches. One difference involves the sorts of statistical assumptionswe make, and it is worth contrasting our assumptions to those of statisticalapproaches like VAR models. VAR models are commonly used to model inter-dependent time series in econometric studies (Asteriou & Hall, 2015; Belsley &Kontoghiorghes, 2009; Kilian & Lütkepohl, 2017). They allow for interpretationof the time series based on impulse response functions, which measure the reac-tion of target variables to impulse shocks in one or more of the input variables,and historical and predictive decomposition of error variances based on theimpulse response functions. Impulse functions can be analytically determinedbecause the change in the target variable as a result of a change in one of theinput variables is linearly separable. The model developed here does not need tomeet standard statistical assumptions about the time series. For example, ourmodel does neither assume that the time series is stationary, that the Poissonincrements are independent, nor that model residuals are normally distributed.In addition, our model allows for the probability of violence based on the buildup to be non-linear, and does not assume the effects of violence in previous peri-ods are linearly separable, as is often assumed. While the repetition and retalia-tion components that are central to our model are similar to the reactionfunctions used in previous VAR modeling of the intifada data (Haushofer et al.,2010; Jaeger & Paserman, 2006), they are constrained and simplified in ourmodel based on assumptions drawn from psychological theory.

This psychological perspective means that, while our model ultimately can beconceived as a set of statistical assumptions, it makes theoretical commitmentsand has parameters with interpretations that are more grounded in psychologi-cal theory. We believe that this approach leads to a model that is both statisti-cally simpler, and psychologically more structured, than previous models of theintifada data. For example, some approaches allow a free parameter quantifying


the weight given to each previous time, often by choosing a finite window thatlimits the time-lag over which previous violence can exert an influence. Ourapproach, in contrast, makes a theoretical assumption about constraintson social memory that lead to a single interpretable recency-weighted parameterfor each period, controlling the influence of past violence on current build up.

The other major methodological differences in our approach relate to ourlack of emphasis on analytic solutions. It is common for models in the cognitivesciences to resist almost any sort of analytical tractability. There are, of course,exceptions, and there are certainly classes of models developed in mathematicalpsychology from the 1960s onward that involve insightful analytic results. But,even in those early times, foundational models like multidimensional scaling(e.g., Kruskal, 1964; Shepard, 1962) relied on computer-based numerical optimi-zation to be applied to data. Thus, while cognitive science recognizes the scal-ability, efficiency, and insight that can accompany analytic solutions, it hasalways allowed for the possibility that the most interesting and useful models ofhuman behavior might require computational methods of analysis.

Our methods represent a modern form of this reliance on computation, usingthe graphical model formalism and the MCMC computational methods it sup-ports, to provide numerical approximations to the full Bayesian joint posteriordistributions used for inference. One advantage of this approach is that it allowsmodels to be developed that follow the theoretical assumption the modeler wantsto make, without having to compromise to facilitate analytic tractability. Thegoal of a generative model is to formalize the probabilistic processes, and con-trolling parameters, that produce observed behavioral data. Once this is done,Bayesian inference automatically defines inferences in terms of the posterior dis-tributions over parameters, and the posterior predictive distributions over data,that follow from conditioning on data. Relying on computational methodsmeans that these inferences can be approximated for any generative model ofinterest. This allows theoretical freedom and creativity in the expression of themodel, with statistical inference simply serving to find the inferences that followfrom the proposed model and available data. We think that this is the right divi-sion between theory and methods, with theory being primary, and methods serv-ing a supporting role. Of course, the loss of analytic tractability can sometimeslimit the feasibility of analyses, but our current results show that a reasonablycomplicated model scales to the level of the real-world intifada data.

7. CONCLUSIONUnderstanding the pattern of violence in the Second Intifada presents both amodeling challenge that is both statistical and psychological. The time series ofobserved violence has regularities and variability that can be described, predict,and understood in terms of statistical processes. In this chapter, however, weattempted to lay a foundation of psychological understanding beneath the statis-tical modeling. We proposed that a latent psychological construct of build up isestablished by previous patterns of violence, and plays a key role in determiningthe probability and intensity of future violence. To formalize these ideas, we had


to specify a generative probabilistic model that went beyond standard statisticaltime-series models, and use computational approaches to perform Bayesianinference. We hope the descriptive and predictive performance of our model,and the psychological insights it provides into the patterns of violence in theintifada data, has demonstrated the usefulness of this psychological theoreticalperspective and computational methodological approach.

ACKNOWLEDGMENTSThere is a project page associated with this chapter on the Open Science Frameworkhttps://osf.io/rvt4q/. The project pages contains code, data, and other supplementaryresults and information. We thank Ivan Jeliazkov for helpful discussions.

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AUTHOR BIOGRAPHYPercy K. Mistry is a Postdoctoral Scholar at the Stanford Cognitive and SystemsNeuroscience Laboratory at Stanford University since 2018. He has a PhD inCognitive Psychology from the University of California, Irvine. He has a degreein electronics engineering and an MBA, after which he worked in the financialindustry in India and Hong Kong. He has a diverse range of interests and lovesto develop highly interdisciplinary approaches. His research focuses on under-standing and modeling the latent cognitive processes that drive human behavior.He works with both experimental and real-world data, with a special focus onbehavior that involves how people learn, change, and adapt. He develops novelapproaches to incorporate structural models of change, learning, and adaptivityinto applied problems that span behavioral economics, econometric analysis,computational psychiatry, and computational neuroscience. His methods ofteninvolve computational and statistical modeling, and Bayesian inference.

Michael D. Lee is a Professor of Cognitive Sciences at the University ofCalifornia, Irvine. He is a former President of the Society of MathematicalPsychology, and winner of the William K Estes Award from that Society. Hisresearch involves the development, evaluation, and application of models of


cognition including representation, memory, learning, and decision-making, witha special focus on individual differences and collective cognition. His researchemphasizes the use if naturally occurring behavioral data, and tries to pursue asolution-oriented approach to empirical science, in which the research questionsare generated from real-world problems. His research methods focus on probabi-listic generative modeling and Bayesian methods of computational analysis.


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