Political Science“Turning Points” in the Iraq Conflict: Reversible Jump Markov
Chain Monte Carlo in Political Science
Arthur S PIRLING

We consider and explore structural breaks in a day-by-day
time series of civilian casualties for the current Iraq conflict:an undertaking of potential interest to scholars of international
relations, comparative politics, and American politics. We review Bayesian change-point techniques already used by political methodologists before advocating and briefly describing theuse of reversible-jump Markov chain Monte Carlo techniques
to solve the estimation problem at hand. We find evidence offour change points, all associated with increasing violence, approximately contemporaneous with some important state building events. We conclude with a discussion of avenues for future
research.
KEY WORDS: Change points; Political methodology; Sectarian violence;

1. INTRODUCTION

for political scientists lies in examining the time series for (potentially multiple) structural breaks and their effects. For scholars of American politics and public policy, the way that thesechange points correspond with administration statements on the
progress of the war may be particularly intriguing. This notion
extends to comparative institutions scholars interested in the potentially pacifying effect of various post-war “state-building”
activities. In keeping with the increasing acceptance and popularity of Bayesian methods in political science, in undertaking
our study we justify and adopt a novel (to political science) approach that uses a more general form of Markov chain Monte
Carlo (MCMC) techniques, well known to statisticians as “reversible jump” MCMC (Green 1995). We do so primarily forcomputational reasons.
Examining civilian casualty data from the official cessationof hostilities (May 2003) to May 2007, we find evidence offour change points. These breaks are approximately contemporaneous with (1) the capture of Saddam Hussein, and the emergence of the Abu Graib scandal (late 2003 to Spring 2004); (2)the installation of the Iraqi Interim Government, and the subsequent handover of power to the Iraqi Transitional Government(Summer 2004 to early 2005); (3) the legislative elections for,and negotiations to form, the first full-term Iraqi government(the early months of 2006); (4) the assumption of security andsome military responsibilities by the Iraqi government (August–
September 2006) . In every case, the frequency with which such
incidents occur is increasing after the break.

The study of inter- and intra-state conflict is a mainstay ofpolitical science. As an international conflict that increasingly
resembles a civil war, the current situation in Iraq provides botha testing ground for theories on the duration and termination ofdifferent types of conflicts (e.g., Filson and Werner 2004; Stamand Bennett 2006), as well as a rich source of data for empirical work. This is quite separate from its obvious importanceas a political, military, and economic event in progress. In partdue to its contemporaneous nature, political scientists have ac2. BACKGROUND AND DATA
cess to carefully, daily recorded, military and civilian casualty
information: an unusual and excitingly fine level of detail. Of
The United States and allied forces attacked Iraq with aerial
course, the utility of any data is only as good as the way it is ex- bombardments, followed by a land invasion, on March 20, 2003.
plored and analyzed. Here, we suggest that a fruitful approach By mid-April, Iraq’s capital city, Baghdad, and Saddam Hussein’s home region of Tikrit were under allied control—bringing
Arthur Spirling is a Graduate Student, Department of Political Science, Uni- a de facto end to the war. A fortnight later, on May 3, 2003,
versity of Rochester, Rochester, NY 14627 (E-mail: spln@mail.rochester.edu).
For comments and advice, I thank Jeff Gill, David Hastie, Michael Herron, President Bush declared that allied combat operations would
Tasos Kalandrakis, Andrew Martin, Michael Peress, Matthew Platt, Bing Pow- now officially cease. As with all conflicts, the war has not beenell, David Primo, Larry Rothenberg, Curtis Signorino, an anonymous referee, costless. What marks the Second Iraq War though, is the conand the associate editor at TAS. The Comparative Politics Working Group provided helpful feedback. Funding and technical support from the star lab is grate- tinued loss of life after the Iraqi army was formally defeated.
At the time of writing, some 3,000 coalition force members had
fully acknowledged. The usual caveat applies.
c 2007 American Statistical Association

DOI: 10.1198/000313007X247076

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Figure 1. Iraq casualty incidents, May 2003–May 2007. Solid line represents cumulative counts; solid dots are jittered incident occurrences;various dates of interest are demarcated.

died in addition to at least 57,000 civilian fatalities since military operations began. [Sources are http:// icasualties.org/ oif/and http:// www.iraqbodycount.org/ , respectively. Some studies have placed the number of civilian fatalities at a muchhigher number. For example, Burnham et al. (2006) claim up to600,000 deaths.] Violence has not yet abated despite the passingof some presumably important landmarks in what some characterize as the development of Iraq’s polity and stability. For
example, the capture of Saddam (December 2003), the placing of the former dictator on trial from “crimes against humanity” (July 2004), and his execution (December 2006); the killingof Saddam’s sons, Uday and Qusay (July 2003); National Assembly elections (January 2005); the drafting (December 2003–
March 2004) and subsequent referendum approval (October2005) of a constitution; the election of a new president (April2005) and the forming of a governing coalition (May 2006); theexecution of an Al-Qaeda ringleader, Abu Musab al-Zarqawi,
thought responsible for planning many terrorist attacks (June2006); and the assumption of security responsibility by the Iraqi
government (September 2006). We are interested in violence
for the post-(official) war period: although we certainly cannot
make firm causal claims, our study will enable us, for example,
to make statements about the plausibility of various events as“turning points” and allows us to pass some exploratory comments on how new democratic institutions and state apparatus
developments are affecting Iraqis. Hence our study focuses on
May 3, 2003, through to the present time (May 2007).
Our data are drawn from iraqbodycount.org an online
database that records civilian deaths in Iraq “that have resulted
2Interdisciplinary: Political Sciencefrom the 2003 military intervention by the USA and its allies.
The count includes civilian deaths caused by coalition military
action and by military or paramilitary responses to the coalition
presence (e.g., insurgent and terrorist attacks)” (Dardagan andSloboda 2006). The data in raw form record deaths at the day
level, from January 2003 through to the present and are compiled from (primarily Western) media reports and other sources.
Since uncertainty often exists on precise numbers, especiallywhen different agencies have conflicting figures for the same
incident, the database reports a range of possible death numbers
from a “minimum” to a “maximum.” Potential “over-counting”
is a concern, so we use the “minimum” and define a “casualty
incident” as involving five deaths or more (our findings below
are similar when we define the incident threshold at 10 or 20
deaths). For the purposes of this article, we focus on the (changing) frequency of attacks, rather than their size (above our minimum). In part this is a behavioral assumption: we would contend that, at least initially, terrorists were able to control how
often they planned to inflict casualties, rather than how many.
There were 1,682 such incidents in our time series, and wegraph their occurrence in Figure 1; there, the solid line is the
cumulative incident count, the solid dots are simply jittered incident occurrences (for which the y-axis is not the scale). We
also report various dates that may of interest and to give readersa sense of timing perspective.
Although univariate time series work is not regularly encountered in political science, it is valuable in the current context as a“first glance” exploration before covariate information becomes
available. We think that such work helps to prompt both theo-rizing and data gathering for more nuanced and sophisticatedanalysis.
3. ESTIMATION PROBLEM
The single change-point problem, estimated using Markov
chain Monte Carlo techniques, has been discussed for and bypolitical scientists elsewhere (see Western and Kleykamp 2004).
That treatment is similar to the (hierarchical) presentation given
by Carlin et al. (1992): suppose y = (y1 , . . . , yT ) is a vector of observations of the random variable Y (casualty incidents) over time and let f and g be unknown densities in thesame parametric family with yi ∼ f (Y |λ1 ), i = 1, . . . , k,yi ∼ g(Y |λ2 ), i = k + 1, . . . T . We wish to estimate k the (single) change-point which takes (discrete) values in {1, 2, . . . , T }.
A frequentist approach proceeds by maximizing
k
Y

T
Y

after them. That is, we have a set of Mk = {1, . . . , K } candidate models of our data-generating process, each with a different
number of parameters. Otherwise put, the number of parameters
is, of itself, a parameter. More formally, the kth model in Mk
has associated parameter vector θk which contains n k parameters such that θk ∈ Rn k .
Continuing to denote our data vector y, the joint distribution
becomes:p(k, θk , y)==p(y|k, θk ) p(k, θk )p(y|k, θk ) p(θk |k) p(k).

(2)

Since we have a constant of proportionality we can rearrange
and reexpress (2) into the more familiarp(k, θ |y) ∝ p(k) p(θk |k) p(y|k, θk ) .
{z} | {z }| {zk } |(3)

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posterior

prior(s)likelihood

Notice that p(y|k, θk ) is simply the likelihood, while p(θk |k)(1) is the prior for the parameter vector, given a particular datai=1
i=k+1
generating process and p(k) is the prior on the model itself. We
to obtain k and the parameters λ1 and λ2 (which for the count wish to generate samples from (3). Setting up a Markov chain tocase are arrival rates for a Poisson) if they are of interest. A do this may be difficult though, because it is required not simBayesian approach proceeds by placing a prior τ (k) on the ply to move around the parameter space for any particular θk ,change point. There are computational advantages of a Bayesian but to also “jump” from space to space (from model to model)
MCMC approach here since (a) maximizing (1) requires opti- depending on the k in question.
mization in a space that is not continuous (recall that k is disThis type of problem was given a general formulation bycrete) which, say, Gibbs sampling does not; (b) the resultant Green (1995), known as reversible jump MCMC (RJMCMC),
non-nested models may be straightforwardly compared using of which standard MCMC algorithms are special cases. Green
Bayes factors (Chib 1998); and (c) missingness in y is handled explicitly discussed a Poisson count change-point problem andsystematically. This is quite apart from the philosophical appeal we followed his approach for our application (though we varof Bayesian approaches of which political scientists are increas- ied the priors somewhat to ensure that our results were robust to
ingly aware (e.g., see Gill 2002, 1–6 and Jackman 2004, p. 486). such alternative specifications). Although well known to statistiHere, we are interested in exploring multiple change point cians, the details are somewhat technical, and readers are guided
and such work (Bayesian or otherwise) is much less common in to Brooks (1997) who gave an accessible overview for politicalpolitical science. In part this is because, with respect to the logic scientists.
above, there are profound computational difficulties in generatThe implementation of RJMCMC, in particular the efficiency
ing proposals for situations where we suspect there are more of proposals, can be problematic in practice and Hastie (2005)
than a couple of change points. One approach, suggested by devoted considerable attention to designing a technique to do
Chib (1998) and applied to American politics by Park (2006), this. We used his Automix sampler (with a maximum of ten
treats the change-point model as a type of time series Markov possible change points) for our estimation. Though the full demixture model, where the observations are (assumed) drawn tails are somewhat technical, drawing on Hastie (2005, 202–
from latent state variables. Notice that this approach requires 203), it is instructive to summarize the way that the model of theseparate Markov chain Monte Carlo runs for the different num- data-generating process is selected. The first two stages of thebers of change points hypothesized (Leonte et al. 2003). An al- sampler produce a Normal mixture distribution for every possiternative solution is to use reversible jump Markov chain Monte ble value of k. In the third stage, assuming the Markov chain is
Carlo which allows us to complete the computational operations currently in state (k, θk ), Automix allocates the parameter vecin one “go” as well as allowing us to be a priori agnostic over tor θk to a component lk of the mixture and uses it to standardize
the number of parameters to be estimated.
θk . Then a new model k 0 is proposed, along with a commensuTypically when MCMC is used in political science the pa- rate (new) mixture which has component lk0 . To obtain the newrameter vector θ has a known number of components, denoted state vector θk0 0 , the standardized vector is transformed using then. For the single change-point problem n = 3 (these are k, λ1 , mean and the covariance matrix of the mixture component lk0 0 .
and λ2 ). Now consider a very different scenario which arises for Automix then accepts the proposed state (k 0 , θk0 0 ) with somean unknown number of k change points: for every possible k, we specified acceptance probability. A particularly pleasing featureneed to estimate 2k + 1 parameters—the change points them- of this software is that issues such as burn in and the requisite
selves and then parameters of the densities before, between and number of post-burn iterations are handled automatically.
L(y) =f (yi |λ1 )g(yi |λ2 )Figure 2.

Posterior of k: number of change points, Iraq casualty data.

4. RESULTS
There are three sets of (posterior) distributions that interest
us here:1. The posterior of k: this enables us to answer the question“how many change points in the data?” This will have support
k = 1, . . . , kmax where kmax = 10.
2. The posterior of change-point positions conditional onsome estimated k. More intuitively, this enables us to answer
the question “given a particular number of change points, when
did they occur in the data?”
3. The posteriors of the rates for each period, conditional onsome estimated k: that is, given the number of change points,
and when they occurred, we can answer “what were the effectsof the change points?”
In Figure 2 we display the posterior for k, the number ofchange points. The strongest evidence (in the sense of Kassand Raftery 1995) is for k = 4 and we will explore this possibility exclusively. In Table 1 we summarize the results fork = 4 model in a way that answers questions 2 and 3 above.
The first break, in late January 2004 occurs between incidents
that may be of import. The first was the capture and arrest of
Saddam at a farmhouse near Tikrit in December 2003. The subsequent months saw both an insurgency uprising lead by rebel
Shia cleric Muqtada al-Sadr in Baghdad and the diffusion ofabusive photographs taken at the Abu Graib prison where coalition forces were holding Iraqi detainees. The political falloutof the latter was profound, and criticism of the Bush administration by allied, Arab, and other politicians was widespread.
This event, arguably, rallied and spurred sectarian hatreds and4Interdisciplinary: Political Science

violence. The break marks a sharp increase in the casualty rate,
doubling from one incident every four days, to one every twodays.
The second break occurs in August of 2004, a little while
after the Iraqi Interim Government assumed power from the
Coalition Provisional Authority (in June 2004). This new entity, under the Premiership of Iyad Allawi, was subsequently
recognized as the legitimate sovereign government of Iraq byboth the United Nations and the Arab League (an importantregional player). Allawi quickly announced new security measures to tackle insurgency forces and was criticized by some for
their draconian nature. As part of this offensive, the Iraqi Interim Government began to censor the critical reports of media
outlet al Jazeera. Included in the highest posterior density interval for this break is the January 2005 democratic elections for
the Iraqi Transitional Government. This change point saw anincrease in violence from one incident every two days, to four
incidents every five days.
The third break itself, in February 2006, occurs not long after
the elections for the first full term Iraqi government (December2005) and at around the time of the protracted negotiations toform a new coalition government. These talks were deadlocked
for some time, lasting from December through to April of thefollowing year. Jawad al-Maliki, leader of the Islamic Dawa
Party would become Prime Minister after the original candidate
Ibrahim al-Jaafari proved unacceptable to the Sunni and Kurdish representatives in parliament. Once again, violence surged
after this point with, on average, five incidents occurring every
three days.
The fourth and final break occurs in September of 2006, atime when the Iraqi government assumed control of national
security for approximately 70% of the country. The first specifictask of the Iraqi Security Forces was, and is, to tame insurgency(with coalition logistical and medical support). By now incident
rates were approaching three per day.
In Figure 3 we summarize our findings in a different way:
the open circles represent the median incident rate between therelevant breaks which are demarcated by the broken lines. For
reference, we again draw the jittered incidents themselves on
the plot.
5. DISCUSSION
Our study—to our knowledge the first that uses RJMCMC
in a political science context—suggests that violence is increasing and that important state-building activities, like democraticelections, are contemporaneous with upticks in casualties. Apartfrom this rather grim substantive conclusion, we found that investigating time series on violence to be an interesting and fruitful exercise. If a Bayesian approach is pursued, then reversible
jump techniques seem most helpful. We hope that our brief article will encourage others in political science to consider suchmethods in future.
As noted earlier, we do not establish causation in any sense:
the events we noted were simply occurring at around the sametime as the breaks in the time series and it is speculative thatthey may be of direct importance. This suggests some interesting avenues for future research: for example, one possibility isTable 1. Change-point median dates [90% HPD] in Iraq casualty data, with rates between breaks. Comments note political and other activities
approximately contemporaneous to break.
PeriodTime

before break 1
break 1

0.283
Jan 26, 2004
[Jul 18, 2003–Apr 30, 2004]

btwn break 1 and 2
break 2Aug 7, 2004
[Jul 10, 2004–May 4, 2005]

break 4after break 4

Figure 3.

1 incident every 4 days

1 incident every 2 days
Iraqi Interim Government assumes power (Jun 2004)
elections for Iraqi Transitional Govt (Jan 2005)

0.814
Feb 22, 2006
[Feb 16, 2006–Mar 24, 2006]

btwn break 3 and 4

Comments

Capture of Saddam (Dec 2003)
Abu Graib scandal (Apr 2004)
0.476

btwn break 2 and 3
break 3Rate (λ)

4 incidents every 5 days
elections for first full term Iraqi government (Dec 2005)

1.660
Sept 4, 2006
[Aug 23, 2006–Sept 20, 2006]

5 incidents every 3 days
Iraqi govt assumes control of counter-insurgency
operations for much of country (Sept 2006)

2.850

8 incidents every 3 days

Change in rate of casualty incidents over time: rate on y-axis, actual incidents as jittered points. Dates correspond to breaks found.

The American Statistician, November 2007, Vol. 61, No. 4

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that increasing violence is a product of an increasingly organized insurgency. On this point, notice that the solid line in Figure 1 resembles a exponential curve of form F(t) = aer t + (we are indebted to an anonymous TAS referee for this observation). Modeling—both theoretical and statistical—of this apparent pattern would allow us to think about the development of
the conflict in Iraq in a more systematic way. We leave this for
future research.

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[Received February 2007. Revised June 2007.]

Jackman, S. (2004), “Bayesian Analysis for Political Research,” Annual Review
of Political Science, 7, 483–505.

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