Bayesian hidden Markov models for latent variable labeling assignments in conflict research: application to the role ceasefires play in conflict dynamics

Submitted to the Annals of Applied Statistics B A YESIAN HIDDEN MARK O V MODELS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH: APPLICA TION TO THE R OLE CEASEFIRES PLA Y IN CONFLICT D YN AMICS B Y J O NAT H A N P W I L L I A M S 1 , 5 , G U D M U N D H H E R M A N S E N 2 , 3 , 5 , H Å V A R D S T R A N D 2 , 3 , 5 , G OV I N D A C L A Y T O N 4 , A N D H Å V A R D M O K L E I V N Y G Å R D 3 1 North Car olina State University , jwilli27@ncsu.edu 2 University of Oslo 3 P eace Resear ch Institute Oslo (PRIO) 4 ETH Zurich & Centr e for Humanitarian Dialogue 5 Centr e for Advanced Study , Norwegian Academy of Science and Letters A crucial challenge for solving problems in conﬂict research is in lever - aging the semi-supervised nature of the data that arise. Observed response data such as counts of battle deaths over time indicate latent processes of in- terest such as intensity and duration of conﬂicts, but deﬁning and labeling in- stances of these unobserved processes requires nuance and imprecision. The av ailability of such labels, ho wever , would make it possible to study the ef fect of interv ention-related predictors — such as ceaseﬁres — directly on conﬂict dynamics (e.g., latent intensity) rather than through an intermediate proxy like observ ed counts of battle deaths. Moti vated by this problem and the ne w av ailability of the ETH-PRIO Civil Conﬂict Ceaseﬁres data set, we propose a Bayesian autoregressiv e (AR) hidden Markov model (HMM) frame work as a sufﬁciently ﬂe xible machine learning approach for semi-supervised regime labeling with uncertainty quantiﬁcation. W e motiv ate our approach by illus- trating the way it can be used to study the role that ceaseﬁres play in shaping conﬂict dynamics. This ceaseﬁres data set is the ﬁrst systematic and globally comprehensiv e data on ceaseﬁres, and our work is the ﬁrst to analyze this ne w data and to explore the effect of ceaseﬁres on conﬂict dynamics in a compre- hensiv e and cross-country manner . 1. Introduction. W ithin the conﬂict research community HMMs have been studied from a v ariety of different perspectiv es, and with v arying degrees of sophistication. Early applications of HMMs in the literature are inv estigated as case studies for various countries. They were largely motiv ated by a perceived need for the conﬂict research community to explore its data beyond what can be provided by linear models, arguing that the dynamics exhibited by these data are complex, non-linear political systems ( Petrof f, Bond and Bond , 2013 ; Schrodt , 1997a , b , 2006 ). A fe w years after the preliminary in vestigations of HMMs in the conﬂict research literature, the book chapter Petrof f, Bond and Bond ( 2013 ) summarized the best practices with particular emphasis on forecasts/predictions of violence. Overall, the ideas about, and implementations of, HMM strategies for explaining/predicting conﬂict data are outdated and hav e not advanced beyond the ideas and strategies prescribed in the classical tutorial paper , Rabiner ( 1989 ). See Anders ( 2020 ), Besle y , Fetzer and Mueller ( 2021 ), and Randahl and V e gelius ( 2022 ) for more recent accounts. T ypically , there has been a proposed set of theoretically motiv ated (unobserved) conﬂict states (in the range of 3- 6) linked by way of an HMM for pre-processing and organizing sequences of event-coded K e ywor ds and phrases: state space model, multistate model, discrete-time Markov process, discrete-valued time series, count-valued time series. 1 2 symbols from a large repository of international news summaries (provided by an agency such as Reuters). Locally maximum likelihood estimates are obtained from a Baum-W elch algorithm, the most plausible (unobserved) states-sequences are inferred via the V iterbi algo- rithm, and the ﬁtted HMMs hav e been regarded as predominantly uninterpretable but useful for forecasts/predictions of future data. Furthermore, that it is not clear whether data has been properly discretized in the exist- ing studies. For example, Petroff, Bond and Bond ( 2013 ) describes the ability to tailor the length/number of time intervals to the precision of the data av ailable, and discourages ag- gregation of data (i.e., hours, days, weeks, etc.). While it is true that a discrete-time Markov process can be deﬁned on any time grid, the a priori chosen grid must apply to all observed and future data sequences. The use of time stamps of observed data sequences, as they were actually recorded in time, requires adherence to a continuous-time Marko v process. Such a process can be modeled with a continuous-time HMM, and has been studied extensi vely in the disease progression literature; e.g., Satten and Longini Jr ( 1996 ) and W illiams et al. ( 2020 ). Our contributions are the following. W e propose a discrete-time Bayesian HMM to make inferences on how violence dynamics ev olve in time over a latent, discrete, conﬂict-inferred state space, moti v ated by the new ETH-PRIO Civil Conﬂict Ceaseﬁres data set ( Clayton et al. , 2021 ), combined with the violence data from the Uppsala Conﬂict Data Program’ s (UCDP) geo-referenced event data set ( Sundberg and Melander , 2013 ). W e use weekly battle death counts as the emitted response variable, combined with conﬂict-domain-theory motiv ated, country-speciﬁc cov ariates. In particular , we demonstrate how the semi-supervised deﬁning and labeling of conﬂict-inferred states is methodologically essential to dev eloping fundamen- tal insights into some of the most challenging contemporary questions in conﬂict research, such as the effect of ceaseﬁres on conﬂict dynamics. The utility of HMM frame works for deﬁning and assigning labels for latent variables has also been exhibited in the recent article Anders ( 2020 ) to identify territorial control during a civil war , that is intrinsically dif ﬁcult (if not impossible or infeasible) to manually label. Arguably , due to a deﬁciency in meaningful labels, HMM-based semi-supervised data labeling strategies could pa ve the way for the ne xt decade of conﬂict research progress. Additionally , we offer a variety of inferential analyses and conclusions that can be drawn from ﬁtting conﬂict data within this framework, as well as graphical tools and algorithms that could be used from a policy making perspectiv e for predicting or characterizing intensity of violence. W ith suf ﬁcient data and intervention-rele vant predictors it is possible to conduct analysis using the state space sampler to quantify the effect of changes in policy (e.g., how the risk or duration of a conﬂict w ould be predicted to change if interventions are implemented). The weekly battle death count data we consider are modeled, conditional on the under- lying latent state, using a negati ve-binomial distribution with an AR mean structure. This is a natural choice because conﬂict-related death count data are time series that are commonly characterized by both over -dispersion and zero-inﬂation; both are common features of many of the battle death series (there are several illustrations belo w). F or a general introduction and ov ervie w of count time series, see for example the recent re view paper Davis et al. ( 2021 ). W e implement a Markov chain Monte Carlo (MCMC) algorithm to ﬁt the HMM, and the re- peated sampling cov erage of all HMM parameter estimates is ev aluated via the construction of posterior credible sets. Ceaseﬁres are arrangements through which conﬂict parties commit to stop ﬁghting, and all ceaseﬁres share the same immediate objecti ve: to stop violence ( Clayton, Nathan and W iehler , 2021 ). They are a common part of intra-state conﬂict, each year occurring in about HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 3 a third of all conﬂicts. 1 Between 1989 and 2020 there were at least 2202 ceaseﬁres across 66 countries, in 109 ci vil conﬂicts ( Clayton et al. , 2021 ). Surprisingly , despite their frequency , it remains unclear to what extent ceaseﬁres really work, i.e., it is not known to what extent they shift a conﬂict from a more violent to a less violent state. T o illustrate this point, in early 2003 a ceaseﬁre between the gov ernment of Sudan and the SPLA/M mark ed a transition from a long period of sustained violence into a relati vely non-violent state that remained in place until the parties reached a comprehensiv e peace agreement in 2005. Y et in Syria, for example, where there have been more than 130 ceaseﬁres in the ongoing ci vil conﬂict, many ceaseﬁres seem to have produced an escalation rather than deescalation in violence, or had no effect at all ( Karakus and Svensson , 2020 ). From existing research it is not possible to determine if the Syrian or Sudanese case are indicative of the general effect of ceaseﬁres on conﬂict violence. 2 The lacuna in understanding the role ceaseﬁres play in conﬂict is a result of conﬂict re- searchers lacking not only the necessary data but also the statistical tools to properly model violence dynamics, and to study and understand the covariates that inﬂuence these dynam- ics. T o date, ceaseﬁre research is largely limited to case studies ( Palik , 2021 ; Pinaud , 2020 ; Åkebo , 2016 ), or analysis tailored for the policy and practice community (e.g. Brickhill , 2018 ; Buchanan, Clayton and Ramsbotham , 2021 ). The research in this area details a number of cases in which ceaseﬁres ha ve ultimately prov ed successful (e.g., De Soto , 1999 ), b ut also sho ws that in many cases violence does not end with the onset of a ceaseﬁre ( Kolås , 2011 ; Jarman , 2004 ; Höglund , 2005 ; Akebo , 2016 ), and some ceaseﬁres ev en mak e violence worse ( Luttwak , 1999 ; K olås , 2011 ; Mahieu , 2007 ). In-depth qualitative analysis has many advan- tages (see, George and Bennet , 2005 ), but is ill-suited to systematically identifying broad trends, such as whether ceaseﬁres in general produce a signiﬁcant shift in violence dynamics. This instead requires comparativ e quantitativ e analysis which has, to date, been limited for questions surrounding ceaseﬁre onset ( Clayton et al. , 2019 ) and design ( Clayton and Sticher , 2021 ). There is some evidence that ceaseﬁres stop violence ( Fortna , 2003 , 2004 ), but this is limited to inter-state conﬂict and the analysis suf fers a number of serious methodological lim- itations. Accordingly , perhaps the most fundamental question on ceaseﬁres remains largely unanswered: Do ceaseﬁres stop violence? Furthermore, stopping violence can serve various purposes. Firstly , ceaseﬁres can help to support conﬂict management efforts: creating breaks in the ﬁghting to facilitate humanitar- ian assistance ( Aary , 1995 ); helping to contain conﬂict when resolution is not yet possible ( Clayton et al. , 2020a ); and terminating violence in such a way that does not require the resolution of the incompatibility ( Hanson , 2020 ; Kreutz , 2010 ). Second, ceaseﬁres can also help with conﬂict resolution efforts: helping to build trust ( Åkebo , 2016 ); signalling control and cohesion ( Höglund , 2011 ); and creating an environment more conduci ve to negotiations ( Smith , 1995 ; Mahieu , 2007 ; Chounet-Cambas , 2011 ; Clayton and Sticher , 2021 ). Third, not all ceaseﬁres are concei ved for peaceful purposes. Ceaseﬁres can be used to gain some strate- gic adv antage, including b uying time to rearm and regroup ( Clayton et al. , 2020b ; Smith , 1 For simplicity we refer to ‘armed conﬂict’ as conﬂict or armed conﬂict. W e follow the UCDP/PRIO Armed Conﬂict Database and deﬁne an armed conﬂict as: ‘a contested incompatibility that concerns government and/or territory where the use of armed force between two parties, of which at least one is the go vernment of a state, results in at least 25 battle-related deaths in one calendar year’ ( Gleditsch et al. , 2002 ). In this article we only focus on internal armed conﬂicts. 2 In contrast, a bur geoning body of literature explores the dri vers of ceaseﬁre onset, and the impact that cease- ﬁres have on other outcomes such as peace processes, crime, and state-building; see, Åkebo ( 2016 ); W aterman ( 2020 ); Bara, Clayton and Rustad ( 2021 ); Clayton et al. ( 2020a ). 4 1995 ), to support state building efforts ( W oods , 2011 ; Sosnowski , 2020 ), or undertaking il- licit acti vity ( K olås , 2011 ; Dukalskis , 2015 ). 3 Ne vertheless, in almost all cases, it is logical to assume that in order to achiev e their purpose, ceaseﬁres must ﬁrst achieve the immediate objecti ve, i.e., shift conﬂict from a violent to a non-violent state. Building on existing conﬂict research literature, we discuss how our analyses and infer- ences are motiv ated from and translate to the existing theory for how ceaseﬁres shape conﬂict violence. Most notably , a major ﬁnding of ours is surprising e vidence for an escalation in the state of violence in the pre-ceaseﬁre period (i.e., the two weeks prior to a ceaseﬁre) of a conﬂict. The remainder of the paper is organized as follows. In the next section, we describe the data set that moti vates our work, along with a discussion of why we focus on a weekly resolution of battle deaths. After commenting on limitations and challenges of using these data for modeling conﬂict dynamics, Section 3 moti vates the speciﬁcation of our count- v alued time series model for weekly battle deaths. The technical details of the discrete- time HMM and its implementation are provided in Section 3.2 . W e illustrate why simpler statistical models are inadequate for our methods with various empirical studies in Sec- tion 4 , along with a simulation study on synthetic data for the model we propose. Re- sults, analyses, and robustness considerations are pro vided in Section 5 . The paper con- cludes in Section 6 by moti v ating a v ariety of open problems in conﬂict research to at- tract the attention of other statisticians. Documented R code for the use of our methods by other researchers and policy analysts on their data, along with the workﬂow for reproduc- ing our results are av ailable in the Supplementary Material ( Williams et al. , 2024 ) and at https://jonathanpw.github.io/research.html . 2. Data. Our goal is to build a model that is able to capture and re-create the intensity of conﬂict, with its spik es and lulls, as well as more enduring patterns of violence. The violence data we utilize come from the UCDP geo-referenced ev ent data set ( Sundber g and Melander , 2013 ), which reports all events with at least one battle-related casualty . Each ev ent record sho ws where and when an e vent took place, which actors where in volved, and ho w many battle-related deaths ensued. The response variable that we consider records the number of people killed due to intrastate conﬂict and/or internationalized intrastate conﬂicts per week in each country . This also means that the data for countries with sev eral parallel on-going conﬂicts are collapsed into one country time series. For ceaseﬁres, we rely on the ETH-PRIO Ci vil Conﬂict Ceaseﬁres data set ( Clayton et al. , 2021 ), which represents the ﬁrst systematic and globally comprehensi ve data on ceaseﬁres. Our work is the ﬁrst to use this ne w data to explore the effect of ceaseﬁres on conﬂict dy- namics in a comprehensiv e and cross-country manner . The ceaseﬁres data deﬁnes a ceaseﬁre as ‘an arrangement that includes a statement by at least one conﬂict party to stop violence temporarily or permanently from a speciﬁc point in time’. This broad conceptualization of a ceaseﬁre captures the full range of security arrangements through which belligerents might agree to temporarily suspend and/or terminate hostilities. W e include unilateral and bi/multi- lateral ceaseﬁres. A unilateral ceaseﬁre occurs if one group alone declares the cessation of hostilities. For example, in December 2018 the T atmadaw (army) in Myanmar declared a unilateral ceaseﬁre towards a number of armed ethnic or ganizations that was not recipro- cated. A bi/multi-lateral ceaseﬁre occurs when two or more parties jointly declare a ceaseﬁre 3 A ceaseﬁre might prove to be successful according to one purpose (e.g., humanitarian aid), but unsuccessful in another (e.g., promoting peace talks), or successful in the eyes of one conﬂict party , while representing an abject f ailure in the eyes of another . Ceaseﬁres might also achie ve their purpose, but produce other unintended effects (e.g., promoting the splintering of a non-state group ( Plank , 2017 )). HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 5 to wards one another . For example, in Nov ember 2018, Israel and Hamas jointly agreed to simultaneously halt hostilities. Focusing on the impact of ceaseﬁres on violence dynamics, rather than peace agreements, we consider only non-deﬁnitiv e ceaseﬁres (i.e., ceaseﬁres that attempt to suspend rather than permanently terminate a conﬂict). Moreover , since we are focused on country lev el dynamics, we also exclude ceaseﬁres that only cover a part of the conﬂict area (i.e., so-called local ceaseﬁres), as these agreements seek to reshape violence in a limited area, and so it does not make sense in our context to assess their impact on the conﬂict as a whole. Finally , we exclude ceaseﬁres that extend or renew prior agreements, based on the assumption that any shift in violence dynamics is likely to ha ve occurred in response to the original agreement. The ceaseﬁres data includes the date on which the arrangement enters into effect. W e deﬁne the week containing the start date of the ceaseﬁre, together with the follo wing four weeks (i.e., ﬁ ve weeks in total) as a ceaseﬁr e period . W e do this to focus the analysis and the attention of the model on the dynamics that we are most conﬁdent relate to the ceaseﬁre. This helps to mitigate the record-keeping uncertainty surrounding the effecti ve duration of a ceaseﬁre. Further , we deﬁne the two weeks prior to the start of a ceaseﬁre (i.e. the two weeks before the week that contains the start date) as the pr e-ceaseﬁr e period . Ceaseﬁres tend to be negotiated fairly quickly , and once agreed to often take a few days to implement. Thus, we believe two-weeks represents a suf ﬁciently long period so as to capture the direct period in which the ceaseﬁre is under consideration, but short enough to av oid picking up other conﬂict-related factors. 4 There are se veral countries in the data set that do not contain any battle deaths or ceaseﬁres. Out of the 170 countries, there are 74 without any battle deaths, and 124 countries with less than 1000 battle deaths throughout the entire time period 1989–2018. These countries are important for estimating the baseline state of ‘non-violent’, also referred to as ‘state 1’. W e partially label the data using the following deﬁnition: a week is labelled as state 1, without error , if it is at least 60 days (in both directions in time) remov ed from an observation of at least one battle death, and if it is also part of a consecutive period of at least 2 years without any battle-related deaths. This is the only state labeling assignment used for model estimation. 2.1. Contr ol covariates. There are a collection of standard cov ariate types that are known in the conﬂict research community to affect the likelihood of conﬂict, such as political regime, economic de velopment, and population. For these cov ariate types we use the follo wing. Po- litical regime is measured with the polyarchy index from the V -Dem project ( Coppedge et al. , 2019 ), used to control for alternative conﬂict management opportunities within Dahl’ s Dahl ( 1971 ) conceptualization of democracy . It is understood that countries some where in the middle of the polyarchy spectrum are most vulnerable to violent conﬂict, and so we include polyarchy in nominal value, as well its squared and cubed v alues, as cov ariates in our analy- sis. The V -Dem indices consist of a mix of variables measured either at the end of a year, at the maximum value throughout a year , or as the average o ver a year . That being true, we use lagged polyarchy v alues to pre vent mixing cause and ef fect. Economic de velopment is measured as gross domestic product (GDP) per capita. GDP re- lates directly to the feasibility of armed conﬂict because potential rebels in wealthy countries hav e more to lose from a rebellion, and wealthy gov ernments ha ve a larger capacity to co-opt 4 Since we are aggreg ating the data at a country level, it means that it is possible for a country to simultaneously be in a pre-ceaseﬁre and a ceaseﬁre period. This is relatively rare (only 244 out of a total 3872 weeks ha ve multiple ev ents) thus we leav e modeling this challenge to future work (see Section 6 for additional discussion). 6 a population through public goods and coerce a potential rebel through a strong security ap- paratus. Poor countries are more lik ely to ha ve poor citizens, often more willing to engage in risky w arfare, less ability to co-opt, and weaker militaries. Population affects both the risk of conﬂict in the ﬁrst place and the likelihood of a cease- ﬁre in a speciﬁc conﬂict. Larger countries are more likely to have conﬂicts simply because of the larger number of people able to start a rebellion. Consequently , larger countries are more likely to ha ve se veral parallel conﬂicts, which creates a complex situation where a rebel group might seek a ceaseﬁre with the government to actively harm competing rebel groups. The economic dev elopment and population variables are obtained from the W orld Dev elop- ment Indicators ( http://data.worldbank.org/indicator ), and are included in log-lag-v alues in our analyses. Lastly , we include as cov ariates, indicators for ceaseﬁre and pre-ceaseﬁre periods (as deﬁned in Section 2 ) for each country/week. 2.2. Limitations of the data and weekly r esolution. The UCDP geo-referenced event data set has a number of kno wn limitations (see also Raleigh, Kishi and Linke , 2023 ). The main problem is missing data, as some e vents are undetected and some not found newsw orthy or politically useful ( Dawkins , 2021 ). Signiﬁcant biases arise in UCDP data sets because they are based on secondary sources. W eidmann ( 2015 ) is the best reference for this; the bias is substantial and correlated with cell phone coverage. Sources are sometimes uncertain, conﬂicting, or partially overlapping, and measurement error can occur on a number of di- mensions. An “ev ent size bias” as in vestigated in Price and Ball ( 2014 ) is also an important artifact that arises when the probability of an e vent being reported is associated with the size of the e vent. Further , reports may be conﬂicting about the se verity of a giv en ev ent; reports may be partially overlapping, raising questions about whether there were actually two different ev ents or imprecise reports of a single event. Methods and case studies for determining the unique reported e vents (i.e., “unique entity estimation”) and linking duplicate reported e vents (i.e., “duplicate detection” or “entity resolution”) hav e been published in the statistics literature (e.g., Sadinle , 2014 , 2018 ; Chen, Shri vasta va and Steorts , 2018 ). All three papers, Sadinle ( 2014 ), Sadinle ( 2018 ), and Chen, Shri vastav a and Steorts ( 2018 ), are indeed very interesting and address a major problem in the collection of conﬂict data, namely the potential ov erlap between reports from different sources. See Brunborg, L yngstad and Urdal ( 2003 ) for a more in depth discussion of this problem in the context of Srebrenica and the requirements for legal documentation used in an international court of law . Y et the paper only addresses one dimension as it focuses on individual, civilian casualties collected by individual traits. The v ast majority of casualties in conﬂict remain anon ymous and are referred to in relation to the location and organizations in volved, which introduces two dimensions not discussed by any of the papers Brunborg, L yngstad and Urdal ( 2003 ), Sadinle ( 2014 ), Sadinle ( 2018 ), or Chen, Shri vasta va and Steorts ( 2018 ). Despite these limitations, UCDP has decades of experience in consistently coding such ov erlaps and has fairly good routines to handle data reporting challenges. As a result, their data set contains, in addition to the best estimate, a high and lo w estimate. In about 80% of the cases these are the same or very close, but sometimes there is a substantial variation. While this does not address missing data, it does offer an opportunity to assess the rob ustness of empirical ﬁndings. Moreover , it is reasonable to believ e that larger ev ents are more likely to be precisely reported than smaller events ( Price and Ball , 2014 ), and that more violent periods in general are more likely to be reported. These presumptions sufﬁce for reasoning that the UCDP data reﬂect, to some extent, true tr ends in conﬂict intensity , and our studies support our assumption that ov erall trends in conﬂict dynamics are consistently measured by UCDP . The same logic is used in ( T ai, Mehra and Blumenstock , 2022 ) to support similar use HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 7 of ev ent data. Additional issues arise from conﬂicting reports about or ganizational afﬁliation, which is a major problem in many situations but not ours. Since we aggre gate fatalities at the country le vel, the or ganizational aspect is not rele v ant ( Lacina and Gleditsch , 2012 ). Aggregation to a weekly resolution of data is commonly used for the study of conﬂict dy- namics ( W ood , 2014 ; Krtsch , 2021 ; Holtermann , 2021 ), as well as for studying other aspects of contentious behaviors such as electoral violence ( Reeder and Seeberg , 2018 ) and denial of service attacks ( Lutscher et al. , 2020 ). Studying conﬂict data on a weekly resolution bal- ances sev eral concerns. The UCDP geo-referenced ev ent data set is often coded on a daily precision le vel, but a signiﬁcant number of observations hav e some lev el of imprecision. A country-day data structure would saturate the data set with large number of artiﬁcial-zero observ ations, which would add more noise than information. On the other hand, a monthly aggregation would obscure much of the dynamics we aim to capture. In our aim to trace an ef fect of ceaseﬁres on conﬂict dynamics, the weekly resolution provides enough precision and enough time between observations. The ceaseﬁre data set is also reliably av ailable at a weekly le vel of resolution. There sometimes exist conﬂicting reports on the timing of a gi ven e vent, b ut again, UCDP will inform about the uncertainty of temporal data. Since a week is some what arbitrary one could imagine that our analysis would look slightly different if we deﬁned a week as Thursday–W ednesday rather than Monday–Sunday , but based on a rudimentary analysis, we do not believe this particular problem to be very signiﬁcant. In the UCDP geo-referenced e vent data, the e vent start and end dates cross over into the next week in less than 10% of the cases. Finally , there is a limitation from only relying on incidence of battle deaths as a response v ariable associated with underlying conﬂict dynamics. Conﬂict dynamics surely e xhibit mul- tifaceted manifestations, including but not limited to deaths, e.g., injury and disability . In fact, there has been much debate on the topic of conﬂict related injuries; it is ar gued in Fazal ( 2014 ) that a decline in war casualties has been due to medical improvements rather than a decline in the number of wars, but adequate data does not e xist to fully in vestigate the issue. A paper published in 2003 — Ghobarah, Huth and Russett ( 2003 ) — did report long- term ef fects from civil wars on a measurement called disability adjusted life years (DAL Y) , but D AL Y is a theoretically constructed index that is not regularly av ailable (i.e., weekly , monthly , etc.). W e do not consider counts of injuries and disabilities due to conﬂict simply because of a lack of adequate data on such incidences. W e admit that there are signiﬁcant problems with the UCDP data, and UCDP uncertainty is sometimes, perhaps ev en often, present on se veral dimensions at once. They ha ve sometimes resolved this by clustering a number of unclear ev ents together to a super-e vent that may last for quite some time, raising obvious issues for our weekly model. W e believ e, howe ver , that we use the best data av ailable, and that the conceptual validity of focusing on fatalities as an indicator of conﬂict intensity is defensible as the best option av ailable. Moreov er , we trust that the experts at UCDP are handling these issues in a manner which we cannot impro ve on. 2.3. Challenges of modeling conﬂict dynamics. Figure 1 illustrates the statistical chal- lenge at hand. It sho ws weekly aggre gated number of battle deaths in internal armed conﬂicts ov er the 1989 to 2018 period for the Democratic Republic of Congo “Congo” and Colombia. Once again, our goal is to build a model that is able to capture and re-create the intensity of conﬂict, with its spikes and lulls, as well as more enduring patterns of violence. The typical statistical models employed by conﬂict researchers, overwhelmingly logistic or ordinary least squares regression, count models such as Poisson and negativ e-binomial, and time-to-ev ent models such as Cox proportional hazard models, are not particularly well suited for such purposes (a recent re vie w is Dav enport et al. , 2019 ). In contrast to the literature on conﬂict 8 dynamics, the literature on the onset of armed conﬂict has beneﬁted tremendously from hav- ing a, more or less, standard statistical model (a logistic cross-section time series model) and a standard set of cov ariates (in particular related to socio-economic de velopment, political institutions, and demography). This standard model has allowed the literature to accumulate kno wledge as more and more pieces of the puzzle have been added. Unfortunately , this has not been the case for the conﬂict dynamics literature, which has de veloped in a much less coordinated fashion. Congo 0 100 200 300 1992−01−01 1997−01−01 2002−01−01 2007−01−01 2012−01−01 2017−01−01 Number of Battle Deaths (Weekly) Colombia 0 100 200 300 1992−01−01 1997−01−01 2002−01−01 2007−01−01 2012−01−01 2017−01−01 Number of Battle Deaths (Weekly) F I G 1 . W eekly number of battle deaths in the Democratic Republic of Congo “Congo” and Colombia, 1989– 2018. Note that the Democratic Republic of Congo suffer ed 3,000 deaths the week of December 14, 1998, a value be yond the plot range c hosen for illustration. 3. Statistical methodology . HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 9 3.1. Response variable model. Fitting a Poisson model is perhaps the simplest approach for inference on count-valued data, such as the number of battle deaths in a week, for a gi ven country . A suggested by Figure 1 , howe ver , weekly battle deaths in a given country exhibit obvious patterns of zero-inﬂation and over -dispersion, and so a more ﬂexible (i.e., two parameter) negati ve-binomial model is a standard, pragmatic choice. Furthermore, beyond issues relating to zero-inﬂation and over -dispersion, temporal correlations in the week-to- week fatality counts cannot be ignored. For instance, in Jakobsen ( 2021 ), it is observed that se veral countries have battle death time series with fairly strong autocorrelation, motiv ating the application of a count time series model. An introduction to count time series models can be found in W eiß ( 2018 ), and the recent re view paper Da vis et al. ( 2021 ) is a more general revie w . In short, count time series models are a rich class of models, well-suited for data that are zero-inﬂated, over -dispersed, and with heteroscedasticity . A v ariety of such models are e xplored in Jakobsen ( 2021 ) in the context of battle deaths time series. From this, there is not a clear “best” model for such time series; different types of models work well, provided they have sufﬁcient capacity to model autocorrelation, zero-inﬂation, and ov er-dispersion. Indeed, a Poisson-based model is found to be lacking a sufﬁcient degree of dispersion, while a negati ve-binomial is recognized as more appropriate. Moreover , it reasoned that an AR type of (mean) structure is sufﬁcient for capturing the rele vant autocorrelation across time. Accordingly , to model the battle death counts Y i,k for week k of country i , we specify a simple negati ve-binomial model with an AR mean structure: (1) Y i,k | Y i, ( k − 4):( k − 1) ∼ neg ativ e-binomial ( r i,k , p ) , with p = c ( c + 1) − 1 and r i,k := a + ρ · Y i, ( k − 4):( k − 1) , where Y i, ( k − 4):( k − 1) := 1 4 P 4 l =1 Y i,k − l and c, a, ρ > 0 . This is actually a special case of a so-called “NB-DIN ARCH(4)” model, i.e., negati ve-binomial dispersion inte ger AR conditional heteroscedasticity model, introduced in Xu et al. ( 2012 ). In experimenting, we determined that specifying the rate r i,k as an AR function of the av erage battle death counts — over the pre vious four weeks — balances the noise in the weekly data v alues with their monthly composite. Further , this AR structure allo ws for meaningful interpretability of parameters with respect to the underlying conﬂict intensity; for example, E { Y i,k | Y i, ( k − 4):( k − 1) } = a c + ρ c · Y i, ( k − 4):( k − 1) , meaning that we can view a/c as a type of baseline intensity and ρ/c as an escalation pa- rameter . The time series is weakly stationary if 0 ≤ ρ/c < 1 , and explosi ve otherwise ( Xu et al. , 2012 , see Theorem 4.1), which gives a natural characterization of the non-escalatory or escalatory dynamics of conﬂict violence. Assuming no battle deaths occurred in a giv en four consecutive weeks, small values of the c parameter and large v alues of the a parameter are associated with an increased likelihood of violence on the ﬁfth week: Pr { Y i,k > 0 | Y i, ( k − 4):( k − 1) = 0 } = 1 −  c c + 1  a . Model ( 1 ), howe ver , is not fully adequate for characterizing a time series of battle deaths ov er the entire time period of the data (1989-2018) because it assumes that the v alues for c, a, ρ are ﬁxed ov er time and across countries. This assumption is clearly violated for coun- tries like Congo, where, as observed in Figure 1 , the country transitions in and out of re gimes of violence and peace. In fact, sev eral countries in the data set do not exhibit any conﬂict violence for all of 1989-2018. What is necessary is for the response variable model ( 1 ) to be able to adapt to latent regime changes between violence and peace, as well as to incorpo- rate country-speciﬁc covariates. These necessary extensions motiv ate our construction of an HMM, introduced next. 10 3.2. HMM likelihood function. In this section we dev elop our discrete-time Bayesian HMM. As described previously , we regard the data resolution on a weekly grid, but the details would be the same for any discretization ov er time. Note that if the data cannot be meaningfully organized on a grid of time points (no matter how precise), then a continuous- time HMM must be developed for the transition matrix to be properly computed (e.g., see W illiams et al. , 2020 ). For a data set consisting of N countries, denote each country by an index i ∈ { 1 , . . . , N } , and let y i,k be the number of observed battle deaths for week k , for k ∈ { 1 , . . . , n i } , where n i is the number of weeks included in the data set for country i . The v alue of y i,k results from a multitude of circumstances, and we summarize these circumstances by a latent state s i,k ∈ { 1 , 2 , 3 } , corresponding to the ‘state’ of conﬂict dynamic at week k . These three un- derlying conﬂict-related states are deﬁned as ‘non-violent’, ‘stable violence’, and ‘intensiﬁed violence’, respecti vely; their names/interpretations are deduced from our inferences on the HMM ﬁtted to the real data, as discussed in Section 5 . W e limit our focus to three states, as Petroff, Bond and Bond ( 2013 ) argues that beyond three states [in an HMM of conﬂict dynamics] it becomes exceedingly difﬁcult to interpret the empirical results, and ﬁnds dis- tinctions accounted for by including additional states to be v ague at best. The underlying state sequence, s i, 1 , . . . , s i,n i , deﬁnes a stochastic process, and we assume for computational feasibility , as required for an HMM, that it is a Marko v chain in that the state of the process at week k only depends on the state of the process (and possibly co- v ariates) at week k − 1 . That being so, the conditional distribution of the random variable S i,k | S i,k − 1 is determined by a 3 × 3 transition probability matrix P ( i,k ) , which we express as, P ( i,k ) :=     1 1+ e q ( i,k ) 1 + e q ( i,k ) 2 0 0 0 1 1+ e q ( i,k ) 3 + e q ( i,k ) 4 0 0 0 1 1+ e q ( i,k ) 5 + e q ( i,k ) 6        1 e q ( i,k ) 1 e q ( i,k ) 2 e q ( i,k ) 3 1 e q ( i,k ) 4 e q ( i,k ) 5 e q ( i,k ) 6 1    , where q ( i,k ) 1 , q ( i,k ) 2 , q ( i,k ) 3 , q ( i,k ) 4 , q ( i,k ) 5 , q ( i,k ) 6 are real-valued parameters that determine the rates of their respective state transitions. Note that the matrix on the left simply re-scales (as row- wise multiv ariate logistic function transformations) the matrix on the right to have unit row sums, making it a proper transition probability matrix. The column and row indices corre- spond to the state space { 1 , 2 , 3 } . For example, the (1,2) component of P ( i,k ) expresses the v alue of the probability of transition from state 1 to state 2, for any two successive weeks. Furthermore, we express the transition probability parameters as,  q ( i,k ) 1 q ( i,k ) 2 q ( i,k ) 3 q ( i,k ) 4 q ( i,k ) 5 q ( i,k ) 6  := ( x ( i ) k ) ′ ζ , where x ( i ) k is a column vector of the geopolitical, re gion speciﬁc control cov ariates from Sec- tion 2.1 as well as the indicators for ceaseﬁre and pre-ceaseﬁre periods, for country i at week k , and ζ is a coefﬁcient matrix. The feature-rich coefﬁcient matrix ζ is a crucial component of the HMM for the purpose of studying dynamics of conﬂict. In theory , the transition rate parameters can be made arbitrarily conﬂict speciﬁc by including as many features (i.e., co- v ariates and coefﬁcient parameters) as necessary . These features determine the rate at which the underlying state of conﬂict e volv es or ceases to progress at all. Accordingly , for weeks 1 , . . . , n i the probability mass function of the latent state sequence s i, 1 , . . . , s i,n i has the form, (2) ℓ  { s i, 1 , . . . , s i,n i } | P ( i,k )  = π s i, 1 · n i Y k =2 P ( i,k ) s i,k − 1 ,s i,k , HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 11 where P ( i,k ) s i,k − 1 ,s i,k denotes row s i,k − 1 and column s i,k of the matrix P ( i,k ) , and π s i, 1 is the initial state probability for state s i, 1 . This Markov process deﬁned for the latent conﬂict state sequences is then embedded as a structural component of the response variable model ( 1 ), such that the parameters a and ρ are allowed to depend on the latent-state and to be country-speciﬁc, as follows. For week k ∈ { 1 , . . . , n i } and country index i ∈ { 1 , . . . , n } , conditional on the latent process S i,k , the data-generating model is expressed as (3) Y i,k | Y i, ( k − 4):( k − 1) , S i,k ∼ Neg ativ e-Binomial ( r i,k , p ) , where p := c (1 + c ) − 1 , (4) r i,k := a i,k + ρ i,k · Y i, ( k − 4):( k − 1) , a i,k := a 1 1 { s i,k = 1 } + a 2 1 { s i,k = 2 } + a 3 1 { s i,k = 3 } , and (5) ρ i,k := 1 { s i,k  = 1 } · e ( β 1 1 { s i,k =2 } + β 2 1 { s i,k =3 } ) ′ x ( i ) k , where a 1 , a 2 , a 3 , and c are positive parameters, β 1 and β 2 are coef ﬁcient column vectors, and 1 {·} is the indicator function. Denote a := ( a 1 , a 2 , a 3 ) and β := ( β 1 , β 2 ) . Recall that for this model to be weakly stationary , it suf ﬁces that 0 ≤ ρ i,k < 1 . An implication of this data-generating process speciﬁcation is that if s i,k = 1 , then Y i,k ∼ negati ve-binomial ( a 1 , p ) . Accordingly , the parameters a 1 , a 2 , a 3 , and c describe the model for rare conﬂict related deaths that may occur during weeks when a country is not experiencing a substantial conﬂict (e.g., isolated terrorist attacks). Moreov er , this forces the rate parameter ρ i,k to be identiﬁed with an increased mortality rate relating speciﬁcally to a deﬁned period of conﬂict (i.e., when the system is in state 2 or state 3). W ith the negati ve- binomial rate parameter deﬁned as in ( 4 ), (6) E { Y i,k | Y i, ( k − 4):( k − 1) , S i,k } = a i,k c + ρ i,k c · Y i, ( k − 4):( k − 1) . From the deﬁnition of ρ i,k in ( 5 ), this implies that the number of conﬂict deaths during state 1 has a mean of a 1 /c , whereas in the conﬂict states the mean structure is AR. T o distinguish between states 2 and 3, the constraint that β 11 ≤ β 12 is imposed (i.e., base- line ρ i,k for state 2 does not exceed that for state 3). This constraint helps to facilitate the identiﬁcation of states 2 and 3, respectiv ely , as associated with ‘stable’ versus ‘intensiﬁed’ conﬂict violence. Furthermore, we also impose the constraints that a 1 ≤ a 2 ≤ a 3 for the pur- pose of state space identiﬁcation. Finally , combining components ( 2 ) and ( 3 ) giv es a full likelihood function for the HMM for each country i ∈ { 1 , . . . , N } . F or ef ﬁcient estimation of the parameters, the likelihood can be expressed as a marginal mass function, resulting from integrating ov er all possible state space sequences. That is, p ( y i, 5 , . . . , y i,n i ) = 3 X s i, 1 =1 · · · 3 X s i,n i =1 X y i, 1 ≥ 0 · · · X y i, 4 ≥ 0 p ( y i, 1 , . . . , y i,n i , s i, 1 , . . . , s i,n i ) = 3 X s i, 1 =1 p ( s i, 1 ) · · · 3 X s i, 5 =1 p ( s i, 5 | s i, 4 ) · p ( y i, 5 | y i, 1:4 , s i, 5 ) · · · × 3 X s i,n i =1 p ( s i,n i | s i,n i − 1 ) · p ( y i,n i | y i, ( n i − 4):( n i − 1) , s i,n i ) = π ′ · P ( i, 2) · · · P ( i, 4) · P ( i, 5) D ( i, 5) · · · P ( i,n i ) D ( i,n i ) 1 , (7) 12 where π is the common initial state probability column vector , D ( i,k ) :=    p ( y i,k | y i, ( k − 4):( k − 1) , s i,k = 1) p ( y i,k | y i, ( k − 4):( k − 1) , s i,k = 2) p ( y i,k | y i, ( k − 4):( k − 1) , s i,k = 3)    , and using the negati ve-binomial mass function, p ( y i,k | y i, ( k − 4):( k − 1) , s i,k ) =  r i,k + y i,k − 1 y i,k  p r i,k (1 − p ) y i,k . Note that if further state space information is av ailable, such as partial labels, then the number of state sequences that are integrated ov er is reduced. F or example, if it is kno wn that s i,k ∈ { 2 , 3 } for some k ∈ { 1 , . . . , n i } , then P 3 s i,k =1 · reduces to P 3 s i,k =2 · within expression ( 7 ). Equi valently , the (1 , 1) component of D ( i,k ) is set to zero. Finally , the joint posterior density including the data from all countries i ∈ { 1 , . . . , N } then has the form, (8) π ( ζ , β , a, c | { y i,k } ) ∝  N Y i =1 p ( y i, 5 , . . . , y i,n i )  · π ( ζ , β , a, c ) · 1 { β 11 ≤ β 12 , a 1 ≤ a 2 ≤ a 3 } , where π ( ζ , β , a, c ) is a prior density . 3.3. Remarks on implementation. W e implement a Metropolis-within-Gibbs MCMC al- gorithm to draw posterior samples from ( 8 ). W ith these posterior samples, we then estimate the conditional posterior distribution of the latent state space for a gi ven country in our data set; we refer to this algorithm as the state space sampler algorithm. The state space sampler algorithm is a tool for ev aluating or predicting the most likely state (i.e., non-violent, stable violence, or intensiﬁed violence) at any giv en country/week, based on the HMM ﬁt to the real data. In Section 5 , we present a visual representation of the posterior distribution of the latent state sequence for countries both in our training data and in our held-out test data. 4. Empirical studies. 4.1. Illustration of why simpler statistical models ar e inadequate. As motiv ated by Sec- tions 3.1 and 3.2 , the negati ve-binomial response model for battle death counts with an AR mean structure, conditional on a latent state space process, is essential for capturing features of these data that are critical for pursuing conﬂict research questions. Moreov er, assuming the latent state space process is Markovian, as in the HMM framework we proposed, is the sim- plest and most computationally pragmatic approach. T o exemplify the importance of these methodological features, in this section we examine the limitations of simpler approaches to analyzing these data. Such simpler approaches may lead to similar, albeit more limited, conclusions as those that can be drawn from ﬁtting our full HMM model (e.g., as in Section 5 ). Simpler statistical approaches in volv e numerous subjecti ve choices that are not necessary easy to substantiate. Many of these subjectiv e choices become data dri ven in the context of our HMM approach. A most simple approach is to compare the number of battle deaths before and after many records of ceaseﬁres. If ceaseﬁres work as intended, we would expect, on av erage, a statis- tically/practically signiﬁcant reduction in the magnitude of violence after a ceaseﬁre is in ef fect. Figure 2 plots the weekly average number of battle deaths in the 8 weeks surrounding each ceaseﬁre recorded in the data set, over all countries; it is seen that the av erage number of battle deaths tend to be higher in the weeks preceding a ceaseﬁre than those that follow . HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 13 These data, ho we ver , exhibit a right sk e w due to the inﬂuence of a fe w countries with excep- tionally high battle death counts, and so we instead reconstruct Figure 2 and use a trimmed mean for the robustness of our subsequent analyses in this illustrati ve section; see Figure 3 . 15 20 25 30 −4 0 4 8 Week Battle Deaths (A verage) Ceasefire − Mean Battle Deaths per W eek (mean) F I G 2 . Displayed is the mean number of battle deaths per week, of the 4 weeks before each ceaseﬁr e declaration and the 4 weeks after each ceaseﬁre is in effect. All battle deaths series are center ed at the week of the ceaseﬁre, which is then deﬁned as week 0. The two vertical dashed lines mark the period deﬁned as declar ed and ceaseﬁre , r espectively . 6 7 8 9 −4 0 4 8 Week Battle Deaths (A verage) Ceasefire − Mean Battle Deaths per W eek (10% tr immed mean) F I G 3 . Displayed is the 10% trimmed mean number of battle deaths per week (i.e., the mean after remo ving the 10% highest and lowest values for each week surrounding all ceaseﬁres, for all countries in the data set), of the 4 weeks befor e each ceaseﬁr e declaration and the 4 weeks after each ceaseﬁr e is in effect. Compar e to F igur e 2 . A further complication is that the number of weeks to include in the befor e and after cease- ﬁre periods are indeterminate. Including too many weeks before or after the ceaseﬁre will pull do wn the av erage battle death count, as illustrated by Figure 4 , simply because conﬂicts 14 typically expire, eventually . The diminishing battle death counts near the right and left bound- aries reﬂect increasingly more weeks associated with periods of peace. Next, since there are sometimes less than, for example, 50 weeks between two ceaseﬁres, [subjecti ve] choices are required to avoid weeks that overlap between ± 50 weeks of more than one ceaseﬁre; e.g., keeping or removing ceaseﬁres with overlapping intervals or reducing the number of weeks to include before and after – all options will likely introduce a systematic bias into the anal- ysis. Moreover , it is unrealistic to assume a ﬁxed number of weeks (e.g., be it 4, 50, etc.) would apply to all conﬂicts, across all countries and time. The utility of the HMM frame- work introduced in Section 3 is that the model determines the weeks most likely associated with conﬂict and peace and at the country-speciﬁc resolution, so that the ef fects of ceaseﬁres on violent conﬂict can be estimated without conﬂating weeks of peacetime in the estimation. 4 6 8 −30 0 30 Week Battle Deaths (A verage) Ceasefire − Mean Battle Deaths per W eek (10% tr immed mean) F I G 4 . The 10% trimmed mean number of battle deaths per week, of the 50 weeks before each ceaseﬁre declaration and the 50 weeks after each ceaseﬁr e is in effect. Compar e to F igure 3 . For the sake of more precise illustration, we will proceed to compare the 10% trimmed mean number of battle death counts before and after the ceaseﬁres as in Figure 3 , i.e., includ- ing the 4 weeks before each ceaseﬁre declaration and the 4 weeks after each ceaseﬁre is in ef- fect. Let W j,k be the number of fatalities associated with week k ∈ {− 6 , . . . , − 3 } ∪ { 5 , . . . , 8 } for ceaseﬁre j ∈ { 1 , . . . , J } , where J is the total number of ceaseﬁres observed in the data set, within and across all 170 countries. For the simple approach of in vestigating the role cease- ﬁres play in conﬂict dynamics by comparing battle death counts before and after a ceaseﬁre (again for now , assuming exactly 4 weeks of counts before and after are appropriate), all that is necessary is to ﬁt the parameters of an analysis of v ariance (ANO V A) model, such as: independently for j ∈ { 1 , . . . , J } and k ∈ {− 6 , . . . , − 3 } ∪ { 5 , . . . , 8 } , (9) W j,k = δ · 1 { k < − 2 } + µ j + U j,k , where δ + µ j represents the expected number of fatalities before a ceaseﬁre has been declared, and U j,k is a mean zero inno v ation. In this speciﬁcation, δ is the theoretical ef fect of the ceaseﬁre on the expected number of battle deaths. Next, denote (10) W j, before := 1 4 − 3 X k = − 6 W j,k and W j, after := 1 4 8 X k =5 W j,k , HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 15 and V j := W j, before − W j, after = δ + ( U j, before − U j, after ) , with U j, before and U j, after deﬁned analogous to the averages in ( 10 ). In this formulation, E ( V j ) = δ and the 10% trimmed mean of the observed v j = w j, before − w j, after for j ∈ { 1 , . . . , J } is a seemingly robust estimate for δ . Accordingly , we ﬁnd the estimated δ from the data is P J 1 v j /J = 2 . 36 with a corresponding 0.95 bootstrap conﬁdence interv al of (1.08, 3.79), indicating a signiﬁcant decrease in fatalities after a ceaseﬁre is in effect. Note that re- moving ceaseﬁres that hav e ov erlapping weeks in the 4 weeks before a ceaseﬁre is declared, or the 4 weeks after a ceaseﬁre is in ef fect, does not change these estimates signiﬁcantly . W e hav e not yet, howe ver , controlled for any ceaseﬁre-speciﬁc covariates, such as GDP , population, or polyarchy (democracy score). T o do so, a simple extension to model ( 9 ) is (11) W j,k = ( δ + x ′ j γ ) · 1 { k < − 2 } + µ j + U j,k , where x j is a vector of ceaseﬁre-speciﬁc cov ariates consisting of GDP , population, and pol- yarchy , and γ is a corresponding coefﬁcient parameter vector with 3 components. In T able 1 , we present the trimmed 10% least squares estimates (c.f., Rousseeuw , 1984 ) of the parame- ters in model ( 11 ). Estimate 0.025 0.975 Ceaseﬁre δ 2.26 1.09 3.87 GDP γ 1 -0.02 -1.57 1.76 population γ 2 -0.21 -2.08 1.21 polyarchy γ 3 -2.16 -3.98 -0.17 T A B L E 1 P arameter estimates for model ( 11 ) ar e based on trimmed 10% least squar es, after normalizing the explanatory variables, i.e., mean center ed and scaled to have unit standard de viation. The 0.025 and 0.975 columns display the corr esponding bootstrapped quantiles. The estimates displayed in T able 1 for the δ parameter are more or less unchanged from those without including cov ariates, and polyarchy is the only explanatory variable with evi- dence of a signiﬁcant association, at the 0.95 le vel. If we remove ceaseﬁres that have ov er- lapping time periods, ho wev er , then the ef fect of the democrac y score polyarchy is no longer signiﬁcant. Moreover , Figures 5 and 6 demonstrate ho w the estimate of δ as in model ( 9 ) will change if more or less weeks are included in the periods before a ceaseﬁre is declared or after it has gone into ef fect. These ﬁgures sho wcase ho w an in vestigation of the role ceaseﬁres play in conﬂict dynamics depends on the number of weeks to include – whether ceaseﬁres with ov erlapping time periods are included or not; it is clear that some choices will lead to the ﬁnd- ing of a δ signiﬁcantly different from zero while other choices will not. Beyond this ﬁnding, such an analysis does not allow for the number of before/after weeks to be ceaseﬁre-speciﬁc. The HMM framew ork we propose in Section 3 avoids these limitations because the HMM itself will quantify the uncertainty , in a data-driv en manner , for the number of before/after weeks to include for each observed ceaseﬁre in estimating the ef fect of a ceaseﬁre, and at the country-speciﬁc resolution (i.e., using country-speciﬁc co variates). Moreov er, our HMM for - mulation estimates the effect of the presence of a ceaseﬁre on a rolling basis, thus bypassing the need for justiﬁcations about dropping/including weeks with ov erlapping ceaseﬁres that would be necessary in the simpler , ANO V A approach with predeﬁned before/after weeks. A problem with the ANO V A approach is that, unlike the HMM approach, it treats each ceaseﬁre as an independent e vent, e ven if it overlaps a simultaneously occurring ceaseﬁre. 16 −3 0 3 6 9 −200 −100 0 100 200 Week Since Ceasefire Estimated Change Estimated Eff ect of a Ceasefire on Battle Deaths F I G 5 . The estimated effect, δ as in model ( 9 ), is plotted as points, each corresponding to a differ ent number of weeks included in the periods befor e a ceaseﬁre is declared and after it has gone into effect. Zer o “W eeks Since Ceaseﬁr e” corr esponds to including 4 weeks befor e and after , as in the discussion above; the magnitude of ne gative (positive) values of “W eeks Since Ceaseﬁr e” correspond to how many additional weeks are included in the period befor e (after) a ceaseﬁre. The solid lines correspond to 0.025 and 0.975 bootstrapped quantiles. All ceaseﬁr es that have overlapping ceaseﬁre time periods wer e r emoved. It is clear that the choice of the number of weeks to compar e befor e and after a ceaseﬁr e will inﬂuence the signiﬁcance of the effect of a ceaseﬁr e. −3 0 3 6 9 −200 −100 0 100 200 Week Since Ceasefire Estimated Change Estimated Eff ect of a Ceasefire on Battle Deaths F I G 6 . Same as F igur e 5 , but without r emoving ceaseﬁr es that have overlapping ceaseﬁr e time periods. 4.2. Simulation study of synthetic battle death data. The purpose of this section is to verify that synthetic data generated by our HMM resembles important features of the real battle death count data, and that our Bayesian estimation procedure produces credible sets HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 17 for the HMM parameters that achiev e their corresponding repeated sampling cov erage. The ‘true’ HMM parameter values used to generate the synthetic data are set as the posterior means estimated from the real data set; those v alues are presented in T able 3 . W e generate synthetic data for each of N = 167 countries in our training data set (lea ving data from 3 countries as test data), based on the following procedure. For each country/week with recorded co variates a ‘true’ latent state s i,k is sampled from either the initial state prob- ability vector π if k = 1 or P s i,k − 1 , · if k > 1 , and a count of battle deaths y i,k is sampled from ( 3 ). Note that the ﬁrst four values, y i, 1:4 , are sampled from ( 3 ) with ρ i,k = 0 . Further- more, the maximum number of battle deaths in any one week is restricted to not exceed the highest week-death-count to country-population proportion of any country in the data set. The highest such proportion is approximately 0.0006 for Congo. If a simulated death count y i,k exceeds 0.0006 times the country population, then the generated data for that country is discarded and re-simulated until the restriction is satisﬁed. This was not a problem for any of the 167 countries in the data set, with the notable exception of India. For India, it took an excessiv e amount of time to generate battle death data that satisfy the maximum weekly death restriction, and so India was omitted from our simulation study . trans. rates baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop ζ ′ 1 ( 1 → 2 ) .66 .83 .91 .92 .93 .94 .90 .91 ζ ′ 2 ( 1 → 3 ) .94 .97 .93 .94 .92 .94 .95 .94 ζ ′ 3 ( 2 → 1 ) .93 .96 .94 .97 .99 .94 .95 .92 ζ ′ 4 ( 2 → 3 ) .95 .96 .97 .93 .91 .97 .95 .93 ζ ′ 5 ( 3 → 1 ) .96 .96 .93 .93 .96 .95 .92 .95 ζ ′ 6 ( 3 → 2 ) .95 .93 .97 .94 .91 .94 .97 .98 AR coef. baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop β ′ 1 (state 2) .90 .97 .95 .96 .95 .96 .92 .94 β ′ 2 (state 3) .96 .97 .97 .96 .95 .93 .97 .93 other a 1 a 2 a 3 c π 2 π 3 .91 .66 .95 .96 .90 .95 T A B L E 2 Pr oportion of 100, .95 posterior credible sets that contains the true parameter value for eac h of the HMM parameters (constructed by e xcluding the upper and lower .025 tails of each marginal posterior distrib ution). Synthetic data for N = 166 countries ar e generated for eac h of the 100 data sets in this simulation study . Note that parameter values r eﬂect covariate values for lag v2x polyar chy (linear , quadractic, and cubic), la g and log GDP per capita, and lag and log population. These variables have all been center ed and scaled to have unit variance. See the Supplementary Material ( W illiams et al. , 2024 ) for box plots over the 100 posterior medians for each par ameter . India is an outlier country in our data set in the sense that it has an uncharacteristically large population size which, based on the ﬁtted HMM parameters (see T able 3 ) is associated with markedly less time spent in state 1, and battle death sequences represented with an explosi ve or non-stationary series (i.e., ρ i,k > 1 ). Additionally , India is often in a state of ceaseﬁre which is associated with increased instability (i.e., states 2 and 3). One possible explanation for why conﬂict dynamics are not explained so well by our ﬁtted HMM for countries with populations as large as India is the greater possibility for numerous unrelated conﬂicts ongoing at any point in time. For the av erage country , conﬂict dynamics are more likely limited to a single conﬂict at a time. As with our real data set, for our synthetic data sets we apply a single rule-based partial label. That is, any week that is at least two months after or two months before a week with one or more battle deaths, and is within a two year sequence of weeks with no battle deaths 18 is labeled as state 1, without error . A total of 100 synthetic data sets are generated based on the described procedure. W e implement a simple independent components Gaussian prior density with mean zero and excessi vely diffuse standard deviation 20 for all parameters. T o enforce the constraint that a 1 , a 2 , a 3 , and c are positiv e-valued, we place the Gaussian prior on log( a 1 ) , log ( a 2 ) , log ( a 3 ) , and log ( c ) . Similarly , the Gaussian prior is placed on the logit transforms of the components of the initial state probability vector π . An MCMC algorithm is used to estimate the posterior distributions for all 70 HMM parameters, for each of the 100 synthetic data sets. The co verage at the 0.95 lev el of signiﬁcance for each parameter is stated in T able 2 . Box plots of the 100 posterior medians for each parameter are presented in our Supplementary Material ( W illiams et al. , 2024 ). Figure 7 giv es a visual representation of the synthetic data we generated for the held-out test set countries, Sudan and Afghanistan. Sudan 0 500 1000 1500 1992−01−01 1997−01−01 2002−01−01 2007−01−01 2012−01−01 2017−01−01 Synthetic Battle Deaths 0 500 1000 1500 1992−01−01 1997−01−01 2002−01−01 2007−01−01 2012−01−01 2017−01−01 Number of Battle Deaths (Weekly) Afghanistan 0 300 600 900 2007−01−01 2012−01−01 2017−01−01 Synthetic Battle Deaths 0 300 600 900 2007−01−01 2012−01−01 2017−01−01 Number of Battle Deaths (Weekly) F I G 7 . The top panels for each country display 1000 synthetic realizations of the r esponse variable sequence simulated fr om the HMM ﬁt with the posterior means of all parameters pr esented in T able 3 , the maximum a posteriori latent state sequence, and all covariate values observed in the real data set. The triangle points r epr esent the upper 0.025 percentile while the cir cle points r epresent the lower 0.025 percentile . F or r efer ence, the bottom panels for each country display the r eal numbers of observed battle deaths. HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 19 5. Results and analyses. For our analysis of the real data, we focus on three inferential aspects of the ﬁtted model. First, the posterior mean estimates of all 70 HMM parameters described in Section 3 are summarized in T able 3 . The MCMC trace plots and histograms of the posterior samples are provided in the Supplementary Material ( Williams et al. , 2024 ). The posterior means presented in T able 3 quantify the effect of the v arious parameters/features in the model, but they are not dynamic in the sense that they represent the model at a weekly resolution whereas the HMM is ﬁtted to the data as a system that e volves over many years of accumulated weeks. For this purpose, second, we present probability ev olution curves for a small selection of countries in Figures 8 and 9 . For these ﬁgures, the transition probabilities are computed and plotted ov er the same time periods observed in the training data, using the cov ariate v alues associated with each country/week. Furthermore, these probability ev olution plots ignore the HMM response data (i.e., the counts of conﬂict deaths each week) to provide inference purely on the effects of the state transition probability co variates. In particular , they demonstrate the role that ceaseﬁres have played in the de-escalation of violence for the countries in our data set. Third, we use the posterior mean estimates from T able 3 to sample state sequences via the state space sampler discussed in Section 3.3 . These are displayed, for the same small selection of countries, in Figures 10 and 11 . The probability ev olution and state space sampler plots for all 170 countries included in our data set are av ailable with our Supplementary Material ( W illiams et al. , 2024 ). trans. rates baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop ζ ′ 3 ( 2 → 1 ) − 4 . 714 ⋆ − 0 . 322 1 . 243 ⋆ − 0 . 938 ⋆ 1 . 748 ⋆ − 0 . 899 ⋆ − 0 . 554 ⋆ − 0 . 588 ⋆ ζ ′ 4 ( 2 → 3 ) − 5 . 965 ⋆ 1 . 693 ⋆ 0 . 524 ⋆ − 0 . 375 − 0 . 572 ⋆ 0 . 421 − 0 . 486 ⋆ − 0 . 516 ⋆ ζ ′ 5 ( 3 → 1 ) − 0 . 986 ⋆ − 1 . 550 ⋆ 0 . 367 − 1 . 617 ⋆ 0 . 153 0 . 317 − 0 . 554 ⋆ − 0 . 005 ζ ′ 6 ( 3 → 2 ) 0 . 993 ⋆ − 0 . 289 0 . 161 − 0 . 734 ⋆ 0 . 134 0 . 245 0 . 143 0 . 305 AR coef. baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop β ′ 1 (state 2) − 4 . 228 ⋆ 0 . 078 − 0 . 099 ⋆ 1 . 784 ⋆ − 3 . 945 ⋆ 2 . 389 ⋆ 0 . 238 ⋆ 0 . 337 ⋆ β ′ 2 (state 3) − 3 . 849 ⋆ 0 . 660 ⋆ 0 . 061 − 0 . 986 ⋆ − 0 . 851 ⋆ 0 . 392 0 . 724 ⋆ 0 . 948 ⋆ other a 1 a 2 a 3 c π 2 π 3 0 . 0004 ⋆ 0 . 0911 ⋆ 5 . 8714 ⋆ 0 . 0246 ⋆ 0 . 0279 ⋆ 0 . 0140 ⋆ T A B L E 3 P osterior means of the HMM parameter s. Note that parameter values r eﬂect covariate values for lag v2x polyar chy (linear , quadractic, and cubic), la g and log GDP per capita, and lag and log population. These variables have all been center ed and scaled to have unit standar d deviation. Boldface ⋆ indicates that the .95 cr edible r e gion, formed by excluding posterior samples in the upper and lower .025 tails, e xcludes the value 0. Note that we omit the state 1 → 2 transitions fr om this table because the inferential focus of our application to conﬂict r esear ch is r estricted to the other transitions. See the Supplementary Material ( W illiams et al. , 2024 ) for the MCMC trace plots and histogr ams of the posterior samples. The parameter estimates in T able 3 suggest a v ariety of interesting ﬁndings. First, we note that the state 2 (stable violence) to state 1 (non-violent) transition probability increases by about a factor of 3.5 when a ceaseﬁre is in effect, taking all other covariates at mean value. Note that while there is also found to be a statistically signiﬁcant, positive coefﬁcient for the ceaseﬁre indicator variable associated with the 2 → 3 transition, the larger magnitude of the 2 → 1 coefﬁcient indicates that 2 → 1 transitions will occur with higher probability than 2 → 3 transitions. Nonetheless, the ﬁnding of positi ve coef ﬁcients for both the 2 → 1 and 2 → 3 transitions is evidence that ceaseﬁres are directly associated with some change in the underlying conﬂict dynamics, most likely a cessation of violence. Over time, the factor of 3.5 effect of ceaseﬁres is visually displayed in the top panels of Figures 8 and 9 , within the 20 vertical bars that indicate ceaseﬁres are in effect. Note that it is also observed in the ﬁgures that this effect carries momentum for diminishing state 2 or 3 transition probabilities even after the ceaseﬁre period. Second, observe the statistically signiﬁcant, positi ve pre-ceaseﬁre indicator variable coefﬁcient appearing for transition 2 → 3 , as well as the statistically sig- niﬁcant, negati ve pre-ceaseﬁre indicator v ariable coefﬁcient for the 3 → 1 transition. Such ﬁndings suggest heightened or sustained lev els of violence in the weeks associated with the pre-ceaseﬁre period. This is a major ﬁnding of our analysis. It highlights the lag time between when a ceaseﬁre is negotiated and when it actually be gins, and that negotiating and preparing for a ceaseﬁre is associated with an immediate short-term escalation in violence (which in turn is likely to also increase the likelihood of a ceaseﬁre). Parties have incentiv es to ﬁght harder to gain the strongest relati ve position prior to the ceaseﬁre suspending the violence. Escalated violence also increases the incentives for a ceaseﬁre. Once a ceaseﬁre enters into effect, we ﬁnd that conﬂict dynamics tend to transition from a violent to a non-violent state in the weeks that follow the ceaseﬁre. An explanation is that the beneﬁts accrued from a ceaseﬁre, whether peaceful or military/strategic, require some immediate shift in violence dynamics. Finally , we ﬁnd e vidence for three underlying states of conﬂict, which we describe as ‘non-violent’, ‘stable violence’, and ‘intensiﬁed vio- lence’. W e illustrate the utility of the constructed HMM for both inferential purposes and as a tool for predicting intensity and violence in conﬂict. South Sudan 0.00 0.25 0.50 0.75 1.00 2012−01−01 2013−01−01 2014−01−01 2015−01−01 2016−01−01 2017−01−01 T ransition Probability State 2 or 3 State 3 Declared Ceasefire F I G 8 . Displayed is the evolution of the pr obabilities of transition on a week-by-week r esolution. Counts of conﬂict r elated deaths ar e omitted fr om the computation of these probabilities. Instead, the pr obabilities e xclusively reﬂect the effects, on the tr ansition rates, of the covariates observed for the labelled country , over time . Israel 0.00 0.25 0.50 0.75 1.00 1992−01−01 1997−01−01 2002−01−01 2007−01−01 2012−01−01 2017−01−01 T ransition Probability State 2 or 3 State 3 Declared Ceasefire F I G 9 . Displayed is the evolution of the pr obabilities of transition on a week-by-week r esolution. Counts of conﬂict r elated deaths ar e omitted fr om the computation of these probabilities. Instead, the pr obabilities e xclusively reﬂect the effects, on the tr ansition rates, of the covariates observed for the labelled country , over time . HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 21 Furthermore, the estimated mean AR coef ﬁcient, ρ i,k /c (recall equation ( 6 )), increases from 0.8659 in state 3 to 1.6753, taking all other cov ariates at mean value, for all pre-ceaseﬁre weeks. Howe ver , in either case, this coefﬁcient will be explosiv e (i.e., the AR process is not stationary) for countries with lar ger GDP per capita and/or lar ger population, as demonstrated by the signiﬁcant coefﬁcient estimates 0.724 and 0.948, respecti vely . The explosi ve value of this coef ﬁcient is consistent with our interpretation of state 3 as ‘intensiﬁed’ violence. Con versely , the ‘stable’ violence interpretation for state 2 comes from the fact that it has an AR coefﬁcient, taking all other cov ariates at mean v alue, estimated to be less than 1, which describes a weakly stationary process. Such processes revert to a stationary mean, and it is in this sense that the ‘stable’ violence state describes both non-escalating and de-escalating violence. South Sudan 0 100 200 300 400 2012−01−01 2013−01−01 2014−01−01 2015−01−01 2016−01−01 2017−01−01 Number of Battle Deaths (Weekly) Declared Ceasefire 0.00 0.25 0.50 0.75 1.00 2012−01−01 2013−01−01 2014−01−01 2015−01−01 2016−01−01 2017−01−01 P osterior State Probability State: 1 2 3 F I G 1 0 . The top panel displays the observed data for South Sudan, and the bottom panel displays the estimated posterior pr obability of each state for each week. One ﬁnal important ﬁnding is the statistically signiﬁcant effect for the third degree polyno- mial coef ﬁcient for the v2x polyarchy v ariable for the state 2 to 1 transition probability (with a leading negati ve coef ﬁcient), and for the state 2 AR coefﬁcient. Both democracies and au- tocracies are less likely than countries in the middle to enter into conﬂict, but when they do their trajectories differ dramatically . Democracies are known to have less fatal conﬂicts than other regimes ( Lacina , 2006 ), but also more durable conﬂicts ( Crisman-Cox , 2022 ). Conﬂicts are more likely to erupt in the hybrid regimes between pure dictatorships and democracies ( Hegre et al. , 2001 ). The signiﬁcant ef fect of a third de gree polynomial for the v2x polyarch y v ariable is e vidence for the hypothesized non-monotonic nature of the association between measure of democrac y and violent conﬂict dynamics. Our results suggest that low-le vel con- ﬂicts are most lik ely to terminate for autocratic regimes, and most likely to escalate for hybrid regimes. Democracies are least likely to mo ve in either direction, which is consistent with the literature. F or instance, consider the Middle East. Israel, a democrac y , has been in volved in a 22 series of armed conﬂicts with v arious non-state org anizations. These conﬂicts usually remain acti ve but do not escalate beyond a certain lev el because Israel puts a limit on its own use of force. Its most proximate neighbors, howe ver , have not shown the same restraint. Jordan used maximum force in September 1970 to expel PLO from Jordan, and were successful. Four decades later, Syria tried the same strategy , and failed, with the Syrian Civil W ar as a consequence. Israel 0 200 400 1992−01−01 1997−01−01 2002−01−01 2007−01−01 2012−01−01 2017−01−01 Number of Battle Deaths (Weekly) Declared Ceasefire 0.00 0.25 0.50 0.75 1.00 1992−01−01 1997−01−01 2002−01−01 2007−01−01 2012−01−01 2017−01−01 P osterior State Probability State: 1 2 3 F I G 1 1 . The top panel displays the observed data for Israel, and the bottom panel displays the estimated posterior pr obability of each state for each week. GDP per capita and population show very similar effects. It is important to keep in mind that conﬂicts tend to happen in poorer countries. Hence, the negati ve effect of GDP per capita on transitions and the positiv e effect on fatalities must be interpreted in the context of the sample of countries in the data set that had conﬂicts. The very poorest countries are more likely to have very intermittent conﬂicts, which causes the country to move often between state 1 and 2, and sometimes state 3. Richer countries are, once they ﬁnd themselves in a state of conﬂict, more capable of persistently ﬁghting these conﬂicts. Similarly , more populous countries are also more likely to maintain a strong military and are as such more likely to ﬁeld a sustained campaign. This does not mean that the most f atal conﬂicts necessarily are in richer country , but the conﬂicts with one or two deaths per week, scattered over time tend to be in the poorest and smallest countries. The fact that the data exhibit patterns of both stable and unstable models is a strong indi- cation that conﬂicts endure distinct phases, and that intensiﬁed phases marked by explosi ve violence are only sustained for short periods of time. Namely , observe that, taking all other cov ariates at mean v alue and in the absence of ceaseﬁres or negotiations, the baseline transi- tion probabilities from 3 → 1 , 3 → 2 , and 3 → 3 are 0.0916, 0.6628, and 0.2456, respectiv ely; once in state 3 (intensiﬁed violence) it is about 0.75 probability of transitioning to another state in the next week. HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 23 Our last remark on the estimates in T able 3 is that the ﬁtted expected number of battle deaths per week, in the absence of battle deaths in pre vious weeks, during state 1, state 2, and state 3 are 0.0163, 3.7033, and 238.6748, respecti vely . Recall from the expected value of the negati ve-binomial distribution in ( 6 ) that these are computed as a j /c , for j ∈ { 1 , 2 , 3 } . This statistic serves as a simple, rudimentary description of how the HMM has ﬁtted the mean behavior of each of the three states. Finally , we note that (although not displayed in T able 3 ) the baseline probabilities for tran- sition from state 1 → 2 and 1 → 3 , in one week, are estimated to be 0.0006276 and 6 × 10 − 7 , respecti vely . This is consistent with the fact that periods of violent conﬂict are rare e vents, and are not part of the natural progression of the state of aff airs for the av erage country , (a verage in the sense of the cov ariates we consider). Ho wev er , we ﬁnd that the transition probability from state 1 → 2 increases by a factor of about 52 for a country with a declared ceaseﬁre, and by a factor of about 18 for a country with a ceaseﬁre in effect, taking all other covariates at mean v alue. Mostly , this reﬂects the fact that ceaseﬁres occur in the midst of violent conﬂict, and so the likelihood of future violence is higher than if a country was in state 1 not having recently transitioned from a state of violent conﬂict. 5.1. Addr essing limitations of the data: model r e-estimation using low and high estimates of weekly battle death counts. UCDP states that the fatality estimates in the UCDP geo- referenced ev ent data set used in this analysis are fraught with uncertainty 5 . T o quantify the uncertainty expressed in the source material, they publish three estimates: best, high, and low . T o clarify the UCDP geo-referenced e vent coding scheme ( Högbladh , 2023 ), geo- referenced event e vents are deﬁned based on a subjectiv e judgement of one or more sources with at least one trustworthy claim that at least one person was killed. The estimates of best, high, and lo w reﬂect the v ariance reported in the number of casualties from sources deemed trustworthy ( Högbladh , 2023 ). UCDP has a conservati ve policy , by which they will report the lowest number supported by a source. The variability between best, high, and low can arise either from a range reported by a source or different estimates given by different sources. If a partially corresponding aggre gate report contains a single fatality estimate, then the corresponding record for this ev ent will be best = high = low . There are generally four dif ferent types of uncertainty: 1. Some ev ents are reported by se veral independent sources. The September 11 terrorist attach in the United States, for instance, is reported by a large number of sources, which all report the same number of casualties (2,986). This is an example of a credible and precise number . 2. For some e vents, the best, high, and low estimates dif fer , sometimes by a lot. There might be conﬂicting reports that claim either 2 dead or 25 dead. Both of these are precise, but they cannot both be credible. Nonetheless, we hav e to choose between 2 or 25, not any random number in-between. 3. There might be sev eral sources that report, e.g., between 10 and 20 casualties. This is not precise, but can be credible. In this case any random number between 10 and 20 is as likely as an y other . 4. There might be a single report of an imprecise number , such as “at least 50,000 persons are killed”. This is not very precise, and it is hard to adjudicate the credibility . W e cannot report any number other than 50,000 without making a number of additional assumptions, but sole y reporting 50,000 creates a false impression of precision. 5 See https://www .pcr .uu.se/research/ucdp/methodology/ for a discussion. 24 W e use the best estimates in our main analysis in the previous section, but we check the robustness of the ﬁtted model using both the high and low estimates, here; see T ables 4 and 5 . W e additionally ﬁt the model on another 100 data sets, where the number of fatalities each week is taken as a mixture drawn from all three categories to assess the robustness of the current model to the unknown variability across these categories; see T able 6 . The HMM ﬁt to the best–high–low mixture is largely consistent with the model ﬁt using the UCDP best estimates of f atalities, e ven more so than the parameters ﬁt using exclusi vely the UCDP high or lo w estimates. trans. rates baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop ζ ′ 3 ( 2 → 1 ) − 4 . 572 ⋆ − 0 . 354 0 . 947 ⋆ − 0 . 954 ⋆ 1 . 924 ⋆ − 1 . 069 ⋆ − 0 . 620 ⋆ − 0 . 619 ⋆ ζ ′ 4 ( 2 → 3 ) − 6 . 141 ⋆ 1 . 458 ⋆ 1 . 195 ⋆ − 0 . 894 ⋆ − 0 . 104 0 . 115 − 0 . 490 ⋆ − 0 . 472 ⋆ ζ ′ 5 ( 3 → 1 ) 2 . 427 ⋆ − 1 . 239 ⋆ − 0 . 641 ⋆ 0 . 205 − 0 . 618 1 . 507 ⋆ 0 . 040 − 0 . 385 ζ ′ 6 ( 3 → 2 ) 1 . 372 ⋆ 0 . 046 0 . 201 − 0 . 143 0 . 145 − 0 . 352 0 . 501 ⋆ 0 . 501 ⋆ AR coef. baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop β ′ 1 (state 2) − 4 . 404 ⋆ − 0 . 008 − 0 . 219 ⋆ 2 . 133 ⋆ − 4 . 543 ⋆ 2 . 732 ⋆ 0 . 239 ⋆ 0 . 321 ⋆ β ′ 2 (state 3) − 2 . 102 ⋆ − 0 . 753 ⋆ − 0 . 421 ⋆ − 1 . 004 ⋆ 0 . 778 0 . 543 0 . 615 ⋆ 0 . 655 ⋆ other a 1 a 2 a 3 c π 2 π 3 0 . 0004 ⋆ 0 . 0937 ⋆ 5 . 7008 ⋆ 0 . 0201 ⋆ 0 . 0396 ⋆ 0 . 0208 ⋆ T A B L E 4 P osterior means of the HMM parameter s using high counts of battle deaths . Compar e with T able 3 . trans. rates baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop ζ ′ 3 ( 2 → 1 ) − 4 . 773 ⋆ 0 . 079 1 . 228 ⋆ − 1 . 328 ⋆ 2 . 618 ⋆ − 1 . 506 ⋆ − 0 . 508 ⋆ − 0 . 560 ⋆ ζ ′ 4 ( 2 → 3 ) − 6 . 323 ⋆ 1 . 078 ⋆ 1 . 169 ⋆ − 1 . 030 ⋆ 0 . 440 − 0 . 111 − 0 . 402 ⋆ − 0 . 384 ⋆ ζ ′ 5 ( 3 → 1 ) 3 . 289 ⋆ − 1 . 244 ⋆ 0 . 742 − 0 . 807 ⋆ 1 . 062 ⋆ 0 . 357 0 . 129 − 1 . 519 ⋆ ζ ′ 6 ( 3 → 2 ) 1 . 898 ⋆ − 0 . 332 1 . 126 ⋆ − 0 . 004 0 . 091 − 1 . 035 ⋆ 0 . 776 − 0 . 204 AR coef. baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop β ′ 1 (state 2) − 4 . 160 ⋆ 0 . 116 ⋆ − 0 . 102 ⋆ 2 . 232 ⋆ − 5 . 035 ⋆ 3 . 101 ⋆ 0 . 223 ⋆ 0 . 317 ⋆ β ′ 2 (state 3) − 2 . 731 ⋆ 1 . 675 ⋆ − 1 . 198 ⋆ − 1 . 072 ⋆ − 0 . 235 1 . 069 ⋆ 0 . 673 ⋆ 1 . 001 ⋆ other a 1 a 2 a 3 c π 2 π 3 0 . 0005 ⋆ 0 . 0891 ⋆ 6 . 4379 ⋆ 0 . 0252 ⋆ 0 . 0600 ⋆ 0 . 0007 ⋆ T A B L E 5 P osterior means of the HMM parameter s using low counts of battle deaths . Compare with T able 3 . As pointed out above, the interpretation of isolated coefﬁcients is dif ﬁcult. The intended ef fect of a ceaseﬁre can both be to reduce the fatalities in a giv en state of conﬂict and to increase the transition probability to a less intensiﬁed state of conﬂict. The AR coefﬁcients for ceaseﬁre are strongly neg ativ e in T ables 4 , 5 , and 6 for state 2, and as such pro vide stronger support than the main results. The ﬁnding that ceaseﬁres are preceded by an increased intensity does not ﬁnd similarly robust support. Using the UCDP lo w estimate counts, the pre-ceaseﬁre period is associated with a strong increase in the number of fatalities, whereas the high estimate counts result in the opposite association. This ﬁnding is hard to explain as an ything other than a reﬂection of the uncertainties in the data. The model ﬁt on both the high and lo w estimate counts as well as the best–high–low mixture, howe ver , ﬁnds support for an increased transition probability HMMS FOR LA TENT V ARIABLE LABELING ASSIGNMENTS IN CONFLICT RESEARCH 25 from state 2 to state 3 prior to ceaseﬁres, as well as a reduced transition probability from state 3 to state 1, all consistent with the ﬁndings in T able 3 . The UCDP high estimates hav e associated baseline transition rates consistent with less duration in state 3 than those of the best estimates, and while the estimate for a 3 based on the high estimates is similar to that based on the best estimate, the baseline AR coefﬁcient for state 3 is markedly higher based on the UCDP high estimates. This reﬂects UCDP’ s tendency tow ards conserv ativ e estimates. Alternativ ely , using the UCDP low estimate counts compared to the best, the estimates for a 3 and the baseline AR coef ﬁcient for state 3 are both higher , but the low estimates hav e associated baseline transition rates consistent with even less duration in state 3. This suggests more pronounced dif ferences between the distributions of battle death counts for states 2 and 3, associated with the UCDP lo w estimates (i.e., battle death counts that are more ﬂat ov er time with abrupt but short-li ved spikes). trans. rates baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop ζ ′ 3 ( 2 → 1 ) -4.73(.23) -.37(.80) 1.08(.33) -1.02(.45) 1.78(.66) -.90(.58) -.55(.13) -.59(.15) ζ ′ 4 ( 2 → 3 ) -5.93(.45) 1.60(.45) .50(.51) -.45(.57) -.66(.76) .52(.56) -.44(.19) -.59(.21) ζ ′ 5 ( 3 → 1 ) .30(.63) -1.42(.75) .13(.77) -1.33(.56) .44(.63) .84(.76) -.52(.53) -.30(.59) ζ ′ 6 ( 3 → 2 ) .58(.45) -.28(.76) .13(.74) -.68(.58) .12(.80) -.18(.69) .16(.39) .43(.47) AR coef. baseline pre-ceaseﬁre ceaseﬁre v2x v2x 2 v2x 3 GDP pop β ′ 1 (state 2) -4.31(.05) .07(.05) -.13(.05) 1.93(.26) -4.13(.63) 2.48(.41) .25(.02) .34(.02) β ′ 2 (state 3) -3.83(.48) .64(.85) -.09(.59) -.96(.56) -.60(.78) .46(.70) .66(.25) 1.08(.43) other a 1 a 2 a 3 c π 2 π 3 .00(.00) .09(.00) 6.18(.32) .02(.00) .04(.03) .04(.04) T A B L E 6 A verage of posterior means of eac h HMM parameter over 100 data sets, eac h data set constructed by uniformly sampling over the best, high, or low count of battle deaths, when variation e xists. Standar d deviations of the posterior means ar e given in par entheses. Compar e with T able 3 . 6. Concluding remarks. There are many research directions we hope to in vestigate in furthering this work. Namely , we hope to distinguish between ceaseﬁres of dif ferent types and scope, separating those ceaseﬁres that hav e markedly different characteristics (e.g., ces- sation of hostility arrangements vis-a-vis preliminary ceaseﬁres). A complimentary research direction is to include a cov ariate to account for the duration of a giv en conﬂict (i.e., time in state 2 and/or state 3). This would allo w us to address questions relating to whether con- ﬂicts tend to hav e limited persistence. Finally , spatial correlations in conﬂict data (e.g., due to geopolitics, etc.) are well established ( Gleditsch , 2007 ). In part, they stem from the fact that certain conﬂict-inducing factors are present in the same area. Pov erty , for instance, is geographically clustered. But they also relate through conﬂict dynamics. V iolent conﬂict can spill ov er from one country to another , such as between Rwanda and the Democratic Repub- lic of Congo. This might be because ethnic groups cohabit both sides of a border , or it might be because a government intervenes in another country to pre vent the organization of armed resistance, such as the Israeli intrusion of Lebanon. In any case, spatial correlations are a feature of these data that we plan to address in a subsequent, more complex formulation of our HMM approach, in our future research. Further open research problems include the follo wing: 1. W e in vestigate the patterns that civil conﬂicts share across countries. Howe ver , if labels were on the dyad lev el (i.e., pairs of armed and opposing actors) rather than the country le vel, would the same patterns emerge? Are there additional patterns that would emerge? What about patterns on the resolution of conﬂict labeled data? 26 2. In our analysis, we are able to estimate the ef fect of ceaseﬁres for static time periods immediately surrounding the agreement. Further methods are needed to determine the length of persistence of a gi ven ceaseﬁre. 3. What are the ef fects of peacekeeping ef forts on the sustainability of ceaseﬁres? 4. What are the ef fects of seasonality on the sustainability of ceaseﬁres? Certain countries and regions simply cannot maintain violence during certain parts of the year (e.g., due to seasonal weather patterns, the necessity of farming, etc.). SUPPLEMENT AR Y MA TERIAL Code and data to r eproduce the empirical results presented in the manuscript. The sav ed MCMC output for the real data analysis and the simulation study are provided, but they can of course be reproduced by re-running the provided script ﬁles and data. The ﬁgures presented in the paper , along with the ﬁgures delegated to the Supplementary Material are contained in these ﬁles. 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Bayesian hidden Markov models for latent variable labeling assignments in conflict research: application to the role ceasefires play in conflict dynamics

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