arxiv: 2604.25139 · v1 · submitted 2026-04-28 · 📊 stat.ME · stat.AP· stat.ML

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Conflict Forecasting via Conformal Prediction for Markov Processes

Aditya Basarkar , Emmett B. Kendall , David Randahl , Jonathan P. Williams , Gudmund H. Hermansen

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Pith reviewed 2026-05-07 15:44 UTC · model grok-4.3

classification 📊 stat.ME stat.APstat.ML

keywords conformal predictionMarkov processconflict forecastingprediction setstime seriesuncertainty quantificationstate sequencesdiscrete states

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The pith

Conformal prediction on Markov processes produces valid sets of possible future conflict sequences.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to apply conformal prediction to time series of conflict states that follow a Markov process, creating sets of possible future sequences rather than single-point forecasts. These sets come with a coverage guarantee that the true sequence will be included at a chosen probability level, and the method stays valid even if the fitted model is not exactly correct. The authors compare this directly to likelihood-based predictions under the Markov assumption and demonstrate the approach on real data for multiple countries. They also note practical limits that arise because sequential dependence breaks the exchangeability usually required for conformal methods.

Core claim

Conformal prediction can be used on temporally dependent data assumed to arise from a discrete-state Markov process to obtain prediction sets of possible future conflict state-sequences, yielding valid uncertainty quantification that is robust to model misspecification and outperforming point predictions when the cost of error is high, as shown by comparisons with likelihood-based strategies and by producing real forecasts across countries.

What carries the argument

Conformal prediction sets built from nonconformity scores on observed Markov trajectories to produce multi-step-ahead regions for discrete state sequences.

If this is right

Policy decisions receive intervals of possible conflict paths with explicit coverage guarantees instead of single guesses that carry extreme penalties when wrong.
The same conformal procedure remains usable when the Markov model is only approximately correct, unlike likelihood methods that can degrade under misspecification.
Real-data forecasts for multiple countries illustrate how the sets can be computed and interpreted for actual policy-relevant horizons.
The approach highlights the need to adjust standard conformal techniques when data exhibit temporal dependence rather than full exchangeability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same construction could be tested on other sequential categorical data, such as economic regime switches or ecological state changes, where dependence is also present.
Efficiency gains might come from developing Markov-specific nonconformity scores that exploit the transition structure rather than treating trajectories as generic sequences.
If the coverage guarantee holds in practice, the sets could serve as inputs to downstream decision models that optimize actions over ranges of possible futures.

Load-bearing premise

The sequence of conflict states for each country follows a discrete-state Markov process so that conformal prediction still achieves its nominal coverage despite the dependence that violates exchangeability.

What would settle it

Future observed conflict sequences falling outside the conformal prediction sets at a rate substantially above the nominal level, for example more than 5 percent of the time when 95 percent coverage is claimed.

Figures

Figures reproduced from arXiv: 2604.25139 by Aditya Basarkar, David Randahl, Emmett B. Kendall, Gudmund H. Hermansen, Jonathan P. Williams.

**Figure 1.** Figure 1: All allowable state transitions between the four conflict-states (left), and the corresponding adjacency matrix (right), where states 1, 2, 3, and 4 correspond to peacetime, escalation, war, and deescalation, respectively. observations of states 2, 3, and/or 4), and the proportion of instances of peacetime (state 1) does not exceed 0.99 for a given country. In summary, the goal of this rule is to exclude … view at source ↗

**Figure 2.** Figure 2: Reliability curves illustrating the relationship between the empirical coverage and the target coverage for both the CP approach (left) and likelihood-based prediction approach (right) for the simulated data. The empirical coverage is then determined as the proportion of the 500 prediction sets that contain the true forecasted state-sequence view at source ↗

**Figure 3.** Figure 3: Visualization of the prediction sets for a single simulated state-sequence resulting from the conformal (left) and likelihood-based (right) prediction approaches. Note that the state at time T (i.e., XT ) is not being forecasted; however, the realized value of XT is being plotted because the possible forecasted state-sequences (XT +1, . . . , XT +T1 ) depend on XT by way of the Markov assumption. Next, view at source ↗

**Figure 4.** Figure 4: Reliability curves illustrating the relationship between the empirical coverage and the target coverage for both the CP approach (left) and likelihood-based prediction approach (right) for the real conflict data. application of CP leads to an empirical coverage quite close to the nominal coverage for all T1. Contrarily, the likelihood-based approach tends to over-cover for all values of T1 at lower confide… view at source ↗

**Figure 5.** Figure 5: Composition of the prediction sets of forecasted state-sequences using CP (top row) and likelihood-based prediction (bottom row), where T1 = 6 and 1 − α = 0.80. retain greater coverage of plausible state-sequences across all forecast steps, reflecting the underlying trajectory uncertainty rather than steady-state behavior. In practical terms, this means CP sets are better positioned to capture the true dyn… view at source ↗

**Figure 6.** Figure 6: Composition of the prediction sets of forecasted state-sequences, where T1 = 12 and 1−α = 0.80. 5 Limitations Recall the data cleaning procedure from Section 4.1. In particular, we exclude countries where a single state perpetuates for the entire duration of the observed time frame. What happens to the results when we do include such countries? Consider Sweden, for example, where the country is in peacetim… view at source ↗

**Figure 7.** Figure 7: Composition of prediction sets for the special case where all of the calibration data are state 1. These data correspond to Sweden. S(I) will be large whereas S(π) will be small for almost all π ∈ Π. In other words, when X1 = . . . = XT = 1 and i = 1, the number of i-blocks (and therefore the number of possible permutations) is large enough to reject the inclusion of state-sequences in the prediction set t… view at source ↗

**Figure 8.** Figure 8: Illustration of how the number of permutable i-blocks changes depending on the last state, in the special case that all of the training/calibration data are the same state. Note that depending on the realizations for (XT +1, . . . , XT +5), it is possible for I0 to extend to further time points (i.e., beyond time T) for cases 2, 3, and 4. based approach where the prediction set contains one state-sequence,… view at source ↗

read the original abstract

Whether or not a country is at war, or experiencing escalating or deescalating levels of conflict, has massive ramifications on a country's national and foreign policy. Given a country's history of conflict, or lack thereof, future predictions about the war-status of a country are valuable information. In this paper, we present the use of conformal prediction on temporally-dependent data to obtain prediction sets of possible future conflict state-sequences. More specifically, we compare the results of conformal prediction to a likelihood-based prediction strategy when the data are assumed to come from a discrete-state Markov process. A point-prediction may not supply sufficient information because the penalty for a wrong prediction is extreme, and so we consider a machine learning alternative that gives valid uncertainty quantification and is robust to model misspecification. In the data analysis, we present real forecasts of conflict dynamics across multiple countries. Lastly, we comment on the possible limitations of existing approaches for applying conformal prediction to Markovian data, where the exchangeability assumption is violated.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies conformal prediction to get sets for Markov conflict sequences and compares it to likelihood methods on real data, but the dependence fix for coverage is not shown clearly enough.

read the letter

The core move is taking conformal prediction, which normally needs exchangeable data, and trying to use it for future conflict state sequences under a discrete-state Markov assumption. They run it on real country data and pit the resulting sets against a likelihood-based predictor. That domain application is the main thing on offer, and the real-data forecasts give a concrete sense of what the sets look like in practice. The motivation is also reasonable: point forecasts are too brittle when the cost of being wrong is high, so sets with some robustness to misspecification make sense for this setting. The comparison to likelihood methods is at least a direct benchmark rather than an abstract one. The soft spot is exactly the one the stress-test flags. Standard split conformal gets its finite-sample guarantee from exchangeability of the scores, but a Markov chain introduces dependence, so the scores are not exchangeable unless the procedure conditions on the last state, blocks the calibration set, or uses some other adjustment. The abstract notes that existing approaches have limitations under this violation but does not say what construction they actually use here. Without that detail the coverage claim and the robustness statement rest on an unverified step. The paper is aimed at applied statisticians and conflict researchers who already work with Markov models and want uncertainty sets instead of single predictions. A reader in that group could extract the data example and the basic comparison, but anyone wanting to implement or extend the method would need the missing technical piece. I would send it to referees so they can check whether the dependence adjustment actually delivers valid sets; the work is not incoherent on its own terms, just underspecified on the load-bearing part.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes the use of conformal prediction on temporally dependent data to generate prediction sets for sequences of future conflict states across countries. It assumes the underlying process is a discrete-state Markov chain, compares the conformal approach to likelihood-based prediction, presents real-data forecasts, and discusses limitations arising from the violation of exchangeability.

Significance. If a valid finite-sample coverage guarantee can be established for the conformal sets under Markov dependence, the work would provide a useful non-parametric tool for uncertainty quantification in high-stakes forecasting where point predictions are insufficient. The real-data application to conflict dynamics adds practical value and allows direct comparison to parametric alternatives.

major comments (2)

[Abstract and §2] Abstract and §2 (Method): the central claim of 'valid uncertainty quantification' for future state-sequences rests on an unspecified adaptation of conformal prediction to non-exchangeable Markov data. Standard split conformal requires exchangeable calibration scores for the coverage bound, yet no blocking, state-conditioning, or mixing adjustment is described; without this construction the robustness-to-misspecification claim cannot be evaluated.
[§4] §4 (Data Analysis): the reported real forecasts and comparison to likelihood methods are presented without accompanying coverage diagnostics or simulation checks under the Markov assumption. If the procedure does not explicitly restore exchangeability (e.g., via last-state conditioning), the empirical results do not substantiate the theoretical validity asserted in the abstract.

minor comments (2)

[§2] Notation for the nonconformity score and the resulting prediction set C(X_{n+1}) should be introduced with an explicit equation rather than described only in prose.
[§5] The discussion of limitations of existing conformal approaches for Markovian data would benefit from citing the specific references being critiqued.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed and insightful comments, which have helped us identify areas for improvement in our manuscript. Below we provide point-by-point responses to the major comments and indicate the revisions we intend to make.

read point-by-point responses

Referee: [Abstract and §2] Abstract and §2 (Method): the central claim of 'valid uncertainty quantification' for future state-sequences rests on an unspecified adaptation of conformal prediction to non-exchangeable Markov data. Standard split conformal requires exchangeable calibration scores for the coverage bound, yet no blocking, state-conditioning, or mixing adjustment is described; without this construction the robustness-to-misspecification claim cannot be evaluated.

Authors: We thank the referee for highlighting this important point. Our manuscript applies the standard split conformal prediction procedure to the Markov chain data without introducing a new adaptation such as blocking or state-conditioning to restore exchangeability. The discussion section explicitly comments on the limitations arising from the violation of the exchangeability assumption. The robustness claim refers to the non-parametric nature of conformal prediction, which does not require correct specification of the transition probabilities, unlike the likelihood-based approach. However, we agree that the finite-sample coverage guarantee does not hold strictly due to dependence. In the revision, we will clarify this distinction in §2 and the abstract, and add a note that the coverage is heuristic under the Markov model. This constitutes a partial revision as the main empirical comparison is retained. revision: partial
Referee: [§4] §4 (Data Analysis): the reported real forecasts and comparison to likelihood methods are presented without accompanying coverage diagnostics or simulation checks under the Markov assumption. If the procedure does not explicitly restore exchangeability (e.g., via last-state conditioning), the empirical results do not substantiate the theoretical validity asserted in the abstract.

Authors: The referee is correct that §4 presents real-data forecasts without simulation-based coverage checks. Because the forecasts are for future unobserved sequences, direct coverage cannot be assessed on the real data. We will add a new subsection with Monte Carlo simulations under controlled Markov processes to evaluate the empirical coverage of the conformal sets and compare it to the nominal level. This will help substantiate the practical performance even if the theoretical guarantee is approximate. We plan to include these diagnostics in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper applies conformal prediction to temporally dependent Markov chain data for conflict state-sequence forecasting and compares the resulting prediction sets against a likelihood-based baseline under the Markov assumption. The abstract explicitly flags the exchangeability violation and comments on limitations of prior approaches without claiming a new theorem or fit that reduces to the inputs by construction. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citations appear in the provided text; the comparison to likelihood methods supplies an independent benchmark. The derivation therefore remains self-contained against external statistical procedures.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on modeling conflict data as a discrete-state Markov process and on the feasibility of conformal prediction under violated exchangeability; no free parameters, invented entities, or additional axioms are mentioned in the abstract.

axioms (2)

domain assumption Conflict dynamics follow a discrete-state Markov process
Explicitly stated as the assumption under which likelihood-based predictions are compared.
ad hoc to paper Conformal prediction remains valid for temporally dependent sequences
Invoked by the proposal to obtain prediction sets despite the paper's own note on exchangeability violation.

pith-pipeline@v0.9.0 · 5486 in / 1225 out tokens · 45476 ms · 2026-05-07T15:44:54.658860+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

11 extracted references · 7 canonical work pages

[1]

The 2023/24 views prediction challenge: Predicting the number of fatalities in armed conflict, with uncertainty.Journal of Peace Research, 62(6):2070–2087,

H˚ avard Hegre, Paola Vesco, Michael Colaresi, Jonas Vestby, Alexa Timlick, Noorain Syed Kazmi, Angelica Lindqvist-McGowan, Friederike Becker, Marco Binetti, Tobias Boden- tien, et al. The 2023/24 views prediction challenge: Predicting the number of fatalities in armed conflict, with uncertainty.Journal of Peace Research, 62(6):2070–2087,

2023
[2]

Bayesian hidden markov models for latent variable labeling as- signments in conflict research: Application to the role ceasefires play in conflict dynamics

Jonathan P Williams, Gudmund H Hermansen, H˚ avard Strand, Govinda Clayton, and H˚ avard Mokleiv Nyg˚ ard. Bayesian hidden markov models for latent variable labeling as- signments in conflict research: Application to the role ceasefires play in conflict dynamics. Annals of Applied Statistics, 18(3):2034–2061,

2034
[3]

Forecasting densities of fatalities from state-based conflicts using observed markov models.arXiv preprint arXiv:2410.12374,

David Randahl and Johan Vegelius. Forecasting densities of fatalities from state-based conflicts using observed markov models.arXiv preprint arXiv:2410.12374,

work page arXiv
[4]

Algorithmic Learning in a Random World

ISBN 9783031066481. doi: 10.1007/978-3-031-06649-8. Anders Hjort, Jonathan P Williams, and Johan Pensar. Clustered conformal prediction for the housing market.Proceedings of Machine Learning Research, 230:1–21,

work page doi:10.1007/978-3-031-06649-8
[5]

Safe merging in mixed traffic with confidence.arXiv preprint arXiv:2403.05742,

Heeseung Bang, Aditya Dave, and Andreas A Malikopoulos. Safe merging in mixed traffic with confidence.arXiv preprint arXiv:2403.05742,

work page arXiv
[6]

Network traffic demand prediction with confidence

Mikhail Dashevskiy and Zhiyuan Luo. Network traffic demand prediction with confidence. InIEEE GLOBECOM 2008-2008 IEEE Global Telecommunications Conference, pages 1–5. IEEE,

2008
[7]

Conformal predictions under markovian data

Fr´ ed´ eric Zheng and Alexandre Proutiere. Conformal predictions under markovian data. arXiv preprint arXiv:2407.15277,

work page arXiv
[8]

Conformal inference for time series over graphs.arXiv preprint arXiv:2510.11049,

Sonakshi Dua, Gonzalo Mateos, and Sundeep Prabhakar Chepuri. Conformal inference for time series over graphs.arXiv preprint arXiv:2510.11049,

work page arXiv
[9]

Bronstein, and Filippo Maria Bianchi

Roberto Neglia, Andrea Cini, Michael M Bronstein, and Filippo Maria Bianchi. Rescp: Reservoir conformal prediction for time series forecasting.arXiv preprint arXiv:2510.05060,

work page arXiv
[10]

Adaptive conformal inference by particle filtering under hidden markov models.arXiv preprint arXiv:2411.01558,

Xiaoyi Su, Zhixin Zhou, and Rui Luo. Adaptive conformal inference by particle filtering under hidden markov models.arXiv preprint arXiv:2411.01558,

work page arXiv
[11]

Organized violence 1989–2024, and the challenges of identifying civilian victims.Journal of Peace Research, 62(4):1223–1240,

Shawn Davies, Ther´ ese Pettersson, Margareta Sollenberg, and Magnus ¨Oberg. Organized violence 1989–2024, and the challenges of identifying civilian victims.Journal of Peace Research, 62(4):1223–1240,

1989