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Conflict Forecasting via Conformal Prediction for Markov Processes

This paper proposes using conformal prediction on temporally dependent Markov process data to generate robust prediction sets for future conflict states, offering a superior alternative to likelihood-based point predictions by providing valid uncertainty quantification despite violations of the exchangeability assumption.

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

Published 2026-04-29

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CC BY 4.0

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

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather for a country, but instead of rain or shine, you are predicting whether that country is in Peace, Escalation (tension rising), War, or De-escalation (calming down).

For a long time, experts tried to give a single, specific answer: "Next month, Country X will be at War." But the authors of this paper argue that this is dangerous. If you are wrong, the consequences are huge. Instead of a single guess, they want to give a list of possibilities that is statistically guaranteed to include the truth most of the time.

Here is a breakdown of their approach using simple analogies:

1. The Problem: The "Crystal Ball" vs. The "Safety Net"

Traditional methods (called Likelihood-Based Prediction) act like a crystal ball. They look at the past, build a mathematical model of how countries usually behave, and predict the most likely future path.

The Flaw: If your model is slightly wrong (which is very likely with complex human conflicts), your "most likely" prediction could be completely off. Also, as you look further into the future, these models tend to get lazy. They stop predicting specific paths and just say, "Eventually, everything settles into a boring, average mix of peace and war." They lose the excitement (or danger) of the specific journey.

The authors propose using Conformal Prediction (CP). Think of this not as a crystal ball, but as a safety net.

The Goal: Instead of guessing the one right path, we want to catch the true path in a net of possibilities. We want to say, "We are 90% sure the future will look like one of these 50 scenarios."
The Benefit: Even if our understanding of how conflicts work is imperfect, this method guarantees that the net is wide enough to catch the truth 90% of the time.

2. The Challenge: The "Chain Reaction"

The tricky part is that conflict isn't random like flipping a coin. It's a Markov Process. This means the future depends heavily on the immediate past.

Analogy: If you are in a "War" state today, you are very likely to be in "War" tomorrow. But if you are in "Peace," you might stay in "Peace."
The Statistical Hurdle: Standard statistical tools (like Conformal Prediction) usually assume that every piece of data is independent (like rolling a die). But in a conflict, today's state is tied to yesterday's. It's like trying to predict a line of dominoes falling; you can't treat each domino as if it's unrelated to the one before it.

3. The Solution: The "Permutation Puzzle"

To fix the "chain reaction" problem, the authors use a clever trick called Partial Exchangeability.

The Metaphor: Imagine you have a long string of colored beads representing a country's history (Red=War, Blue=Peace).
The Trick: Instead of looking at the beads in order, the algorithm looks at the chunks of beads between specific colors. It asks: "If I swap the order of these specific chunks, does the overall 'story' of the conflict change?"
The Result: If the chunks can be swapped without changing the fundamental rules of how the conflict moves, the algorithm treats them as interchangeable. This allows them to build their "safety net" (prediction set) without needing to perfectly understand the complex rules of the conflict. They just need to know the rules of the chunks.

4. What They Found: The "Long-Term Vision"

The authors tested this on real data from 86 countries (using data from the Uppsala Conflict Data Programme). They compared their "Safety Net" (CP) against the traditional "Crystal Ball" (Likelihood).

Short Term: Both methods look similar. They both give good lists of possibilities for the next month or two.
Long Term (The Big Difference):
- The Crystal Ball (Likelihood): As they looked further ahead (6 months, 12 months), the Crystal Ball got "bored." It stopped predicting specific, wild swings in conflict and just predicted a boring, average mix of states. It effectively said, "In the long run, it's just a mix of peace and war."
- The Safety Net (CP): This method kept its edge. Even 12 months out, it maintained a diverse list of possibilities. It didn't collapse into a boring average. It kept the "uncertainty" alive, acknowledging that a country currently in peace could suddenly jump to war, or vice versa, and that this possibility should remain in the prediction list.

5. The One Weakness: The "Stuck Record"

The paper admits one major limitation. Imagine a country that has been in "Peace" for 400 months in a row (like Sweden).

The Issue: When the data is too repetitive, the "Permutation Puzzle" trick gets confused. The algorithm realizes that if the country stays in peace, the math works perfectly. But if the algorithm tries to predict a future where the country suddenly goes to war, the math gets messy, and the "Safety Net" becomes huge and inefficient.
The Fix: They tried a "naive" fix (pretending the country goes back to peace at a fake future time point) to make the math work, but they admit this changes the meaning of the prediction slightly. This is an area they need to work on.

Summary

This paper is about building a better forecasting tool for war and peace.

Old Way: Guess the single most likely path. (Risky if the model is wrong; gets boring over time).
New Way (This Paper): Create a guaranteed "safety net" of many possible paths. It is robust against errors and, crucially, keeps the possibility of dramatic changes alive even when looking far into the future.

It's the difference between a weatherman saying, "It will rain," versus saying, "There is a 90% chance it will be either heavy rain, a thunderstorm, or a sudden flood, and here is the list of those possibilities."

1. Problem Statement

The paper addresses the challenge of forecasting conflict dynamics (e.g., peace, escalation, war, de-escalation) for countries over time.

The Core Issue: Traditional point predictions (predicting a single future state sequence) are insufficient for high-stakes policy decisions because the penalty for error is extreme. Decision-makers require uncertainty quantification in the form of prediction sets (ranges of plausible outcomes).
The Statistical Challenge: Most existing uncertainty quantification methods, such as Conformal Prediction (CP), rely on the assumption that data points are exchangeable. However, conflict data is temporal and modeled as a Markov process, where observations are dependent on previous states, violating the exchangeability assumption.
The Goal: To develop a method that generates valid prediction sets for future conflict state sequences using Markov processes, offering robustness against model misspecification compared to traditional likelihood-based approaches.

2. Methodology

The authors propose a framework that adapts Conformal Prediction for discrete-state Markov processes by leveraging the concept of partial exchangeability.

A. Data Representation

States: Conflict is modeled as a discrete-state Markov process with four states: (1) Peacetime, (2) Escalation, (3) War, (4) De-escalation.
Data: The input is a sequence of observed states $\{X_t\}_{t=1}^T$ . The goal is to forecast a sequence of length $T_1$ into the future.

B. Conformal Prediction (CP) Adaptation

Standard CP assumes exchangeability of individual observations. Since Markov data is not exchangeable, the authors adopt the approach from Nettasinghe et al. (2023):

Partial Exchangeability: Instead of permuting individual time points, the algorithm treats state-sequences as the unit of exchangeability.
i-Blocks: The augmented sequence (calibration data + proposed future sequence) is decomposed into "i-blocks." An i-block is a segment of the sequence starting with state $i$ and containing no further occurrences of $i$ .
Permutation Group: The algorithm generates a permutation group of these i-blocks. If the proposed future sequence, when combined with the calibration data, can be permuted to look like the original data without changing the transition counts, it is considered "exchangeable" in a partial sense.
Algorithm:
- For every possible future sequence, the algorithm calculates a non-conformity score based on the likelihood of the sequence under the estimated transition matrix.
- It permutes the i-blocks of the augmented sequence to generate a distribution of scores.
- A p-value is calculated by comparing the score of the original sequence against the permuted scores.
- Sequences with p-values greater than the significance level $\alpha$ are included in the prediction set $C_{1-\alpha}$ .

C. Baseline Comparison: Likelihood-Based Prediction

The authors compare their CP method against a standard Likelihood-Based approach:

A Markov model is fitted to the calibration data to estimate the transition probability matrix ( $\hat{P}$ ).
Prediction sets are constructed by selecting the highest probability sequences until the cumulative probability mass reaches $1-\alpha$ .
Limitation: This method assumes the model is correctly specified and offers no finite-sample guarantees if the model is misspecified.

3. Key Contributions

Adaptation of CP to Markov Processes: The paper successfully applies Conformal Prediction to temporally dependent data by utilizing partial exchangeability via i-block permutations, bypassing the strict exchangeability requirement.
Robustness to Model Misspecification: Unlike likelihood-based methods, the CP approach provides finite-sample marginal coverage guarantees regardless of whether the underlying Markov model is perfectly specified.
Handling Long-Horizon Uncertainty: The study demonstrates that CP preserves trajectory uncertainty over long forecast horizons, whereas likelihood-based methods tend to collapse into stationary distributions, potentially missing rare but critical events (e.g., sudden escalation).
Empirical Validation: The method is validated through Monte Carlo simulations and applied to real-world conflict data from the Uppsala Conflict Data Programme (UCDP) covering 191 countries (1990–2025).

4. Results

Simulation Study

Calibration: The CP method produced empirical coverage rates that closely matched the target confidence levels (e.g., 90% target $\approx$ 90% actual) across all forecast lengths ( $T_1$ ).
Likelihood Over-coverage: The likelihood-based method tended to over-cover (produce sets larger than necessary) at lower confidence levels because the discrete nature of the state space meant a few high-probability sequences captured most of the mass.
Set Composition: As the forecast horizon ( $T_1$ ) increased, likelihood-based sets converged to the stationary distribution of the Markov chain, losing information about the specific starting trajectory. CP sets remained diverse, reflecting the uncertainty of the specific path.

Real Data Analysis (UCDP)

Coverage: Applied to 86 countries with sufficient conflict data, CP maintained near-nominal coverage. The likelihood-based approach again showed over-coverage at low confidence levels.
The "Zero-Probability" Issue: A critical finding occurred at $1-\alpha = 1.00$ . The likelihood-based method failed to cover true sequences if the estimated transition matrix assigned zero probability to a transition not seen in the training data (under-coverage). CP avoided this because it evaluates sequences based on permutation validity rather than strict model probability, allowing it to include "unseen" but plausible transitions.
Case Studies (Ethiopia, CAR, etc.):
- For countries with ongoing conflict, the likelihood-based method predicted a rapid return to peace (convergence to stationary distribution) as the horizon extended (e.g., 12 months).
- The CP method maintained a high probability of continued conflict, aligning better with the observed persistence of conflict in these regions.

5. Limitations and Future Work

Efficiency in Dominant States: The CP method becomes inefficient when the calibration data consists almost entirely of a single state (e.g., a country in perpetual peace). In these cases, the number of "i-blocks" for non-dominant states drops to zero or one, causing the algorithm to accept a vast number of unlikely sequences to satisfy the coverage guarantee.
Proposed Mitigation: The authors suggest a "naive" fix of adding a synthetic future time point forcing a return to the dominant state to improve efficiency, though this alters the interpretation of the forecast.
Covariates: The current study does not incorporate covariates (e.g., economic indicators, political regime) into the transition probabilities, though the framework is theoretically extendable to such scenarios.

6. Significance

This paper is significant for conflict forecasting and statistical machine learning because:

Policy Relevance: It provides decision-makers with valid, calibrated uncertainty sets rather than single-point guesses, which is crucial for risk management in geopolitics.
Methodological Advance: It solves a fundamental theoretical hurdle (exchangeability in time series) by applying partial exchangeability to Markov chains, offering a robust alternative to parametric models that may be misspecified.
Long-Term Forecasting: It highlights the danger of relying on stationary distributions for long-term forecasts in non-stationary environments (like war), advocating for methods that preserve trajectory-specific uncertainty.