On causal inference with marked point process data

Kjetil R{\o}ysland; Mats Julius Stensrud; P{\aa}l Christie Ryalen

arxiv: 2604.12977 · v1 · submitted 2026-04-14 · 📊 stat.ME

On causal inference with marked point process data

P{\aa}l Christie Ryalen , Mats Julius Stensrud , Kjetil R{\o}ysland This is my paper

Pith reviewed 2026-05-10 14:22 UTC · model grok-4.3

classification 📊 stat.ME

keywords causal inferencemarked point processesdynamic treatment regimesmartingale theoryg-formulasurvival analysiscounting processespotential outcomes

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The pith

Dynamic treatment regimes for marked point process data are identified using martingale analogues of consistency, exchangeability, and positivity.

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

The paper defines dynamic treatment regimes and their associated potential outcomes when data arrives as marked point processes that record timed events and marks in continuous time. It develops counterparts to the familiar consistency, exchangeability, and positivity assumptions, but expressed through martingale theory so that they apply directly to the stochastic-process structure of the data. These conditions suffice to identify causal effects and yield marginal g-formulas that recover the usual discrete-time expressions in special cases yet differ in general continuous-time settings. The construction links the counting-process methods of survival analysis to discrete-time causal inference. Readers working with irregular event data would therefore gain a principled way to estimate effects of time-varying interventions without forcing the data into fixed-time grids.

Core claim

What carries the argument

Martingale-theoretic analogues of the consistency, exchangeability, and positivity conditions for potential outcomes under dynamic treatment regimes in marked point processes.

Load-bearing premise

The martingale-theoretic formulation of the consistency, exchangeability, and positivity conditions is sufficient to identify causal effects for the defined potential outcomes in continuous-time marked point process data.

What would settle it

A data-generating process satisfying the martingale versions of consistency, exchangeability, and positivity yet producing a g-formula value that differs from the true marginal distribution of the potential outcome under the regime would disprove the identification result.

read the original abstract

We define dynamic treatment regimes and associated potential outcomes for data described by marked point processes (MPPs). These definitions motivate MPP analogues of the commonly used consistency, exchangeability, and positivity conditions that are sufficient for identifying effects in MPP data structures. The conditions are formulated based on martingale theory, which allows us to derive explicit identifying assumptions for data described by stochastic processes. The definitions and conditions align with well-established discrete-time results in important special cases. Thus, this work bridges the large literatures on survival (event history) analysis with counting processes in continuous time and causal inference with variables in discrete-time. After formulating a set of identification conditions, we derive and characterize marginal g-formulas. The g-formulas are generally different from those studied in related works, though they coincide in important special cases. We relate our findings to previous work on causal inference with (counting) processes, the classical survival literature, and the discrete-time causal inference literature.

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 defines dynamic regimes and potential outcomes for marked point processes, then uses martingale theory to get identification conditions and g-formulas that match known results in special cases.

read the letter

The main takeaway is that this work gives explicit definitions for dynamic treatment regimes and potential outcomes when the data are marked point processes, then translates the usual consistency, exchangeability, and positivity assumptions into martingale language so they can be checked with compensators and predictable processes. From there they derive marginal g-formulas. The authors show these reduce to the discrete-time versions and to classical survival results when the data are suitably restricted, which is the useful part of the bridge they are trying to build between the two literatures. The martingale setup supplies the predictability needed to make the assumptions operational in continuous time without obvious internal contradictions. That is real work if the derivations are clean. The g-formulas are stated to differ from earlier process-based papers except in the special cases, so the contribution sits in those general expressions rather than in a wholly new idea. On the soft side, the identification claim rests on the martingale analogues being sufficient, and while the stress-test found no circularity or hidden inconsistencies, the abstract alone does not let you verify the steps or check whether extra regularity conditions are required for general marked processes. There is also no mention of simulations or real data examples, so it is not yet clear how the framework behaves in practice or how sensitive the g-formulas are to estimation error. This is for readers who already know counting-process survival analysis and discrete-time causal methods and want to extend them to irregular event data. A serious referee should see it because the technical grounding is there and the gap it targets is genuine, even if the paper will need concrete checks and comparisons added in revision.

Referee Report

0 major / 1 minor

Summary. The paper defines dynamic treatment regimes and associated potential outcomes for data described by marked point processes (MPPs). It formulates martingale-theoretic analogues of the consistency, exchangeability, and positivity conditions as sufficient for identifying causal effects in MPP data structures. Marginal g-formulas are derived under these conditions; the framework is shown to align with discrete-time causal inference results in special cases while differing from some related continuous-time approaches in general, thereby bridging survival analysis with counting processes and discrete-time causal inference.

Significance. If the identification results hold, this work provides a significant extension of causal inference methods to continuous-time marked point process data. The martingale formulation supplies the necessary predictability and compensator structure to operationalize the identifying assumptions without internal inconsistencies. The explicit reduction to known discrete-time and classical survival results in special cases, combined with the derivation of generally distinct g-formulas, strengthens the contribution and offers a rigorous bridge between the survival/event-history and causal-inference literatures.

minor comments (1)

The abstract states that the g-formulas are 'generally different from those studied in related works' yet coincide in special cases; a brief, explicit contrast (e.g., one sentence referencing the key structural difference) would improve immediate clarity for readers familiar with the cited literatures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, as well as for the recommendation of minor revision. We are pleased that the work is viewed as providing a rigorous bridge between the marked point process literature, survival analysis, and discrete-time causal inference. The referee's assessment of the significance of the martingale-based identification conditions and g-formulas is appreciated.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper defines dynamic treatment regimes and potential outcomes for marked point processes, then formulates martingale-based analogues of standard identifying assumptions (consistency, exchangeability, positivity) drawn from established discrete-time causal inference and survival analysis. It derives marginal g-formulas that are shown to coincide with known results in special cases, without any step where a fitted parameter is relabeled as a prediction, a result is defined in terms of itself, or a load-bearing uniqueness claim reduces to an unverified self-citation. The central identification results rest on external martingale theory and prior non-overlapping literature rather than internal construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard causal assumptions adapted to stochastic processes and on martingale theory; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption Martingale theory supplies the correct framework for stating identifying assumptions for data generated by marked point processes.
Invoked to derive explicit versions of consistency, exchangeability, and positivity for continuous-time processes.
domain assumption The defined potential outcomes and dynamic regimes coincide with standard discrete-time definitions in important special cases.
Used to claim alignment with existing causal inference results.

pith-pipeline@v0.9.0 · 5470 in / 1211 out tokens · 31286 ms · 2026-05-10T14:22:45.611458+00:00 · methodology

discussion (0)

Reference graph

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= ( ˜T ′ 1, ˜X ′ 1).(66) Suppose then thatτ ′J > T ′ 1, and note thatC 1 = Ω′ by Algorithm 1. There are two cases to check: •Case I: The first observed event is a treatment event (of some typej∈J), and the regime is followed.By Algorithm 1, the first observed event is a treatment event if, for somej∈J,T j 1 =∧ h∈J T h 1 ∧T \J 1 . The regime is followed wh...

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by [33, II 8 Theorem 37]), Wt = I(τ a > t) P0<s≤t(1−Λ as)

In particular, it is a well-defined semimartingale, and (11) is well-defined, with the solution given in Proposition 2, (e.g. by [33, II 8 Theorem 37]), Wt = I(τ a > t) P0<s≤t(1−Λ as) . This is also a local martingale, as highlighted in Lemma 7. Since the only contri- bution to the product are at the times{θ k}, which are discontinuity points of Λ a, we g...

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Using these properties, the proof of (42) follow the lines of the proof of Theorem 1

Specifically, under the conditions (36), and (i), (ii) and (iii) of Theorem 3, the processE(K J) satisfies properties parallel to those established forW=E(K) in Lemma 7 and Lemma 8, withK J,N J,Λ J, andτ J playing the roles ofK a,N a,Λ a, andτ a, respectively. Using these properties, the proof of (42) follow the lines of the proof of Theorem 1. As in Lemm...

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□ AppendixF.Notation General mathematical notation

The derivation of the distributional representation on the right-hand side of (43) follows step for step the proof of Lemma 1 and is therefore omitted. □ AppendixF.Notation General mathematical notation. (Ω,F, P) The underlying abstract probability space wherePis a fixed, but arbitrary, probability measure x∧y,x∨yThe minimum ofxandy, and the maximum ofxan...

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= ( ˜T ′ 1, ˜X ′ 1).(66) Suppose then thatτ ′J > T ′ 1, and note thatC 1 = Ω′ by Algorithm 1. There are two cases to check: •Case I: The first observed event is a treatment event (of some typej∈J), and the regime is followed.By Algorithm 1, the first observed event is a treatment event if, for somej∈J,T j 1 =∧ h∈J T h 1 ∧T \J 1 . The regime is followed wh...

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by [33, II 8 Theorem 37]), Wt = I(τ a > t) P0<s≤t(1−Λ as)

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Using these properties, the proof of (42) follow the lines of the proof of Theorem 1

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□ AppendixF.Notation General mathematical notation

The derivation of the distributional representation on the right-hand side of (43) follows step for step the proof of Lemma 1 and is therefore omitted. □ AppendixF.Notation General mathematical notation. (Ω,F, P) The underlying abstract probability space wherePis a fixed, but arbitrary, probability measure x∧y,x∨yThe minimum ofxandy, and the maximum ofxan...

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