Dynamic Causal Inference (DCI)

• DCI is a statistical framework that identifies the true causal drivers of time-stamped relational events in network data using likelihood-based invariance criteria.
• It employs a nested case–control partial likelihood and Pearson risk functional to rigorously select minimal causal parent sets from observational data.
• Applied to real datasets like bikeshare trips, DCI uncovers dynamic effects such as time-of-day impact and spatial competition with high stability and predictive accuracy.

Dynamic Causal Inference (DCI) is a principled statistical framework for identifying and estimating the causal drivers of temporal dynamics in network or relational event data. Unlike purely descriptive network models, which capture correlative structure, DCI provides machinery for learning which covariates act as true causal parents for time-stamped relational events, using a likelihood-based approach with population-level invariance criteria. The DCI methodology enables consistent causal discovery from a single dynamic observational environment, without requiring environmental perturbations or interventional data. The framework is grounded in stochastic process modeling and partial likelihood theory, with rigorous identifiability, large-sample properties, and operational algorithms for real-world applications (Lembo et al., 5 Mar 2025).

1. Formal Relational Event Model and Structural Assumptions

DCI models event data as marked point processes

$$\mathcal{M} = \{(t_i, (s_i, r_i)) \mid i = 1, \ldots, n \}$$

where each event $i$ at time $t_i$ is a directed interaction from sender $s_i \in V_1$ to receiver $r_i \in V_2$. For each candidate dyad $(s, r)$, an associated counting process $N_{sr}(t)$ tracks the number of events up to time $t$. Covariates—often $p$-dimensional vectors $X_{sr}(t)$ tracking time-varying, node-level, and dyadic endogenous features—evolve via underlying structural equations permitting exogenous noise.

The central structural assumption is that, conditional on the “risk set” $\mathcal{R}(t) \subset V_1 \times V_2$, each dyad’s event hazard admits a structural form:

$\lambda_{sr}(t) = \mathbbm{1}\{(s,r) \in \mathcal{R}(t)\} \lambda_0(t) \exp\left\{ f_{PA}(X_{sr, PA}(t)) \right\}$

where $f_{PA}$ is the unknown (possibly nonlinear) causal risk function, depending on a subset $PA \subset \{1, \dots, p\}$ of "causal parent" covariates. The model generalizes standard (multiplicative) intensity models by enforcing that only a core, minimal subset of covariates exerts direct causal influence on event hazard.

A discrete-time structural equations picture underlies this setup: each covariate $X_j(t) = g_j(\text{history},\epsilon_j)$ is itself a function of past events and exogenous noise. The augmented dynamic DAG links $\{ X_1, \ldots, X_p, N \}$, with the Markov blanket of $N$ comprising direct causes and children of $N$’s hazard function.

2. Identification Strategy: Population Invariance and Likelihood

DCI advances population-level identifiability conditions for the causal risk function $f_{PA}$ beyond standard maximum likelihood. For any candidate $f_S(X_{sr, S}(t))$ (where $S \subset \{1,\ldots,p\}$), consider the nested case–control partial log-likelihood:

$$\ell(f) = \sum_{i=1}^n \left[ \Delta_i f - \log\left( 1 + e^{\Delta_i f} \right) \right]$$

where $$\Delta_i f = f(X_{s_i r_i}(t_i)) - f(X_{s_i^* r_i^*}(t_i))$$ and $(s_i^*, r_i^*)$ is a risk-set-matched non-event.

Introduce the Pearson risk functional:

$$R^P(f) = \mathbb{E} \left[ \frac{(Y - \dot{b}(\Delta f))^2}{\ddot{b}(\Delta f)} \right], \quad b(\theta) = \log(1+e^\theta), \,\, Y\equiv1$$

DCI’s identification theorem establishes:

• MLE condition: The causal function $f_{PA}$ uniquely (almost everywhere) maximizes expected log-likelihood, but this alone does not distinguish parents from children.
• Invariant Pearson criterion: Only the true causal risk $f_{PA}$ satisfies $R^P(f_{PA}) = 1$. Any candidate $f' \neq f_{PA}$ cannot satisfy both MLE and invariance conditions simultaneously.

By requiring both, DCI ensures unique recovery (up to null-sets) of the causal structuring set $PA$ and its functional effect $f_{PA}$ from population data.

3. Algorithmic Implementation and Statistical Inference

The empirical DCI algorithm operationalizes these criteria using sample-level analogues:

1. Enumerate all non-empty $S \subset \{1, \dots, p\}$.
2. For each $S$, fit a penalized logistic regression (nested case–control likelihood) for the basis-expanded risk function $f_S(x_S) = \beta_S^T \psi(x_S)$.
3. Compute the empirical Pearson statistic $R_n(S)$ (sum over $n$ events).
4. Accept $S$ if $R_n(S)$ falls inside a two-sided $\chi^2$ confidence interval.
5. Among accepted $S$, select the lowest Bayesian Information Criterion (BIC) model.

Worst-case computational cost scales as $O(2^p n d\,\text{iters})$, but forward-backward search or marginal screening is recommended for tractable execution. The algorithm returns the estimated minimal causal parent set $\hat S$ and corresponding risk function $\hat f$.

Statistically, as $n \to \infty$, DCI is consistent for both parent set recovery and risk function estimation under regularity (identifiability, positive-definiteness of Fisher information). Notably, all identification and estimation is based solely on data from a single environment—multi-environment or interventional data is not required.

4. Empirical Illustration: DCI for Dynamic Relational Events

An application to Washington D.C. Capital Bikeshare trip data (July 2023, $n=20{,}000$ events) demonstrates DCI’s practical use. Covariates included global weather, time-of-day, node-level spatial competition, and dyadic endogenous measures (past dyad usage—repetition, reciprocity) and geodesic distance.

Applying DCI yielded:

• Causal drivers: Four covariates—nonlinear time of day; linear sender-station competition ($\beta \approx -0.413, p<10^{-4}$); nonlinear repetition (daily cycle); and nonlinear reciprocity (decaying, secondary midday peak)—formed the minimal invariant parent set.
• Predictive and stability metrics: The causal model’s deviance was within 2% of the full predictive BIC-best model; bootstrap selection frequency of the causal parent set exceeded 90%.
• Interpretation: The selected effects captured rush-hour cycling, station spatial underprovision, regularity of trip repetition, and nuanced patterns of route-return behavior.

The entire model-selection process on 511 submodels completed in ~15 minutes on a standard laptop, demonstrating scalability for moderate $p$.

5. Relationship to Existing Dynamic Network and Causal Models

DCI generalizes relational event hazard models by enforcing causality-justified covariate selection via likelihood invariance, rather than relying on predictive performance alone. Unlike static DAG or graphical model learning, DCI is explicitly tailored for dynamic, marked point processes and directly exploits risk-set sampling analogously to partial likelihood approaches in survival analysis. Unlike synthetic control or time-series methods, DCI is structurally agnostic except for the hazard’s semiparametric exponential form.

This framework is distinct from direct information-theoretic or purely descriptive time-evolving networks, providing causal interpretability grounded in population quantity invariance, which standard dynamic network or Hawkes process models lack.

6. Limitations, Extensions, and Future Directions

While the DCI methodology provides a flexible and consistent mechanism for discovering dynamic causal structure, practical limitations include computational cost for large $p$, reliance on correctly specified risk sets, and potential sensitivity to unmeasured confounding among covariates. The method as formulated assumes no simultaneous events (point-process regularity) and nonparametric smoothness.

Potential extensions include:

• Adaptation to high-dimensional settings via penalized screening or Bayesian model averaging;
• Generalization to multienvironment settings for detecting stable vs changeful causal structure;
• Incorporation of temporal lags, memory effects, or network-specific endogeneity (e.g., feedback loops).

The framework, by providing both formal identification and empirical algorithms from (potentially) single-environment data, represents an advance for dynamic network causal analysis, with applications in temporal relational events across social, biological, and technological systems (Lembo et al., 5 Mar 2025).