Identifiability of total effects from abstractions of time series causal graphs

TL;DR

The paper tackles identifiability of the total effect from observational time-series when only abstractions of the full time-series causal graph are available. It proves that total effects are always identifiable under extended summary causal graphs (ESCGs) and provides explicit adjustment sets for estimation, while offering sufficient conditions and practical adjustment sets for summary causal graphs (SCGs). The approach hinges on comparing all candidate full-time causal graphs compatible with the abstractions and employing backdoor-style adjustments that remain valid across these candidates. This yields actionable guidance for impact analysis in dynamic systems where full temporal graphs are unavailable, with implications for domains such as nephrology, finance, system monitoring, and thermoregulation. Future work could address necessary and sufficient conditions for SCGs, multivariate interventions, and potential hidden confounding.

Abstract

We study the problem of identifiability of the total effect of an intervention from observational time series in the situation, common in practice, where one only has access to abstractions of the true causal graph. We consider here two abstractions: the extended summary causal graph, which conflates all lagged causal relations but distinguishes between lagged and instantaneous relations, and the summary causal graph which does not give any indication about the lag between causal relations. We show that the total effect is always identifiable in extended summary causal graphs and provide sufficient conditions for identifiability in summary causal graphs. We furthermore provide adjustment sets allowing to estimate the total effect whenever it is identifiable.

Figures (7)

• Figure 1.1: Illustration: (a) three FTCGs, (b) three ESCGs derived from them, (c) the SCG which can be derived from any FTCG in (a) and any ESCG in (b). Consider $f(y_t |do(x_{t-1}))$, red vertex: the variable we intervene on, blue vertex: the response we are considering. Bold edges correspond to directed paths from $X_{t-1}$ to $Y_t$, and gray vertices correspond to nodes with different status depending on the FTCG (see Definition \ref{['def:ambiguous_vertices']}).
• Figure 5.1: Three SCGs and a total effect which is identifiable. Each pair of red and blue vertices in the FTCGs represents the total effect we are interested in, and we precise the total effect and the lag considered in the caption. This illustrates Lemma \ref{['lemma:2']} (Figure a-b) and Lemma \ref{['lemma:3']} (Figure c).
• Figure 5.2: An example of an SCG $\mathcal{G}^s_1$ in (a) satisfying Condition 1 in Theorem \ref{['Thm:identification_summary']} and two candidate FTCGs in (b) and (c). Each pair of red and blue vertices in the FTCGs represents the total effect we are interested in. Gray vertices are ambiguous: they are on an active backdoor path in (b) and belong to a directed path in (c) (bold edges indicate direct paths from $X_{t-1}$ to $Y_t$).
• Figure 5.3: An example of an SCG $\mathcal{G}^s_2$ in (a) satisfying Condition 2a in Theorem \ref{['Thm:identification_summary']} and two candidate FTCGs in (b) and (c). Each pair of red and blue vertices in the FTCGs represents the total effect we are interested in. Gray vertices are ambiguous: they are on an active backdoor path in (b) and belong to a directed path in (c) (bold edges indicate direct paths from $X_{t-1}$ to $Y_t$).
• Figure 5.4: An example of an SCG $\mathcal{G}^s_3$ in (a) satisfying Condition 2b in Theorem \ref{['Thm:identification_summary']} with respect to $\Pr(y_t\mid do(x_{t-2}))$ and two candidate FTCGs in (b) and (c). Each pair of red and blue vertices in the FTCGs represents the total effect we are interested in. Gray vertices are ambiguous: they constitute a backdoor path in (b) and belong to a directed path in (c) (bold edges indicate direct paths from $X_{t-2}$ to $Y_t$).

Theorems & Definitions (37)

• Definition 1: Full-time causal graph (FTCG), Figure \ref{['fig:example_FTCG']}
• Definition 2: Extended summary causal graph (ESCG), Figure \ref{['fig:example_ESCG']}
• Definition 3: Summary causal graph (SCG), Figure \ref{['fig:example_SCG']}
• Definition 4: Identifiability of total effects in ESCGs and SCGs
• Definition 5: Backdoor criterion over all candidate FTCGs
• Corollary 1
• Theorem 1
• Proposition 1
• Theorem 2
• Definition 6: Ambiguous vertices