While Directed Acyclic Graphs (DAGs) provide a powerful framework for representing causal assumptions, the "acyclic" constraint is a significant limitation. Many real-world systems, particularly in biology, economics, engineering, and social sciences, feature feedback loops where a variable can influence itself indirectly through a sequence of other variables. Ignoring these cycles can lead to incorrect causal conclusions and flawed model predictions.

Understanding Cycles in Causal Systems

Consider a simple economic scenario: increased consumer Demand might lead firms to raise Prices, but higher Prices could subsequently dampen Demand. Representing this naively might suggest a cycle:

A simple directed graph illustrating a potential feedback loop between Demand and Price.

This representation, a Cyclic Directed Graph (CDG), immediately raises questions. Standard DAG semantics, based on non-recursive structural equations, break down. How do we interpret an intervention, say $do(Demand=d)$ , if Demand is also influenced by Price, which itself depends on Demand? Does the intervention break the incoming edge from Price, or does the system reach a new equilibrium?

The Problem with Simultaneity

The core issue often lies in representing potentially simultaneous influences. If the Demand -> Price -> Demand loop happens extremely quickly, within the same time observation window, it appears cyclic. However, true simultaneous causation is physically problematic and often masks an underlying temporal process.

For instance, the economic feedback loop likely unfolds over time:

Demand(t) influences Price(t) (or perhaps Price(t+1)).
Price(t) (or Price(t+1)) influences Demand(t+1).

Representing this explicitly requires incorporating time.

Temporal Unfolding: Resolving Cycles with Time

A common and often effective approach is to "unroll" the cycle over time, creating a DAG where nodes represent variables at specific time points. This transforms the CDG into a Dynamic Bayesian Network (DBN) structure over discrete time steps.

Let's unroll the Demand-Price example over two time steps:

A temporally unrolled graph (simplified DBN) resolving the Demand-Price cycle over two time steps. Dashed lines indicate potential autoregressive effects.

Now we have a DAG. The feedback loop is represented as a directed path across time slices (e.g., $Demand_t \rightarrow Price_t \rightarrow Demand_{t+1}$ ). Standard identification techniques, like the backdoor criterion or do-calculus, can potentially be applied to this unrolled graph, although it often introduces time-varying confounding. For example, estimating the effect of $Price_t$ on $Demand_{t+1}$ requires conditioning on $Demand_t$ (and possibly other relevant variables at time $t$ ).

Equilibrium Structural Causal Models

In some cases, particularly in econometrics or physical systems, researchers model the equilibrium state of a cyclic system directly using simultaneous equations within an SCM. For example, a linear cyclic SCM might look like:

\begin{align*} X &= \alpha Y + U_X \\ Y &= \beta X + U_Y \end{align*}

Here, $X$ causes $Y$ , and $Y$ causes $X$ . Assuming the system settles into a stable state, we can solve for $X$ and $Y$ in terms of the exogenous noises $U_X$ and $U_Y$ , provided $1 - \alpha\beta \neq 0$ .

\begin{align*} X &= \frac{1}{1 - \alpha\beta} (U_X + \alpha U_Y) \\ Y &= \frac{1}{1 - \alpha\beta} (\beta U_X + U_Y) \end{align*}

Identification of causal effects (e.g., the effect of intervening on $X$ ) in such equilibrium models depends heavily on the assumed functional forms (often linear) and the distributional properties of the noise terms. Do-calculus application requires careful consideration of the intervention's meaning within the equilibrium context. Often, interventions are conceptualized as manipulating the noise term or one specific equation while holding others fixed.

Cycles and Identification Algorithms

Standard identification algorithms based on do-calculus assume DAGs. When applied naively to graphs with cycles, they may fail or produce incorrect results. However, some approaches can handle cycles:

Temporal Unfolding: As discussed, this is the most common resolution, transforming the problem into a DAG setting over time. This relies on having sufficiently granular temporal data.
Constraint-Based Discovery (e.g., FCI): Algorithms like the Fast Causal Inference (FCI) algorithm (covered in Chapter 2) are designed to work under weaker assumptions, allowing for latent confounders and cycles. They typically output Partial Ancestral Graphs (PAGs), which represent a class of DAGs (or sometimes CDGs) compatible with the observed conditional independencies. While FCI doesn't explicitly model the cyclic dynamics, its output acknowledges uncertainty that could stem from cycles or hidden confounding.
Specialized Equilibrium Model Methods: Identification within linear cyclic models sometimes uses techniques related to instrumental variables or specific properties of non-Gaussian noise distributions (as seen in LiNGAM extensions, discussed in Chapter 2).

Practical Considerations

When confronted with potential feedback loops:

Question Simultaneity: Is the cycle truly instantaneous, or does it represent a process unfolding over a timescale relevant to your data collection frequency?
Gather Temporal Data: If possible, collecting data at a higher temporal resolution can help resolve cycles by explicitly modeling the time lags.
Use Appropriate Representations: If modeling equilibrium, be explicit about the strong assumptions involved. If modeling dynamics, use DBNs or related time-series models.
Leverage Domain Knowledge: Understanding the underlying mechanisms (e.g., biological pathways, economic principles) is essential for correctly specifying the model structure, including feedback loops and time dependencies.

Handling cycles requires moving beyond basic DAG assumptions. While temporal unfolding is a powerful tool, understanding equilibrium models and the capabilities of advanced discovery algorithms provides a more complete toolkit for analyzing systems with feedback. Techniques for time-series data, discussed in detail in Chapter 5, are particularly relevant for rigorously addressing these dynamic causal structures.