4.1. Solving the Static Problem

To solve the static problem, we have to modify the definition of an MLCM: instead of requiring that the most detailed causal model M ₁ of the MLCM satisfies CMC, we rather require M ₁ to satisfy the d-connection condition. A model 〈V, E, P〉 satisfies the d-connection condition iff for every dependence of variables X and Y given some $Z\subseteq \{X,Y\}$ there is a d-connection between X and Y given Z (Schurz and Gebharter Reference Schurz and Gebharter2016). Variables X and Y are d-connected given Z iff there is a path π connecting X and Y such that no intermediate or common cause on π is in Z, while every collider on π is in Z or has an effect in Z. Variables X and Y are d-separated by Z otherwise.

The d-connection condition is equivalent to CMC for acyclic causal models (Lauritzen et al. Reference Lauritzen, Dawid, Larsen and Leimer1990). This equivalence reveals the full content of CMC: whenever a causal model satisfies CMC, then every dependence can be explained by some causal connection in the model, and every independence can be explained by missing causal connections in the model. The d-connection condition’s clear advantage over CMC is that it implies the independencies to be expected when applied to causal cycles (Spirtes Reference Spirtes1995; Pearl and Dechter Reference Pearl and Dechter1996). To demonstrate this, assume that the causal model $X_1\to X_2\to X_3\to X_4\to X_1$ discussed earlier in section 4 satisfies the d-connection condition. As we saw in section 4, CMC implies no independencies for this causal model. But since X ₁ and X ₃ are d-separated by {X ₂, X ₄}, the d-connection condition implies the expected independence of X ₁ and X ₃ given {X ₂, X ₄}.

Let us now see how the static problem can be solved for our exemplary mechanism within the modified MLCM approach. The static problem concerns our exemplary mechanism when it has reached equilibrium. The system can be represented by the two-stage MLCM depicted in figure 2. Model M ₂ represents the system at the top level. Variables OT and CK are directly causally relevant to IT. Model M ₁ provides more detailed information about what is happening within the mechanism: the inside temperature is measured by a temperature sensor (S), which is directly causally relevant to an air conditioner (AC), which, in turn, is under direct causal influence of CK.

Figure 2.

Model M ₁ is assumed to satisfy the d-connection condition, and M ₂ is a restriction of M ₁. The MLCM mechanistically explains why OT is relatively insensitive to IT when the cycle $\mathrm{IT}\to S\to \mathrm{AC}\to \mathrm{IT}$ has reached equilibrium and CK = on, that is, why $P\left(\mathrm{it}\right|\mathrm{ot},\mathrm{CK}=\mathrm{on})\approx P(\mathrm{it}|\mathrm{CK}=\mathrm{on})$ holds. If CK is off, then AC is off, and there is no self-regulation due to the causal cycle $\mathrm{IT}\to S\to \mathrm{AC}\to \mathrm{IT}$. Thus, OT will have an influence on IT. But when CK is set to one of its on values, then AC responds to S according to CK’s adjustment. Since AC → IT overwrites OT → IT when CK = on, IT’s value is robust with respect to changes of OT’s value when CK = on. This overwriting property of AC → IT is represented by the bold arrow in figure 2.

Let us finally mention some open problems. First, cyclic models possibly featuring bidirected arrows do not admit the Markov factorization. Since we assume the d-connection condition to hold, they do, however, factor according to the following equation:

(2)$\begin{array}{}P(X_1,\dots ,X_n)=\prod _{i=1}^{n}P\left(X_i\right|d\mathrm{Sep}\left(X_i\right)).\end{array}$

The set dSep(X_i) is constructed as follows: Let Pred(X_i) be the set of X_i’s predecessors in the ordering $X_1,\dots ,X_n$. Now search for sets $d\mathrm{Pred}\left(X_i\right)\subseteq \mathrm{Pred}\left(X_i\right)$ such that $U=\mathrm{Pred}\left(X_i\right)\backslash d\mathrm{Pred}\left(X_i\right)$d-separates X_i from all elements of dPred(X_i). (Note that U may be empty.) If there are no such sets dPred(X_i), then identify dSep(X_i) with X_i’s predecessors Pred(X_i). If there are such sets dPred(X_i), then take one of the largest of these sets and identify dSep(X_i) with the corresponding separator set $U=\mathrm{Pred}\left(X_i\right)\backslash d\mathrm{Pred}\left(X_i\right)$. For the ordering P(OT, CK, IT, AC, S), for example, the joint distribution of M ₁ factors as $P\left(\mathrm{OT}\right)\cdot P\left(\mathrm{CK}\right)\cdot P\left(\mathrm{IT}\right|\mathrm{OT},\mathrm{CK})\cdot P(\mathrm{AC}|\mathrm{OT},\mathrm{CK},\mathrm{IT})\cdot P\left(S\right|\mathrm{CK},\mathrm{IT},\mathrm{AC})$.

Equation (2) has two disadvantages. First, it depends on an ordering of variables. Second, a probability distribution that factors according to equation (2) may not imply all independencies implied by the d-connection condition. For example, it does not imply an independence between OT and S conditional on {IT, AC}, although OT and S are d-separated by {IT, AC}. One open problem is to find out whether there is an order-independent factorization criterion equivalent with the d-connection condition. Another open problem is search. Causal discovery of the latent structure inside a mechanism’s causal arrows in the possible presence of feedback loops can be expected to be an even harder problem than discovery without feedback (cf. Murray-Watters and Glymour Reference Murray-Watters and Glymour2015).

We conjecture that effects of interventions for cyclic graphs possibly featuring bidirected arrows can be computed as usual. To compute postintervention probabilities $P\left(z\right|\hat{x})$ for an instantiation z of a set of variables Z, one needs, first, to delete all the arrows with an arrowhead pointing at X from the graph.Footnote ⁷ Second, use d-separation information provided by the manipulated graph to compute $P\left(z\right|\hat{x})$.

Before we take a look at how to solve the dynamic problem, let us briefly discuss the relationship of the solution to the static problem suggested above with Clarke et al.’s (Reference Clarke, Leuridan and Williamson2014) solution. Although both approaches use Pearl’s (Reference Pearl2000) notion of d-separation instead of CMC to account for cycles, the structures used for probabilistic reasoning differ in the two approaches. Clarke et al. use the “true” cyclic graph to construct an equilibrium network, that is, a BN that is then used “to model the probability distribution of the equilibrium solution” (Reference Clarke, Leuridan and Williamson2014, sec. 6.1). In our view, this move has at least two shortcomings:

• i) Independencies implied by the d-connection condition and the original cyclic causal structure may not be implied by the equilibrium network. We illustrate this by means of our model M ₁, whose equilibrium network could be the one depicted in figure 3. (See Clarke et al. [Reference Clarke, Leuridan and Williamson2014], sec. 6.1, for details on how to construct equilibrium networks.) Now note that OT and CK, for example, are not d-separated in the equilibrium network. So the equilibrium network’s graph does not capture the independence between OT and CK implied by the d-connection condition and the fact that OT and CK are d-separated in M ₁’s graph.

• ii) Since the arrows of the equilibrium network do not capture the “true” causal relations anymore, it cannot be used for predicting the effects of interventions. To illustrate this, assume we are interested in the postintervention probability $P\left(s\right|\widehat{\mathrm{ck}})$ in our model M ₁. In case we use M ₁’s graph for computing this probability, we arrive at $P\left(s\right|\widehat{\mathrm{ck}})=P(s\left|\mathrm{ck}\right)$. If we use the equilibrium network’s graph, however, we arrive at $P\left(s\right|\widehat{\mathrm{ck}})=P(s)$. But since the control knob is causally relevant for the sensor, P(s|ck) will not equal P(s) when intervening on CK.

Figure 3.

4.2. Solving the Dynamic Problem

Solving the dynamic problem requires an extension of the MLCM approach that allows for representing the system’s behavior over a period of time. Clarke et al. (Reference Clarke, Leuridan and Williamson2014) model such behavior by means of DBNs (cf. Murphy Reference Murphy2002). The basic idea behind this move is to roll out the causal cycles over time. We use dynamic causal models (DCMs) that also allow for bidirected arrows.

A DCM M is a quadruple $〈V,E,P,t:V\to {\mathbb{N}}^{+}〉$, where V is a set of infinitely many variables $X_{1,1},\dots ,X_{n,1},X_{1,2},\dots ,X_{n,2},\dots$. The variables X _i,1 (with 1 ≤ i ≤ n) describe the system at its initial state (stage 1), the variables X _i,t+1 (with 1 ≤ i ≤ n), the system at later stages t + 1. The DCMs we consider involve some idealization: directed arrows only connect variables at different stages, and if there is a directed arrow going from a variable X _i,t to a variable X _j,t+u for some stage t, then for every stage t there is such a directed arrow going from X _i,t to X _j,t+u. So the pattern of directed arrows between stages t and t + u is always the same.

What one ideally wants is a DCM 〈V, E, P, t〉 with the following additional properties: (i) arrows do not skip stages, (ii) bidirected arrows occur only between variables of one and the same stage, (iii) every two stages t_i, t_j (with i, j > 1) share the same pattern of bidirected arrows, and (iv) $P\left(X_{i,t}\right|\mathrm{Par}\left(X_{i,t}\right))=P(X_{i,t+1}\left|\mathrm{Par}\right(X_{i,t+1}\left)\right)$ holds for all X _i,t ∈ V with t > 1. For a finite segment of such an “ideal” DCM, see figure 4. The depicted graph’s first stage features more bidirected arrows than later stages. These additional bidirected arrows account for correlations between X _1,1 and X _2,1, X _2,1 and X _3,1, and X _1,1 and X _3,1 because of not-represented past common causes (of the kind described by variables in V).

Figure 4.

Let us now come back to the question of how the dynamic problem can be solved within the MLCM approach. The phenomenon we are interested in is that IT is relatively robust to variations of OT over a period of time when CK = on. Our simple temperature regulation system can be modeled by a two-stage MLCM 〈M ₁, M ₂〉 (see fig. 5 for a finite segment). The mechanism’s top level is represented by M ₂, which is a restriction of M ₁. Model M ₁, which is assumed to satisfy the d-connection condition, provides more detailed information about the mechanism bringing about the phenomenon of interest.

Figure 5.

When adding new intermediate causes, we will typically also add new stages. In our example, we added two new variables (S and AC) and two new stages between consecutive stages of M ₂ arriving at ${\mathrm{IT}}_{t^{*}}\to S_{t^{*}+1}\to {\mathrm{AC}}_{t^{*}+2}$ in M ₁. We assume the intervals between M ₁’s stages to correspond to the time the causal processes we are interested in require to bring about their effects. To guarantee that M ₁’s and M ₂’s probability distributions fit together, we require t* = t, $t^{*}+1=t+1/3$, $t^{*}+2=t+2/3$, $t^{*}+3=t+3/3$, and so on (where t stands for M ₂’s and t* for M ₁’s stages).

Now M ₁ provides information about the causal structure within the temperature regulation system. Variables OT and IT are directly causally relevant to themselves at the next stage. Variable IT also causally depends on OT and AC, while S depends only on IT, AC only on S and CK, and CK on no variable in the model. The bidirected arrows at stage 1 account for dependencies to be expected because of not-represented common causes.

Here we assumed that model M ₁ is especially nice, that is, that it satisfies (i)–(iv) discussed a few paragraphs above. Unfortunately, model M ₂ is not that nice. Since we marginalized out S and AC and there were directed paths from ${\mathrm{OT}}_{t^{*}}$ to ${\mathrm{IT}}_{t^{*}+6}$ and from ${\mathrm{IT}}_{t^{*}}$ to ${\mathrm{IT}}_{t^{*}+6}$ all going through $S_{t^{*}+3}$ or ${\mathrm{AC}}_{t^{*}+3}$ in M ₁, M ₂ features directed arrows ${\mathrm{OT}}_t\to {\mathrm{IT}}_{t+2}$ and ${\mathrm{IT}}_t\to {\mathrm{IT}}_{t+2}$ skipping stages. Since there were paths indicating a common cause path between ${\mathrm{OT}}_{t^{*}}$ and ${\mathrm{IT}}_{t^{*}+6}$ going through $S_{t^{*}+3}$ or ${\mathrm{AC}}_{t^{*}+3}$ in M ₁, M ₂ features bidirected arrows ${\mathrm{OT}}_t\leftrightarrow {\mathrm{IT}}_{t+2}$. Note that there are also bidirected arrows between OT _t and IT _t+1 and between IT _t and IT _t+1.

Now the MLCM mechanistically explains why IT is relatively robust with respect to OT changes when CK = on over a period of time. If CK is off over several stages, then also AC is off, and there is no regulation of IT over paths ${\mathrm{IT}}_{t^{*}}\to S_{t^{*}+1}\to {\mathrm{AC}}_{t^{*}+2}\to {\mathrm{IT}}_{t^{*}+3}$; IT’s value will increase and decrease (with a slight time lag) with OT’s value. If, however, CK is fixed to one of its on values over several stages, then over several stages ${\mathrm{AC}}_{t^{*}+1}$ responds to $S_{t^{*}}$ according to ${\mathrm{CK}}_{t^{*}}$’s adjustment. Now the crucial control mechanism consists of ${\mathrm{IT}}_{t^{*}+1}$ and its parents ${\mathrm{OT}}_{t^{*}}$, ${\mathrm{IT}}_{t^{*}}$, and ${\mathrm{AC}}_{t^{*}}$. The bold arrows ${\mathrm{AC}}_{t^{*}}\to {\mathrm{IT}}_{t^{*}+1}$ in figure 5 overwrite ${\mathrm{OT}}_{t^{*}}\to {\mathrm{IT}}_{t^{*}+1}$ and ${\mathrm{IT}}_{t^{*}}\to {\mathrm{IT}}_{t^{*}+1}$ when ${\mathrm{CK}}_{t^{*}}=\mathrm{on}$; that is, $P_{{\mathrm{CK}}_{t^{*}}=\mathrm{on}}\left({\mathrm{it}}_{t^{*}+1}\right|{\mathrm{ac}}_{t^{*}},u_{t^{*}})\approx P_{{\mathrm{CK}}_{t^{*}}=\mathrm{on}}({\mathrm{it}}_{t^{*}+1}\left|{\mathrm{ac}}_{t^{*}}\right)$ holds, where $U_{t^{*}}\subseteq \{{\mathrm{OT}}_{t^{*}},{\mathrm{IT}}_{t^{*}}\}$. This control mechanism will, after a short period of time, cancel deviations of IT’s value from CK’s adjustment brought about by OT’s influence.

Here are some possible open problems: first, some of the arrows in M ₂ may seem to misrepresent the “true” causal processes going on inside the temperature regulation system. There is, for example, a directed arrow going from CK _t to IT _t+1 but no directed arrow from CK _t to AC _t+1, although CK can clearly influence IT only through AC. This is a typical problem arising for dynamic models. One can, however, learn something about M ₁’s structure from M ₂: the (direct or indirect) cause-effect relationships among variables in M ₂ will also hold for M ₁. Another problem is, again, search. For solutions of several discovery problems involving time series, see, for example, Danks and Plis (Reference Danks and Plis2014). Finally, factorization and interventions: since our DCMs do not feature feedback loops, we conjecture that Richardson’s (Reference Richardson, Bilmes and Ng2009) factorization criterion and Zhang’s (Reference Zhang2008) results about how to compute the effects of interventions in models with bidirected arrows can be fruitfully applied to DCMs.

Let us finally have a look at how our solution to the dynamic problem relates to the one suggested by Clarke et al. (Reference Clarke, Leuridan and Williamson2014). Both modeling strategies use the same basic idea, that is, to roll out the cycles over time. While the arrows of the DCMs we use are intended to capture the “true” causal relations between variables of interest, the directed arrows in Clarke et al.’s DBNs surprisingly are not intended to represent the “true” causal relationships (cf. sec. 6.2). Thus, their models share problem (ii) discussed at the end of section 4.1 with the equilibrium network they use for solving static problems: the model cannot be used to compute the effects of interventions.