SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Consistency of partitioned SCM evaluation. To ensure for the consistent evaluation of all sub SCM  $\mathcal{M}_{\mathbf{A}}$  within a partitioned SCM  $\mathcal{M}_{\mathcal{A}}$  we need to ensure that the evaluation is carried out according to some  $\mathrm{R_X}$  that is compliant according to the base SCM  $\mathcal{M}$ .6 Doing so, guarantees that the value of every exogenous variable  $\mathbf{U}_i$  of a sub SCM  $\mathcal{M}_{\mathbf{A}_s}$  - that is not truly exogenous ( $\mathbf{U}_i \notin \mathcal{M}_{\mathbf{U}}$ ) - is computed as an endogenous variable  $V_j$  inside another  $\mathcal{M}_{\mathbf{A}_t}$ , that is evaluated before  $\mathcal{M}_{\mathbf{A}_s}$  with  $\mathbf{U}_i := \mathbf{V}_j$ . For example  $G_2$  in Fig. 2 (left) computes the values of  $B$  and  $D$ , required as exogenous variables by  $G_3$ . Lastly, during evaluation, all  $\mathcal{M}_{\mathbf{A}}$  need to agree on the same set of applied interventions. This is done by fixing a particular  $\mathbf{I}'$  during evaluation and computing the intervention set  $\mathbf{I}_{\mathbf{A}}' := \psi_{\mathbf{A}}(\mathbf{T}')$  specific to every  $\mathcal{M}_{\mathbf{A}}$ . An algorithm for evaluating partitioned SCM and its proof of consistency are presented in Appendix A.
In particular  $\mathbf{R}_{\mathbf{X}}$  is a strict partial order, if it is asymmetric:  $\forall \mathbf{A}_1, \mathbf{A}_2 \in \mathcal{A} : \mathbf{A}_1 \mathbf{R}_{\mathbf{X}} \mathbf{A}_2 \Rightarrow \neg (\mathbf{A}_2 \mathbf{R}_{\mathbf{X}} \mathbf{A}_1)$ , implying that the evaluation of no two sub SCM mutually depend on each other.
$\tilde{^6\mathrm{As}} <  _{\mathbf{x}}$  is a partial order, there may exist multiple total orders which comply with the partial ordering of  $\mathcal{M}$
Algorithm 1 Consolidation of Structural Causal Models
1: procedure CONSOLIDATE(M,A,E)  
2: for all Ai in A do  
3:  $\mathbf{E}_i\gets \mathbf{A}_i\cap \mathbf{E}$  ▷ Filter aspect variables for the current Ai.  
4:  $\mathbf{E}_i^{\prime}\leftarrow \mathbf{E}_i\cup (\mathrm{pa}(\mathbf{V}\setminus \mathbf{A}_i)\cap \mathbf{A}_i)$  ▷ Add variables that are required by other sub SCM.  
5:  $\mathbf{U}_{\mathbf{A}_i}\gets \mathrm{pa}(\mathbf{A}_i)\setminus \mathbf{A}_i$  ▷ Define exogenous variables and interventions.  
6:  $\mathcal{I}_{\mathbf{A}_i}\gets \{\psi_{\mathbf{A}_i}(\mathbf{I}):\mathbf{I}\in \mathcal{I}\} = \{\{do(X_i = v)\in \mathbf{I}:X_i\in \mathbf{A}_i\} :\mathbf{I}\in \mathcal{I}\}$   
7:  $\rho_{\mathbf{E}_i'}(\mathbf{U}_{\mathbf{A}_i},\mathbf{I})\gets \{\mathbf{F}_j:X_j\in \mathbf{A}_i\}$  ▷ Define a causal compositional variable via  $\rho_{\mathbf{E}_i'}$   
8:  $\rho_{\mathbf{E}_i'}^* \gets \mathrm{argmin}_{\rho_{\mathbf{E}_i'}'}^{\prime}\mathcal{K}(\rho_{\mathbf{E}_i'}')$  ▷ Minimize representation (see Sec. 4).  
9:  $\mathcal{M}_{\mathbf{A}_i,\mathbf{E}}\gets (\mathbf{E}_i',\mathbf{U}_{\mathbf{A}_i},\rho_{\mathbf{E}_i'}^*,\mathcal{I}_{\mathbf{A}_i},\mathrm{P}_{\mathbf{U}_{\mathbf{A}_i}})$  ▷ Define the sub SCM resulting from Ai and E.  
10: end for  
11:  $\mathcal{M}_{\mathbf{A},\mathbf{E}}\gets (\bigcup \mathbf{E}_i',\bigcup \mathbf{U}_{\mathbf{A}_i},\bigcup \rho_{\mathbf{E}_i'}^*,\bigcup \mathcal{I}_{\mathbf{A}_i},\bigcup \mathrm{P}_{\mathbf{U}_{\mathbf{A}_i}}), i\in \{1\dots |\mathcal{A}|\}$  ▷ Merge all  $\mathcal{M}_{\mathbf{A}_i,\mathbf{E}}$   
12: return  $\mathcal{M}_{\mathbf{A},\mathbf{E}}$  ▷ Return the consolidated SCM.  
13: end procedure
Figure 3: CONSOLIDATE Algorithm. The above pseudo-code summarizes the consolidation algorithm as described in this paper by utilizing causal compositional variables and partitioned SCM to obtain simplified SCM. Depending on the use-case Step 11 might be skipped and the partitioned SCM might be returned instead.
Partial consolidation of SCM. Having defined partitioned SCM allows us to selectively swap out arbitrary sub SCM by their consolidated SCM. In Def. 3 we placed no constraints on  $\mathbf{E}$  to allow for arbitrary consolidation of variables. For sub SCM  $\mathcal{M}_{\mathbf{A}}$  that appear within a partitioned SCM  $\mathcal{M}_{\mathcal{A}}$  we need to constrain  $\mathbf{E}$  to additionally include all variables  $V \in \mathbf{U}_{\mathbf{A}}$  such that evaluation of  $\mathcal{M}_{\mathbf{A}}$  additionally computes all variables needed as exogenous by other sub SCMs. Fig. 2 (right) shows an exemplary sub SCM with  $X_{2}$  (green) chosen as a relevant aspect variable by the user, and  $X_{4}, X_{5}$  being required by evaluations of subsequent SCM. Thus  $\mathbf{E}' = \{X_{2}, X_{4}, X_{5}\}$ . Whether to consider  $\mathbf{E}$  or  $\mathbf{E}'$  depends on the standpoint of the user. From a computational perspective  $\mathbf{E}'$  is important as it holds all variables that need to be computed by  $\rho$ . On the other hand, the set  $\mathbf{E}$  captures aspects of the SCM important to the user i.e., variables of interest. We will therefore refer to sub SCM with  $\mathcal{M}_{\mathbf{E}'}$  (and in the same breath write  $\rho_{\mathbf{E}'}$ ) but use  $\mathcal{M}_{\mathcal{A},\mathbf{E}}$  (see the following Def. 5) to retain the initial set of variables chosen by the user. Having defined consolidated SCM  $\mathcal{M}_{\mathbf{E}}$ , partitioned SCM  $\mathcal{M}_{\mathcal{A}}$  and the required constraint on  $\mathbf{E}$  we are now equipped with the tools to define a partially consolidated SCM that yields a consistent  $\mathrm{P_E}$  with the base SCM.
Definition 5 (Partially Consolidated SCM) A partially consolidated SCM  $\mathcal{M}_{\mathcal{A},\mathbf{E}}$  is a partitioned SCM  $\mathcal{M}_{\mathcal{A}}$  such that a subset of sub SCM  $\mathcal{M}_{\mathbf{A}}$  are being replaced by consolidated SCM  $\mathcal{M}_{\mathbf{E}'}$  where  $\mathbf{E}_i' := \{V_i \in \mathbf{V}_{\mathbf{A}} : (V_i \in \mathbf{E}) \vee (\exists \mathcal{M}_{\mathbf{A}'} = (\mathbf{V}', \mathbf{U}', \mathbf{F}', \mathcal{I}', \mathbf{P}_{\mathbf{U}'}') \cdot V_i \in \mathbf{U}')\}$ .
Algorithm 1 summarizes all considerations of this chapter, starting out from a subset  $\mathbf{E}$  and partition  $\mathcal{A}$  up to a (partially) consolidated SCM  $\mathcal{M}_{\mathcal{A},\mathbf{E}}$ . An exemplary step-by-step application of the algorithm can be found in Appendix D.3. The purpose of the argmin operation in Line 8 is to minimize complexity of  $\rho_{\mathbf{E}_i'}'$  by finding a minimal encoding. We discuss this step in more detail in the following section. After formally introducing consolidation, we are ready to illustrate its applicability.
4 Compression of Causal Equations
Model consolidation can lead to compression by reducing the model's graph structure and leveraging redundant computations across equations. This may result in smaller, simpler models that are computationally more efficient and easier to analyze. Compressing structural equations to a minimal representation is highly dependent on the equations under consideration and probably incomputable for most problems. As there is ultimately no way of measuring compressibility of SCM by only considering their connecting graph structure, we provide a discussion with regard to some of the
information-theoretical implications. Specifically, we discuss compression properties for some of the basic structures appearing within SCM; namely chains, forks and colliders. In this section, we, first, analyze how consolidated models may leverage redundant computations for reducing complexity within chained equation in general. Second, we give a condition under which equations, and their interventions can be dropped from the consolidation model altogether. Thirdly, we analyse how interventions within the consolidated model affect our ability to compress equations. Lastly, we will walk through two examples of model compression.
General compression of equation systems. Using our formalization of (partially) consolidated SCM, we now have the chance to replace certain parts of an SCM with computationally simpler expressions. The notion of what a 'simple' expression may be, varies depending on the application and is subjective to the user. To define a measurable metric, we reside to a simplified notion of complexity by measuring the representation length of our consolidated equations. We assume that all structural equations of an SCM can be expressed in terms of elementary operators, where each term contributes the same amount of complexity. As such, we can apply Kolmogorov complexity  $\mathcal{K}$  [Kolmogorov, 1963]. Then a desirable minimal representation of a structural equation  $f_{i}^{\star}$  is one that minimizes  $\mathcal{K}(f_i)$ :  $f_{i}^{\star} \coloneqq \operatorname*{argmin}_{f_{i}'} \mathcal{K}(f_{i}')$  s.t.  $f_{i}'(\mathrm{pa}(X_{i})) = f_{i}(\mathrm{pa}(X_{i}))$ .
Classical marginalization reduces the number of variables in a graph. To keep the model consistent after marginalization, all children  $\mathbf{B} \coloneqq \operatorname{ch}(A)$  of a marginalized variable  $A$  additionally need to incorporate the values of  $\mathrm{pa}(A)$  to accommodate for the causal effects that where previously flowing through  $A$  into  $\mathbf{B}$ . This modifies the structural equations of any  $B \in \mathbf{B}$ ,  $f_B' \coloneqq f_B \circ f_A$ , where  $f_B$  and  $f_B'$  are the structural equations of  $B$  before and after marginalization, respectively. Evaluation of the separate equations  $f_A, f_B$  provides an upper bound on the complexity of the composed representation  $\mathcal{K}(f_B^{\prime \star}) \leq \mathcal{K}(f_A^{\star}) + \mathcal{K}(f_B^{\star})$  [Zvonkin and Levin, 1970]. Since the consolidated system is not required to compute  $A$  explicitly, the encoding length of  $f_B^{\prime \star}$  might resort to directly computing  $B$  from the values of  $\mathrm{pa}(A)$ . Also, the chain rule for Kolmogorov complexity only considers the case of reproducing  $f_A$  and  $f_B$  in their initial forms. In addition to that, we might also use semantic rules to reduce equation length, e.g. by collapsing consecutive additions  $\forall a, b \in \mathbb{R}$ .  $\exists c \in \mathbb{R}$ .  $a + b = c$  and so on. Whether consolidation actually leads to simplified equations depends strongly on the specific equations and their connecting graph structure. No simplification effects occur in cases of already minimal systems, while strong cancellation occurs in the case of  $f_B$ ,  $f_A$  being inverses to each other (see Appendix B.1). Lastly, we want to refer to Appendix B.2, where we showcase the insufficiency of matrix composition to obtain minimal function representations in the case of linear systems.
Marginalizing child-less variables. Regardless of the particular causal graph structure, all equations which do not affect  $P_{\mathbf{E}'}$  can be removed from the model to reduce its overall complexity. In particular we point out that  $P_{\mathbf{E}'}$  is invariant to all  $X \notin \mathrm{an}(\mathbf{E}')$ . By the following deduction we infer that we can always consolidate all child-less variables (if not part of  $\mathbf{E}'$  themselves) from  $\mathcal{M}$ :  $\forall X \in \mathbf{X} \setminus \mathbf{E}'$ .  $[(\mathrm{ch}(X) = \emptyset) \Rightarrow (\forall X' \in \mathbf{X}.X \notin \mathrm{pa}(X')) \Rightarrow (\forall X' \in \mathbf{X}.X \notin \mathrm{an}(X')) \Rightarrow X \notin \mathrm{an}(\mathbf{E}')]$ . Since child-less variables do not affect  $P_{\mathbf{E}'}$ , we can not only consolidate but marginalize them. (Reducing to the same scenario as in Rubenstein et al. [2017, Thm. 9]). Therefore, we are allowed to drop interventions  $d o(X_{i} = c)$  with  $X_{i} \notin \mathrm{an}(\mathbf{E}')$  from the set of allowed interventions. This process can be applied repeatedly until we have pruned the SCM from all child-less variables irrelevant to  $\mathbf{E}'$ .
4.1 Simplifying Graphical Structures
In contrast to marginalization, consolidation preserves the effects of interventions for consolidated variables. This effectively adds conditional branching to every structural equation  $f_{i}$  if some  $\mathbf{I} \in \mathcal{I}$  with  $do(V_{i} = c) \in \mathbf{I}$  exists:
$$
V _ {i} := \left\{ \begin{array}{l l} c & \text {i f} d o \left(V _ {i} = c\right) \in \mathbf {I} \\ f _ {i} (\operatorname {p a} \left(V _ {i}\right)) & \text {e l s e} \end{array} \right. \tag {1}
$$
While conditional branching might prevent us from compressing equations, we consider that not all variables might be affected by interventions. As such, we might be able to utilize local structures within the graph to simplify equations. In the following we briefly discuss the possibilities of simplifying chains, forks and collider structures within the graphs of SCMs:
Simplifying Chains. Consolidating chains of consequent variables corresponds to 'stacking' structural equations and computing the last non-consolidated variable directly. In the general case,
Figure 4: Consolidating a real world mechanism. (Left) The causal time-series model of a milling machine, representing tool length  $L$ , utilization  $U$ , sharpness  $S$  and accuracy  $A$ . (Center-Left) Removing child-less nodes  $L_{t}$  and  $A_{t}$  and controlling for the parents  $U_{t}$  yields a simplified causal structure. (Center-Right) Plots for the consolidated structural equation of  $S$ . Colored areas show the effects of varying  $\bar{U}$  by one and two sigma  $(\pm 0.05, \pm 0.1)$  respectively. Dashed grey lines indicate interventions, which are respected truthfully by the consolidated function. (Right) Marginalization, likewise, simplifies the model, but does not allow us to investigate the effects of interventions.
conditional branching complicates the simplification of the stacked equations into a single closed-form representation. When considering the case of marginalization, that is without considering interventions, as done in Rubenstein et al. [2017], composition of equations turns into direct function composition  $X_{i} \coloneqq f_{i} \circ f_{i - 1} \circ f_{i - 2} \circ \ldots$ . To this end, a complexity bound on chained equations over finite discrete domains is discussed in Appendix B.3, as well as consolidation of the motivating dominoes example in Appendix D.1.
Simplifying Forks. Consolidating the parent node  $B$  of a fork structure,  $A \leftarrow B \rightarrow C$ , might lead to a duplication of  $f_{B}$  into the equations of both child nodes,  $f_{A}^{\prime} := f_{A} \circ f_{B}, f_{C}^{\prime} := f_{C} \circ f_{B}$ . If  $\mathrm{pa}(B) \subset \mathbf{U}$ , then  $A$  and  $C$  will be confounded by exogenous variables. This is the reason why we did not require independence of exogenous variables in Def. 1. Still, consistency with the initial SCM is guaranteed, since we require all structural equations to be deterministic. As a consequence, every evaluation of the duplicated structural equations  $f_{B}$  inside  $f_{A}^{\prime}$  and  $f_{C}^{\prime}$  yields the same value when given the same inputs. While determinism of structural equations is formally required, we illustrate a consistent reparameterization of non-deterministic models in Appendix C.
Simplifying Colliders. Colliders are the most promising graphical structures for simplifying equations. When consolidating  $A$  and  $C$  of a collider  $A \rightarrow B \leftarrow C$ , we might leverage mutual information between  $f_{A}$  and  $f_{C}$  to simplify  $f_{B}$ . Especially in the case of  $\mathrm{pa}(A) = \mathrm{pa}(C)$ , consider  $A \leftarrow X \rightarrow C$  for example, we might be able to discard  $f_{A}$  and  $f_{C}$  altogether and compute  $B$  directly from  $X$ .
4.2 Time Series Example: Tool Wear
We will now demonstrate a simple application of consolidation for a possibly more applied scenario. Imagine that we want to create a causal model of an industrial unit under continuous use, e.g. a milling machine. At the end of every work day the length  $L$  and sharpness  $S$  of the milling cutter are measured. From these measurements other metrics such as the cutting accuracy  $A$  can be derived. Interventions on the process are performed by grinding the cutter, 'resetting' it to a certain sharpness. While every intervention grinds away some material, the weight and size changes are negligible for the considered aspect of accuracy. Throughout our recordings we might encounter multiple such interventions. From the data we fit a 'classical' SCM that models the time series on a day-to-day basis,  $\mathbf{V}_{t-1} \rightarrow \mathbf{V}_t$ . We observe the tool to loose some percentage of its sharpness per day depending on its utilization  $U_t$ . The intervention  $do(S_t = 1)$  resets the sharpness to a constant value, while  $do(S_t = 1, L_t = 1)$  models a tool replacement. As by Def. 1,  $\mathcal{I}$  needs to include  $do(L_t = 1)$ , which might be a recalibration of the machine. Figure 4 (left) shows the initial causal graph of the time series model as defined by the following SCM:
$$
\mathcal {M} = \left\{ \begin{array}{l l} \mathbf {U} & = \{\mathbf {U} _ {t} = \mathcal {N} (0. 5, 0. 0 5 ^ {2}) \} \\ \mathbf {V} & = \{\mathbf {L} _ {t}, \mathbf {S} _ {t}, \mathbf {A} _ {t} \} \\ \mathcal {I} & = \mathcal {P} (\{d o (\mathbf {S} _ {t} = 1), d o (\mathbf {L} _ {t} = 1. 0), d o (\mathbf {S} _ {t} = 1, \mathbf {L} _ {t} = 1) \}) \\ \mathbf {F} & = \left\{ \begin{array}{l l} f _ {l} (l, u) & := (1. 0 - 0. 0 0 2 u) l \\ f _ {\mathbf {s} _ {t}} (s _ {t - 1}, u) & := (1. 0 - 0. 3 u) s _ {t - 1} \\ f _ {a} (s) & := 0. 8 s ^ {2} \end{array} \right. \end{array} \right.
$$
Figure 5: Complex situations can be easy to understand using consolidation. Encoding the behaviour of agents acting in game environments (left) often results in complex causal graphs (right; indeed unreadable due to complexity. A readable version is contained in the Appendix). Even very simple levels with a single agent, coin, power-up and enemy, entail causal graphs that are intuitively non-interpretable. Especially the intertwining of game mechanics and agent behaviour complicates the inference of the agents' actual policy and makes it impossible to judge its performance. In our example a suboptimal greedy policy is embedded within the causal graph, which can be made visible using consolidation (bottom, Eq. (2)). Please note that ps abbreviates 'planning_sequence'. In contrast to marginalization (Eq. (3)), one can still intervene on the consolidated system.
$$
\mathrm {p s} _ {1} := \text {‘ c o i n ’} \text {i f} d o (\text {t a r g e t} _ {\cdot} \text {c o i n} = 0) \notin \mathbf {I} \text {e l s e} ^ {\prime} \text {f l a g} ^ {\prime} \quad \mathrm {p s} _ {3} := ^ {\prime} \text {f i n i s h e d} ^ {\prime} \tag {2}
$$
$$
\mathrm {p s} _ {2} := \text {‘ f i n i s h e d ’ i f} d o (\text {t a r g e t} _ {\cdot} \text {c o i n} = 0) \notin \mathbf {I} \text {e l s e ‘ f l a g ’} \quad \mathrm {p s} _ {4} := \text {‘ f i n i s h e d ’}
$$
$$
\mathrm {p s} _ {1} := \text {‘ c o i n ’} \quad \mathrm {p s} _ {3} := \text {‘ f i n i s h e d ’} \tag {3}
$$
$$
\mathrm {p s} _ {2} := \text {‘ f i n i s h e d ’} \quad \mathrm {p s} _ {4} := \text {‘ f i n i s h e d ’}
$$
Now, we might be interested in extracting a formula for the total tool sharpness  $S_{t}$  at an arbitrary point in time  $t$ . Thus, our consolidation set consists of all  $S_{t}$ ,  $\mathbf{E} = \{S_t\}$ . Since  $U_{t}$  is exogenous, we make the additional assumption that the utilization follows a normal distribution and we simplify to the expected value  $\bar{u} = 0.5$  (Figure 4, center-left). As laid out before, we can marginalize all child-less variables  $L_{t}$  and  $A_{t}$  not part of  $\mathbf{E}'$ . As all  $L_{t}$  are no longer part of  $\mathcal{M}_{\mathbf{E}'}$ ,  $\psi_{\mathbf{E}'}$  maps interventions  $do(S_{t} = s_{t}, L_{t} = l_{t}) \to do(S_{t} = s_{t})$ . Considering the unintervened case, structural equation are now simplified via function composition,  $f_{\mathbf{S}_t} := f_{\mathbf{S}_t} \circ \dots f_{\mathbf{S}_1}$ , which results in the following equation  $f_{\mathbf{S}_t} := (1 - 0.3 \cdot 0.5)^t = 0.85^t$ . According to Eq. (1), we now have to inspect interventions as potential candidates for conditional branching. All remaining interventions are of the form  $do(S_{t} = 1.0)$ . Applying an intervention at time  $t_0$  equals shifting the following equations by the time of that last intervention  $t' := t - t_0$ . Finally, we arrive at the following consolidated equation:
$$
f _ {S} (t) := 0. 8 5 ^ {t - t _ {0}}
$$
$$
\text {w h e r e} \quad t _ {0} = \max  _ {i} \left\{i \mid \exists d o \left(S _ {i}\right) \in \mathbf {I} \wedge i \leq t \right\}
$$
Fig. 4 (center-right) shows the resulting plot of the consolidated model under interventions  $do(S_{12} = 1)$  and  $do(S_{24} = 1)$ . We successfully demonstrated the power of consolidation models for dynamical system while preserving the ability to intervene. In theory more complex dynamical systems could be consolidated. However, as these kind of self-referential models require a more involved discussion, we kindly refer the reader to Bongers et al. [2018, 2021], Peters et al. [2022] for further considerations.
4.3 Revealing Agent Policy
In our second example we apply consolidation to a more complex causal graph relating the game state of a simple platformer environment to the actions of an agent. See Appendix D.4 for the full causal graph and structural equations. Throughout the level the agent ((1) in Fig. 5) can interact with a coin ((2) in Fig.), a power-up (3), an enemy (4) and the finish flag (5) to accumulate a certain reward (6) by doing so. The power-up is required to interact with the enemy. During play, the agent takes the state of the environment as its input and outputs the state of 'towards Coin', 'towards_powerup', etc. The order of the agent actions is then recorded via four 'planning_sequence_i' for  $i \in \{1 \ldots 4\}$  variables. A causal graph, like the one presented in Appendix D.4, might be extracted automatically from observational data, or designed by an expert. Due to the sheer number of variables and edges, dependencies in the obtained SCM are hard to trace. To get a better understanding, we use consolidation to reveal the policy of our agent. We consolidate all endogenous variables except player-
entity 'distance' and the 'planning_sequence' variables. To be able to modify the agents behaviour, we allow interventions by forbidding the agent to target certain entities:  $\mathcal{I} = \mathcal{P}(\{do(target\_coin = 0), do(target\_enemy = 0), do(target\_powerup = 0)\})$ .
Like before, we consolidate equations considering the unintervened case, and then add back in conditional branching for interventions to yield equations (2) in Figure 5. Contrary to the very complex structure of the SCM, the consolidated equation reveals the actually very simple policy of the agent. We find from the consolidated equation that the agent only collects the coin, if not intervened upon, and then heads directly towards the flag. This insight might not be obvious from the initial SCM and, at least, is difficult to spot a priori by looking at the unconsolidated equations. When inspecting the original SCM more closely we find, that a constant factor is added to the calculation of 'targeting_cost_powerup.' This factor might serve to accommodate for the time lost when speeding up or slowing down towards a target. Furthermore, we see that the agent pursues a greedy policy, thus, never considering the overall higher reward of the power-up and enemy together. Instead the policy ignores the power-up, due to its low reward and in consequence also never targets the enemy. This behaviour not only leads to strong simplification of the SCM, but also allows us to discard the imperfect policy, without the need to run possibly costly trials, just to come to the same conclusion.
To summarize, consolidation is a strictly more powerful operation than marginalization. More examples and domains can be found in Appendix D.