On Measuring Influence in Avoiding Undesired Future
Authors: Lue Tao
Tian-Zuo Wang
Yuan Jiang
Zhi-Hua Zhou
Venue: ICLR 2026 Poster (2026)
Keywords: Avoiding Undesired Future
Rehearsal Learning
Influence
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
TL;DR: We provide a principled measure for evaluating influence in avoiding undesired future.

Abstract: When a predictive model anticipates an undesired future event, a question arises: What can we do to avoid it? Resolving this forward-looking challenge requires determining the variables that positively influence the future, moving beyond the statistical *association* typically exploited for prediction. In this paper, we introduce a novel measure for evaluating the *influence* of actionable variables in successfully avoiding the undesired future. We quantify influence as the degree to which the success probability can be increased by altering variables under the principle of maximum expected utility. Our analysis demonstrates a counterintuitive insight: while related to *causality*, influential variables may not necessarily be those with strong intrinsic causal effects on the target event. In fact, it can be highly beneficial to alter a weak causal factor, or even a variable that is not an intrinsic factor at all. We provide a practical implementation for estimating the proposed measure and validate its utility through experiments on synthetic and real-world tasks.

Published as a conference paper at ICLR 2026
ON MEASURING INFLUENCE IN AVOIDING UNDESIRED FUTURE
Lue Tao, Tian-Zuo Wang, Yuan Jiang, Zhi-Hua Zhou
National Key I Laboratory for Novel Software Technology, Nanjing University, China School of Artifcial Intelligence, Nanjing University, China {taol, wangtz, jiangy, zhouzh h amda. www..neeee..... erju.edu.co
ABSTRACT
When predictive model anticipates an undesired future event, question arises: What can we do to avoid it? Resolving this forward-looking challenge requires determining the variables that positively influence the future, moving beyond the statistical association typically exploited for prediction. this paper, we introduce novel measure for evaluating the influence of actionable variables in successfully avoiding the undesired future. We quantify influence as the degree to which the success probability can be increased by altering variables under the principle of maximum expected utility. Our analysis demonstrates a counterintuitive insight: while related to causality, influential variables may necessarily be those with strong intrinsic causal effects on the target event. In fact, it can be highly beneficial alter weak causal factor, or even variable that is not an intrinsic factor all. We provide practical implementation for estimating the proposed measure and validate its utility through experiments on synthetic and real-world tasks.
INTRODUCTION
When an intelligent machine receives a warning from powerful predictive model that an undesired event is expected to occur, an important question arises: What can be done to avoid this future? This is known the avoiding undesired future (AUF) problem (Zhou 2022b), sparking a shift from passively predicting eventualities to proactively shaping the future.
Addressing the AUF problem requires identifying which actions should be taken to realize more desired future. While machine learning (ML) has achieved remarkable success in prediction (Jumper et al. 202 Achiam et al.) 2023] Price al.. 2025) by exploiting statistically asso- ciated variables to predict the target variable, the asso- ciation alone is insufficient for guiding decision actions. Causality (Pearl, 2009 Peters al. 2017) is more in- formative, but should not be regarded as necessary for decision-making (Zhou 2022b) for reasons that (i) making decisions do not need a thorough character- ization of causality, i.c., one can make good decisions without a full or faithful causal understanding: (ii) the real environment is often open and dynamic (Zhou 2022a), so a cause of the target variable in historical data does not imply that altering it will influence the target in the future; (iii) discovering true causal relations from data is inherently difficult (Chickering 1996). and even when identified, they are useless if unactionable. In light of this, /hou (2022)) (023)) proposed the concept of influence, an intermediate that is stronger than association but less demanding than causality (Figure D. forming the foundation for decision- maaking in AUF. Subsequently, a line of rehearsal learning methods have and have proven effective Du et al. 2024 2025a b).
Prediction
Association
Decision
Influence
Actionable
Discovery
Causality
Figure 1: Relationship among association,
influence, and causality (Zhoul 2022b).
been developed to model influence among variables for decision-maaeeee
across a variety of AUF scenarios (Qin al. 2023 2025
Despite these advances, a fundamental question remains open: how to quantify influence in AUF? Namely, how evaluate the influence of actionable variable the future target? Conventional measures of causality (Rosenbaum & Rubin 1983 Holland 1988] Northcott 2008) cannot serve this purpose, because they seek to evaluate the isolated effect of intervention within static environment,

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whereas AUF concerns changeable world potentially reshaped by decisions. Accordingly, a causal strength evaluated on historical data may no longer indicate the actual strength of influence on the future target. Therefore, a principled quantification of influence is required for the AUF problem.
In this paper, we introduce a novel measure for evaluating the influence of actionable variables in successfully avoiding the undesired future. We begin by outlining several intuitive considerations that a measure of influence in AUF scenarios should incorporate. Then, we formulate a novel quantity, termed influence power, defined the degree to which the probability of success can be increased through alteration based on the principle of maximum expected utility. This quantity captures the probabilistic dynamics of the decision process by accounting for the interplay between subsequent alterations and unfolding observations. Consequently, it provides holistic valuation of the influence of alteration, encapsulating both its explicit and implicit impacts on the future target.
Next, we leverage the proposed quantity to elucidate the connection between influential variables and those with intrinsic causal relations to the target variable. Through systematic analysis, we uncover subtle yet important distinction: influential variables are not simply subset of causal ancestors, and vice versa. Specifically, while related to causal effects, variables with positive influence power are not necessarily those with intrinsically strong causal effects. In fact, due to the dynamics of the decision process, it can be highly beneficial to alter a causal ancestor with negligible effects, or even a variable that is not an intrinsic ancestor at all. Another important observation is that not all actionable variables can be safely altered; there exist variables on which any alteration is counterproductive These insights crystallize the Shakespearean quandary for an intelligent machine facing an eventuality: To do, or not to do, that is the question. This work rests on influence power, a principled quantity for measuring influence in AUF, thereby offering rigorous way to answer this question.
Finally, we provide a Monte-Carlo-based implementation for estimating influence power using observational data. We empirically validate the utility of our measure the AUF problem through experiments on synthetic and real-world tasks.
PRELIMINARY
Notation. We represent random variables with capital letters (V), and their realized values with lowercase letters (v). We use bold capital letters (V) to denote set of random variables, and bold lowercase letters (v) to denote their realized values. Let G = (V, E) denote directed graph with nodes V and edges E. In a causal graph G, variable X is causal ancestor of Y, written as X E Anc(Y), if there is a directed path from X to Y in G. When X is binary, its causal strength can be quantified by the average causal effect (ACE) (Holland, 1988, Pearl 2009), defined as (X- = 1) - IE(Y Ido(X = =))). where E(Y do X = x)) denotes the expectation T(X, Y == EY [[]]do
of Y when X is set to the value x. We say that causal ancestor X of Y is weak if the average causal effect of X on Y is zero. Let Ax denote the feasible domain of alteration for a variable X. If
Ax * 0, we call X an actionable variable; otherwise, X is unactionable.
Structural equation models. We employ the language of the structural equation model (SEM) (Pearl 2009) and leverage it describe how nature assigns values variables of interest, i.e., the physical mechanisms governing the generation process of random variables. An SEM is a tuple M (V, N, F, P(N)). where V = .V....., Va} is a set of endogenous variables, N = {NV,, {.NI..... Naa is a set of background noises destributed according to P(N). and F' is a set of deterministic functions fi for each V; E V such that V, == (PA,, N₁) with PA, S V. Throughout this paper, we posit that the natural generation process is governed by an underlying SEM M, though it may remain unknown to the
due to its unobserved nature (Geffner et al.] 2022). }. If V/ has a directed path to V2 in the graph induced by M, we say that V; is an intrinsic ancestor of VG in M. For variable V,, when it actionable, we use the notation V; _ i to indicate that Vú can be altered to v; E Av; in contrast, the notation do(V, v;) is applicable even when V, is not actionable. This operation essentially replaces the structural function of V42 in M with the constant assignment VG2 == Us. The distribution of V, given that V, is altered to U; is then denoted as P(V, V, _ u;).
Problem statement. We consider a setting where data are drawn from a distribution induced by an underlying SEM, which describes the natural generation process of sequence of variables (Vi...., V. ). The final variable V2+1 in the sequence represents the target variable Y. with the set of desired values specified as S. We assume for simplicity that all variables discrete, and the
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variable sequence is pre-specified and consistent with the underlying variable ordering (i.e., variables are topologically ordered with respect to the SEM). The goal of the AUF problem is to maximize the probability of Y falling into S through feasible alterations to the variables Vi...., Va. In general, decision-making occurs after certain variables have already been realized. We denote this subset of realized variables X S (Vi.... Va). Let t = max (s V. E X) be the index of the latest realized variable, and denote the remaining actionable variables as Z = {V: t S d, Av, * 0} Hence, given a realization of X. the AUF problem is addressed by altering variables in Z. Notably, when determining the alteration for specific variable Vi, we do not require all variables preceding V; to be realized. This flexibility allows for certain variables to remain unobserved during the decision process, which is more practical than requiring full observability.
3 INFLUENCE POWER
3.1 MOTIVATION
We review the existing strategies for addressing the AUF problem, describe their limitations, and motivate the considerations that a measure of influence in AUF should incorporate. A primary AUF strategy the literature is to find feasible alteration that directly maximizes the probability that Y lies in the desired domain S (Qin et al.. 2023). This straightforward strategy is expressed as:
Z -,* arg max P(Y E = =
(I)
where is the observed value of X. Z is the set of actionable variables following X, and P(Y S|X x. Z z) is the AUF probability (i.e., the probability of Y E S) after observing X as x and altering Z 2. While intuitive and often effective, Equation (I) evaluates only the explicit impact of altering one variable at time, implicitly presuming a "static" future in which subsequent variables unfold naturally without further alteration. As result, fails to capture the impact stemming from the interplay between altering variables, overlooking situations where no single alteration suffices to achieve the desired target. A simple example illustrates this issue. Consider two binary variables, Zi and Zz. both of which naturally take the value 0 with near certainty, and let Y == Zi A Zz. Altering either variable in isolation is ineffective; only by altering both variables jointly can we ensure Y 1. Thus, when assessing the impact of altering variable in AUF scenarios, one must consider not only the feasible domain of that alteration but also the actionability of other variables.
In light of this insight, the next plausible step is to jointly alter all actionable variables. This approach is considered in prior work (Qin et al. 2025 Du et al. 2025a) and can be written as:
arg 2aX P(Y S|X x, Z z),
(2)
where Az denotes the Cartesian product of the feasible domains of alteration for variables in Z, and Z _ z denotes the joint alteration of variables in Z to the values in z. This strategy also overlooks an important consideration: it is often unnecessary to alter all variables. For example, while both light and water are crucial for crop growth, adding artificial light when sunlight is naturally abundant yields negligible benefits. Thus, when assessing the impact of altering a variable in AUF scenarios, one should account for its naturality, i.e., whether it naturally lies in favorable state. Moreover, will demonstrate later, altering certain variables may be not only unnecessary but counterproductive. no matter how they are altered. These observations underscore the need for a principled measure to determining whether an actionable variable warrants alteration.
3.2 FORMULATION
In this subsection, we present new quantity that assesses whether an actionable variable is worth altering to influence the future. To holistically account for the actionability and naturality of variables, as well as the desirability of the target variable, we need a principled way to rehearse future possibili- ties after alteration. While the Bellman equation (Bellman| (957)) provides a conceptual foundation for this purpose, its standard formulation is not applicable to our context, as it is typically built upon pre-specified separation between state and control variables. In the AUF problem, however, every variable Vú in the sequence (Vi,..., Va) has dual role: it can be proactively manipulated via alteration or passively observed as it unfolds naturally.
As the past cannot be changed, variables preceding V; are considered actionable.
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To this end, we ground our proposal in the principle of maximum expected utility (Russell & Norvig 2020), and recursively define the maximum expected probability (MEP) of achieving the desired future after an alteration observation. Specifically, for 0 < k < d, the MEP after altering V& to Uk. denoted as P(Y E SIVV = uk,...), is given by:
P(Y E V = Uk....) = max
max P(Y E S = U4-11 Vv- Uk.... uk......
(3)
PMM vg.... P(Y E S uk+1, VA - uk uk....)}
where "..." abbreviates an arbitrary sequence of alterations and observations preceding Vv. For the terminal step d, the recursion terminates and no further decomposition is required; the MEP simply reduces to the ordinary probability, i.c., P(Y E SIV _ u.....) .) = P(Y E SVV2 = uk....).
Similarly, for 0 j
j < d. the MEP after observing Vj as by is given by:
P(Y E SIV, = oj....) )) max maxe5 P(Y E SIV _ Uj+1, Vj = Uj....),
(4)
E my.. P(Y E SVV++1 = Uj+1, Vj = Ug ))
where ".. abbbeviaate an arbitrary sequence of alterations and observations preceding Vj. L Likewise, as the sequence concludes at j d, the MEP after observing Vj v, directly equals the ordinary probability, ic., P(Y E SIV, = vy....) .) = P(Y E SVV, = Uy....).
Given the recursive definition of MEP, we present quantity called the influence power (InP), which indicates the ability of an actionable variable to influence the target at a future time.
Definition (Influence Power). The influence power of an actionable variable V, on Y is defined as
(V.,, = max,,eAv P(Y E SVV, = v)) PY SIV = -v)-
Remark. The influence power of V; on Y quantifies the maximum gain in the MEP achievable by altering V,, relative to the expected MEP when V; is observed naturally. Consequently, a positive value indicates that alteration is beneficial, while zero negative value suggests that it is unnecessary or even harmful. By definition, the influence power is bounded in [-1, Because Definition [ recursively applies the principle of maximum expected utility, it be interpreted as a variant of the Bellman equation. Notably, it depends only on probability terms and does not require a fully specified SEM. Furthermore, this definition readily extends to a conditional form: given the observation of X = x, the conditional influence power of V/2 E Z on Y. denoted as p(V,, Y|X = x), is given by ma P(Y E SVV; _ U4. X x) TE,,,-
x) P(Y E V = 2; x).
Relation to Equation (I). We end this subsection by highlighting a connection between Definition I and Equation T Consider scenario with three binary variables: Vi, V2, and Y, where both Vi and V22 are actionable. Suppose an oracle informs us that the structural function f defining the target variable Y depends only on V1 (and not on 122,, i.e., Y = f(Vi). Based on this information, deduce that the solution to Equation T is V1 whenever the following condition holds:
max Edv, P(Y E SIV, _ v1) P(Y S).
(5)
On the other hand, the influence power of V1 Y simplifies to:
p(V1, max,reAv, P(Y E SVV = v1) P(Y E ES).
(6)
Combining Equations 5 and (6) we conclude that the solution to Equation I is 1/2 if p(Vi, Y) >
Thus, Equation ① agrees with Definition on whether V12 should be altered.
Furthermore, let Av; {0. 1} and the condition (VV Y) > 0 reduces to:
(1). By applying the identity max(a, b) = a + + bb b).
(r(Vi,Y) = [Y = 1) BY 0) > 2IE - IV 0) B 1), (7)
where [T( Y) is the absolute value of average causal effect of V1 on Y. This reveals that p(V1, Y) is closely related to 7(V1, Y). and the influence power seems to favor altering variables with strong causal effects. This view, however, is incomplete. In the following section, we demonstrate that the connection between influence power and average causal effect is, in fact, far more nuanced.
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3.3 CONNECTION
In this subsection, we investigate the connection between influential variables and those with intrinsic causal relations to the target variable. Concretely, we analyze how variables with a non-zero influence power (i.e., p(X, Y) # 0) relate to the intrinsic ancestors of the target in the underlying SEM (denoted by AY)(Y)). We also examine the qualitative relationship between the influence power p(X, Y) and the average causal effect r(X, Y), commonly used measure of causal strength. The results formally summarized in the following theorem.
Theorem 1. Let X and Y be two endogenous variables in an SEM. The following statements hold:
1. X E Anc(Y) + p(X,Y)
and i(X,Y) #0 + E Ann((Y))
2. T(X,Y) #0 + p(X,
((X,Y) # + T(X,)) + 0: + and
3. T(X,)) 0
+ p(x, Y) 0 and X Anc(Y) + p(X, Y) 2 0.
Theorem I reveals a nuanced relationship between intrinsic an- cestors and non-zero influence power: neither implies the other. Specifically, causal ancestor of the target can have zero influence power, and conversely, a variable with non-zer influence power not necessarily intrinsic ancestor in the underlying SEM. Similarly, a variable may have a non-zero average causal effect on the target while having zero influence power, and vice versa. Furthermore, neither having a non-zero average causal effect nor being causal ancestor guarantees non-negative influence power. These relationships are visualized in the Venn diagram in Figure In what follows, we demonstrate several key statements from Theorem I with concrete examples, and we defer the detailed proof to Appendix For clarity, these demonstrations focus on binary variables with the desired domain = (1), though these restrictions are not required in general.
Anc
T
P <<
Figure 2: Relationships among
intrinsic ancestors, т, and p.
A causal ancestor can have zero influence power. Altering causal ancestor with strong average causal effect may yield no benefit for the target.
Example 1. Consider the following structural equations and the induced graph:
X Nx.
Z = X N2 + (1 -1-1X11--11
x
Z
Y
Y Z Ny ()- N1-
where Nx, Nz, Ny idd Bern(0.9). Let X and 7 be actionable variables, ler Ax = {0, 1} and
Az = 1) be their feasible domains of alteration, and let the desired domain for Y be S = {1}.
In this example, while X is an ancestor of Y in the SEM, its influence power Y is zero: p(X, Y) max,54x P(Y = 1|X =x) E,-P(x)P(5 = 1|X = x) = max x max,edz P(Y P(x) max P(Y = 1, x) 0.9 0.9 O. This indicates
1|z _ 2, x that altering X yields no improvement in the probability of = 1; rational machine will always maximize the probability of Y = by setting Z to 1, regardless of the value of X. short, altering X in Example is useless because the actionability of Z shields its effect. Thus, causal ancestor does not necessarily have non-zero influence power. In addition, the average causal effect of on Y in the SEM is non-zero: 7(X,)) P(Y 1|X = 1) P(Y 1|X 0) 0.82 0.18 0.64. This shows that a non-zero average causal effect does not guarantee non-zero influence power.
have non-zero influence power. Altering variable that is not intrinsic ancestor of the target in the underlying SEM may still benefit the target.
non-ancestral variable
Example 2. Consider the following structural equations and the induced graph:
U == Nu:
W Nw;
X = U.W. (W (1-Nxx),
U
Z = Nz.
W
Y = Z (1 - U) + (1 ---)- Ny,
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where NV ~ (055( ,w. NN Rid ~ (0rn(0.1), and Ny ~ Bern(0.A).. Let W, X, and Z be actionable variables with Aw Ax Az {0, 1}, and let the desired domain be S {1}.
In this example, W is not an ancestor of Y in the SEM. Notably, the influence power of W on Y is positive: p(W, Y) 0.68 0.518 = 0.162. This indicates that altering W can significantly increase Intuitively, this positive influence manifests because altering W can help X reveal information about U. which facilitates a more informed alteration of Z, ultimately benefiting Y. To provide concrete intuition, us ground the variables from Example 2 in a medical scenario: let U. W, X, Z, and Y denote an allergy gene, skin test, the skin response, a drug injection, and patient recovery, respectively. Performing a skin test (W) has no therapeutic effect; thus, the average causal effect of W on Y is zero. Nevertheless, the skin test is crucial because it informs the doctor's decision to administer the drug (Z), which affects recovery (Y). For instance, if the skin test is positive (observing X = -1 after setting W - 1), the doctor can infer the presence of the allergy and decide not to administer the drug (by setting Z _ 0), thereby maximizing the probability of recovery (Y 1). This shows that while W does not intrinsically cause Y, altering W is instrumental in influencing Y. Influence power successfully captures this implicit benefit, indicating that even non-ancestral variables can be critical for AUF.
the MEP of Y
For completeness, we also examine the conditional influence power of W on Y given U. We find that p(W, Y U = = 0 YU 0) = O. This implies that if the allergy gene (U) is observed, performing the skin test (W) will be unnecessary. In clinical practice, however, directly genotyping the allergy gene (U) for new patient often coneconnu or prohibitively expensive. Thus, the unconditional influence power offers valuable guidance for addressing the AUF problem in practice.
weak ancestor can have positive influence power, Altering causal ancestor with a negligible average causal effect on the target may still benefit the target.
Example 3. Consider the following structural equations and the induced graph:
X = =Nx,
z= (1 - ()-x))-A -N,
Z
x= .Z.
where Nx,NNx1, Ny idd Bern(0.5). Let X and Z be actionable variables with Ax =0, 1} and Az = 10, and let the desired domain for Y be S = (1).
In this example, X is an ancestor of Y, and the average causal effect of X on Y is zero: 7(x, Y) 0. Still, the influence power of X on Y is positive: p(X, Y) = 0.25. Intuitively, this positive influence power manifests through the synergy between X and Z. The benefit of X on Y is elicited when we account for the alteration of Z. This implicit impact is captured by i(X, Y) and missed by T(X, Y).
A strong ancestor can have negative influence power. Altering causal ancestor with non- negligible average causal effect can not only useless but also detrimental for the target.
Example 4. Consider the following structural equations and the induced graph:
U = NU,
X = U - + (1 (x-Nx),
= X. NNN (1- (1)
2
Y = Z 0.11..11110 (1-
where No Bern(0.5), Nx, Nz id Bern(0.9), and Ny Bern(0.4). Let X and Z be actionable variables with Ax = (0, 1} and Az = (0, 1}. and let the desired domain be S = (1).
In this example, X is an ancestor of Y in the SEM with non zero average causal effect: T(X,Y) 0.08, but the influence power negative: p(X, Y) -0.15. This indicates that the MEP after altering X is lower than the expected MEP after observing X. Thus, any alteration on X is counterproducttvv. regardless of the value to which X set. Intuitively, this negative influence manifests because observing X reveals information about U. which is critical in determining the alteration of Z during the computation of p Y). Hence, although altering X can produce a straightforward improvement in Y (as indicated by the non-zero T(X, Y)), this benefit is outweighed by its negative impact on the subsequent alteration of Z, ultimately making the alteration of detrimental. This implicit impact is reflected in the value of influence power.

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4 ESTIMATING INFLUENCE POWER
Influence power is a principled quantity for measuring the influence of actionable variables in AUR. In practice, exact computation can be intractable due to the need to exhaustively evaluate MEP terms; moreover, when the underlying SEM is unknown, the AUF probabilities needed for these terms are not directly available and must be estimated from observational data. To address these challenges, we provide a practical estimation method in this section.
4.1 MONTE- CARTOO APPROXIMATION
Recursively enumerating MEP over all possible alterations can be computationally prohibitive when the number of actionable variables is large. To mitigate this, we cast MEP computation single- player non-deteriinistic game and approximate it using the Monte-Carlo tree search UCT (Upper Confidence Tree) introduced by Koesis & Szepesvári 2006).
Specifically, we incrementally construct a search tree using Monte Carlo simulations. Each node in the tree represents state defined by a sequence of alterations and observations made far, associated with the next variable to be considered. Every iteration starts at the root node No (associated with a pre-specified variable V; E V), traverses its children (associated with Vi+ 1), and continues until reaching a terminal state (associated with the target variable Y). Each edge in the tree represents a choice available at the node, i.e., either an alteration or an observation on the associated variable. The overall construction consists of four steps, iterated until the computational budget is exhausted: (1) Selection: start from the root node and recursively select an edge to child nodes according to the UCT policy until reaching leaf node; (2) Expansion: if the leaf node corresponds to non-terminal state, expand it by randomly adding one child node for each possible choice; (3) Playout: from the newly added node, execute random sequence of choices until reaching terminal state, and compute the AUF probability at that terminal state; (4) Backpropagation: propagate the computed AUF probability back up the tree, updating the statisties of each node along the path. During each iteration, the UCT criterion is used at a node N to select the next edge to traverse:
CN = arg max
& {DN.c + a
Iu
(8)
(N,o
where N = AN U is the set of choices at node N, consisting of feasible alterations the variable associated with N (i.e., AN) and the option to make an observation, (i.c., 0). Here, PN, is the average AUF probability obtained after taking choice at node N, is parameter used to balance between exploration and exploitation (Auer et al.] 2002), fy is the number of times node N has been selected, and NN,e is the number of times choice has been selected node N.
After constructing the search tree, we approximate the MEP terms in the influence power of V on Y by the average AUF probability for each choice the root node No of the search tree. Concretely, we have P(Y E S = c) 2 Noo.c for each e E No. and Eer P(Y E S = v,) N PNo.a. Hence, according to Definition [ the influence power of V; on Y is approximated as:
p(V, Y) R max.cAN, PNo. PN..0-
The quality of this approximation improves over time, as UCT converges to the best choice given sufficient iterations. Moreover, the described procedure is an anytime algorithm, capable of producing an approximate influence power at any point during its computation. We refer the reader to Browne al. (2012) for additional details.
Finally, we emphasize an important practical phenomenon: Equation (9) can serve as a valuable indicator even with limited number of Monte-Carlo simulations. This because AUF does not always require highly accurate estimate of influence power; in many cases, coarse approximation enough. Specifically, if the ground-truth influence power of a variable non-positive (p < 0), the approximation succeeds as long as it correctly indicates that no alteration of the variable is beneficial, which only requires PN..0 2 max.cA N, PNo.e- Similarly, if the ground-truth influence power is positive (p > 0). the approximation succeeds as long as it correctly identifies the optimal alteration c", i.e.. it preserves the correct ordering of the MEP terms; it suffices that PNo.c' 2 maX.EAN, PNo.c- Our experiments provide empirical evidence for this phenomenon: with relatively small number of simulations, the method can often yields the right "alter vs. observe" decision and the best alteration (see Section 5 for details). We believe this phenomenon merits consideration in future studies.
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4.2 AUF PROBABILITY ESTIMATION
While the Monte-Carlo procedure described above can effectively approximate the influence power, terminal states during simulations. it still requires the AUF probability, whose ground-truth value is dictated by the underlying SEM. For situations where the structural equations are unknown, we present an estimator of the AUF probability from observational data.
Specifically, we express the joint probability of (V, Y) as:
Y)V,Y) P(V. Va. Y)= P P(V.[Vi..... ..... Vi-1,,
(10)
where the conditional probabilities VV and P(V. vi-1) can be estimated from observa- tional data D {(v), y'') } using standard ML models. Let A denote the variables in V that are altered. Then the joint probability of (V, Y) given the alteration of A is expressed as:
PIV, = P(Y V) IIve 6(V.) Ilvevi, P(K[V Vi-1),
(11)
where indicates that every variable Vi E A is altered, and 5()) is the Dirac delta function, Let O denote the variables in V that are observed. Then the AUF probability given the alteration of and the observation of O is expressed as:
P(Y E S|A, P(Y ESOOAA Evio VIAA) E(vo((ViA)
P(O)À) Evoo P(Y ESIV) 8(Vi) IIVEV
(12)
EviollviEA 5(V)) I(VVII P(VIVV,..... V/-1)
which is a generic expression of the AUF probability under arbitrary alterations and observations. can be estimated from observational data D and then plugged into the Monte-Carlo procedure described above to approximate the influence power. The following proposition shows the consistency of Equation (12 by applying the manipulation theorem from Spirtes et al. (2000).
Proposition 1. Assume causal sufficiency, i.e., the joint distribution P(V, y induced by an acyclic SEM M with independent noises. Also assume positivity, i.e., P(V.[PA.) > 0 for all 1 S i S. d on the support of P. Then, the expression in Equation (1T consistent with the joint probability dictated by the SEM under the alteration of A. Furthermore, the expression in Equation (12) is consistent with the AUF probubility dictated by the SEM under the alteration of and the observation of O.
Remark. While causal sufficiency is required for Proposition [ it assumed for the remainder of the paper. This assumption is technically needed for reliably estimating the AUF probability in Equation (22 from observational datasets, where relevant variables are assumed to be recorded. Note, however, that we do not assume that the same variables are available at decision time for every incoming instance. Specifically, a variable may be observed in the recorded datasets used to learn the AUF probability, yet be unavailable gunmessured/missingg in the current decision instance (see Example 2}. this sense, learning may rely on richer observability, whereas drecision-making can operate under partial observability. This relaxation of observability is more realistic in practice.
5 EXPERIMENTS
In this section, we conduct experiments to validate the utility of our measure.
Tasks. We simulate three synthetic tasks (TRADER, FARMER, and DOCTOR) and a real-world case study (BERMUDA). We generate 10000 samples from the underlying SEM and repeat each experiment ten times per task. Detailed settings are provided in Appendix A
Baselines. We compare six methods for selecting alterations: (1) OBSERVE, which only observes without altering variables; (2) MAX-ONE, which selects the single variable with the highest AUF probability for alteration, described in Equation I (3) MAX-ALL, which selects all actionable
variables for alteration, as described in Equation 2 (4) CORR, which sequentially selects variables with non-zero correlation and alters them to maximize the conditional AUF probability: (5) ACE, which sequentially selects variables with non- zero causal effects and alters them to maximize the interventional AUF probability; and (6) OURS, which sequentially selects variables with positive
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TRADER
FARMER
DOCTOR
Quantity Devistion Decision Mistaki
Mistakee Rate (%)
Quantity Deviation
Mistakee Rate (%)
Quantity Deviation
Mistakee Rate (%)
Decision Mistake
Dection Mistake
Deviation
Deviation
Deviation
05
0.05
56
250 MCTS Iterations
500
75.0
1000
250
750
1000
250
750
1000
MCTS iterations
MCTS iterations
Figure 3: Convergence of influence power estimates and mistake rates (%) versus the number of MCTS iterations. The deviation of the approximated influence power from its exact value continues to decrease after the mistake rates have converged.
TASK
OBSERVE
MAX-ONE
MAX-ALL
CORR
ACE
OURS
TRADER 37.62 + 4.31
51.01 5.13
50.82 5.00
47.73 6.38
51.13 4.76
60.94 8.43
FARMER
10.05 + 2.87
62.90 + 7.39
63.18 # 7.64
63.88 + 7.59
62.66 * 7.33
63.86 + 7.42
DOCTOR
39.81 4: 5.06
50.76 5.50
51.08 4.80
51.33 5.07
50.93 5.18
65.32 +:2 4.12
Table 1: Success rates (%) of six methods on three synthetic tasks.
TASK
10
100
500
1000
5000
TRADER
42.97 à: 7.24
49.45 + 5.12
51.86 à 9.59
57.63 + 6.64
57.08
8.86
60.34 #: 5.64
FARMER
19.23 11.2
31.80 13.9
60.49 11.9
62. # 8.60
63.22 8.39
63.62 9.38
DOCTOR
44,20 5.07
43.18 + 4.86
46.41 # 6.98
64.96 + 5.43
65.20 + 6.62
65.72 + 4.50
Table Success rates (%) of our method with sample sizes on three synthetic tasks.
TASK
OBSERVE
MAX-ONE
MAX-ALL
CoRR
ACE 69.61
OURS
BERMUDA
2.29 0.49
61.99 1.43 72.71 1.92 19.09 5.79
4.44
75.16 0.89
Table 3: Success rates (%) of six methods on the BERMUDA task.
influence power and alters them to maximize the MEP. Following Kocsis & Szepesvári (2006), the parameter for MCTS is set to 22 by default. For fair comparison, the feasible domain for each actionable variable is set to {0, }} and the number of actionable variables is set to 3 for all methods in synthetic tasks. Each method is evaluated by its success rate, i.c., the frequency with which the target variable falls in the desired domain after applying the suggested alterations.
Figure 3 shows the convergence of approximating influence power. The plot depicts the deviation of the approximated value for the first actionable variable, measured as the absolute difference from the exact value. The mistake rate denotes the frequency of inconsistencies between the suggested alterations based on the approximated value and those of the exact value. In all cases, the mistake rate decreases T increases, demonstrating the effectiveness of MCTS in approximating influence power. Notably, the deviation continues to diminish after the mistake rate has converged to zero; that is, the estimate becomes conssisennnt before it fully converges numerically. This empirically supports the phenomenon discussed in Section 4. a coarse approximation based on a limited number of simulations can still provide useful decision guidance.
Table D compares our method with baselines, showing that our method outperforms existing methods in most cases. These results demonstrate the superiority of the proposed method in guiding alterations for AUF tasks. In the FARMER task, various methods perform comparably. This is because the target variable in this specific task is influenced by single critical variable, which all five methods correctly determined. Table innestigates the impact of sample size the effectiveness of our method. The performance generally improves as the sample size increases. Notably, the success rates exhibit a rapid growth initially and begin to plateau, stabilizing around 1,000 samples across the tasks.
We further validate our method on the Bermuda data to demonstrate its adaptability to real-world constraints involving non-binary variables. As shown in Table 3 our method achieves the highest success rate, outperforming the baselines. These results demonstrate that our measure is valuable and applicable in real-world scenarios with nin-binary variables.

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6 RELATED WORK
This work is grounded in the concept of influence and situated within the rehearsal paradigm (Zhou 2022b; 2023). which advocates for proactively rehearsing future possibilities under various actions to determine alterations that influence the future target before making a final decision-an idea analogous to how human cognition prepares for future events (Driskell et al., 1994). Building upon this foundation. Qin et al. (2023) proposed a rehearsal learning approach that optimizes decisions while learning a structural rehearsal model capable of accommodating bi- bi-directional interactions. Further studies have addressed issues such
-on-staiionarity and non-lin non-linearity in rehearsal learning
(Du et al.
2024 Qin et al., 2025; Tao et al. 2025 Du et al. 2025a lb).
A substantial body of work has focused on inferring causal relations from observational data Vermal
& Pearl 1991] Cooper & Herskovits] 1992; Heckerman et al. [995] Tian & Pearl [002] Shpitser &
Pearl 2006 Huang& & Valtorta| 2006; Zheng et 2018| Lorch et al.] [002]] Various measures for
evaluating the strength of causal relations have also been proposed (Northcott 2008 Janzing et al.
2013 Heskes et al. 2020; Jung et al. 2022 Janzing et al. 2024). This work primarily compares to the ACE (Rosenbaum & Rubin 1983 Holland 1988). as it is among the most widely used measure of causal strength the literature; comparisons with other measures are similar and left to future work. Besides, this work differs from work based on counterfactual reasoning (Halpern 2015 Karimi et al., 2021 Tsirtsis et al. [02T] Beckers et al., 2024). The AUF problem inherently forward-looking. concerning planning for the future, while counterfactual reasoning typically concerns the past, asking what would have happened had we chosen differently at a point in the past (Pearl et al. 2016).
Decision-making problems have been studied from different perspectives. Reinforcement learn- ing (RL) methods (Suttoon & Barto 2018 Lillicrap et al.) 2016 Schulman et al. 2017 Haarnoja et al., 2018) often excel at learning optimal policies through extensive environmental interaction, but such interaction data can be prohibitively expensive or unavailable in many real-world settings. Moreover, unlike common RL formulations in which agent can revisit states, real-world AUF scenarios do not permit "rewind and try again": once realized, past variables are fixed and cannot be changed. Other frameworks, such dynamic treatment regimes (Chakraborty & Moodie 2013 Chakraborty & Murphy 2014 Zhhan & BBreiiboim] 2019: 2020). also address sequential planning, offen via G-estimation or Q-learning; however, they usually enforce a strict distinction between state and action variables, as is standard in RL. In contrast, this work treats all variables uniformly as random variables; one may choose to alter a variable (set a value) or refrain from alteration (letting the variable occur naturally). Besides, causal bandits (Lattimore et al. [016] [ee & Bareinboim] 2018) also address decision-making with causal considerations, though most existing methods require the underlying causal structure to be supplied by domain experts (Bareinboim al. 2015 Lee & Bareinboim 2019)
2020
Aglietti et al., 2020 Lu et al. 2020 Bilodeau et al. 2022; Wang et al. 2023 Varici et al.
2023 Branchini et al.) 2023 Park et al. 2025 Park & Lee 2025 2026). More importantly, these
approaches typically account only for the straighttorward effect of intervention on the outcome, rather than the broader, holistic influence considered in this work. Furthermore, different from previous approaches based on decision networks (Howard & Matheson] 1984 Nilsson & Lauritzen 2000 Everitt et al. [ Van Merwijk et al. 2022). which typically lack natural values for decision nodes, require pre-specified conditional probability distributions, and often operate under Limited-emory constraints, this work acknowledges an underlying SEM as the natural generation process, enabling data-driven learning and accommodating an arbitrary number of actionable variables.
7 CONCLUSION
In this paper, we investigate the challenge of measuring the influence of actionable variables in the AUF problem. We present principled quantity, termed influence power, that evaluates the degree to which altering a variable can increase the probability of a more desired future by accounting for the interplay between potential alterations and subsequent observations under the principle of maximum expected utility. Through a systematic analysis, we demonstrate an insightful distinction between influence and causation: non-ancestral variables can possess non-trivial influence power, and some causal ancestors may be ineffective or harmful to manipulate. We further develop a Monte- basld method to efficiently estimate influence power from observational data without requiring knowledge of structural equations. Experiments on synthetic and real-world tasks validate the utility of the proposed measure in guiding decisions for addressing the AUF problem.
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ACKNOWLEDGMENTS
This research was supported by NSFC (62576165, 62406137) and Jiangsu Science Foundation Leading-edge Technology Program (BK20232003). Tian-Zuo Wang was supported by Xiaomi Foundation. The authors would like to thank Jia-Wei Shan and Tian Qin for their helpful comments.
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A DETAILED SETTINGS
A.1 THE SYNTHETIC TASKS
The underlying SEM for the TRADER task governs the sequence of variables (V1, V2, V3, V4, Y) through the following structural equations:
Vi = Ni:
V/2 = Vi N2 + (1 Vi) (1 - N2),
V/3 = V/2 + (1- 12) -0] Nz),
V1 = V/3 N4 +1 1/) 10-(1 N1),
Y = 14 (1- + (1 V41 Ny,
where the noise terms follow N1 ~ Bern(p1), N22 ~ Bern(p2), N3 ~ Bern(ps), N4 ~ Bern(ps).
and Ny Beru(py). To facilitate diversity across experimental trials, the task parameters are independently and uniformly sampled from the following intervals: f1 E [0.4, 0.6), P. E [0.7, 0.9] for i E {2, 3, 4), and py E [0.3, 0.5]. The variables V22 V3, and V42 are actionable with feasible domains Av, = Av, = A = {0. The variable V1 is not observed at the time of decision. The desired domain is specified
S {1}. Here, Vi represents the economic climate, and the chain V/2 - V/3 + V4 models the progression from consumer demand to the final marketing strategy. The target Y denotes quarterly profit, whose structural equation explicitly encodes the interaction between strategy (V₂) and environment (V1), implying that the profitability of a specific strategy relates to the prevailing economic state. The objective
determine the actionable variables maximize profit.
The underlying SEM for the FARMER task governs the sequence of variables (V1, 12, V3, V4, Y) thorough the following structural equations:
V = N1,
V/2 = (1 - V11 -N:
V/3 = (1 - V/2) -N.
V1 = (1- - Va)= N1.
Y = Vi VV.NY,
where the noise terms follow N₁ (ern(8)), N2 ~ Bern(B2), Ng (arn( Na- Bern(31), and Ny ~ (ern( To facilitate diversity across experimental trials, the task parameters are independently and uniformly sampled from the following intervals: 3 [0.7, 0.9] for E 2,, 3, 4}. and By E [0.7.0.9]. The variables V22 Vs. and Va are actionable with feasible domains Av, = Av, Av, 1). The variable V is not observed at the time of decision. The desired domain is specified as S = (1). In this context, V1 represents sunlight exposure, while the chain 1/2 V/3 Va models the natural water cycle affecting the soil. Specifically, intense sunlight (Vi) naturally reduces precipitation (V2), which in turn increases evaporation (Vg). ultimately leading to low soil moisture (V1). The target Y denotes crop yield. The structural equation for Y explicitly encodes the essential interaction between light (V1) and water (V4), implying that high productivity requires the simultaneous presence of both sunlight and adequate soil moisture. The objective is to determine the actionable variables to maximize crop yield.
The underlying SEM for the DOCTOR task governs the sequence of variables (V,, V2, V3. V4, Y) thorough the following structural equations:
V1 = N1.
V/2 = N2.
V/3 = V1 V/2 - (1-Ns),
V4 = No.
Y = V4 (1- Vi)) + (1- Ny,
where the noise terms follow N1 ~ Bern(>1). N₂ ~ Bern(12), N₂ ~ Bern(1s), N₂ ~ Bern(ya). and Ny Bern(>y). To facilitate diversity across experimental trials, the task parameters are independently and uniformly sampled from the following intervals: 7 E [0.4, 0.6). 7 E [0.1, 0.3] for i (2,3, 4), and yy [0.3, 0.5). The variables Vz, V3. and Va are actionable with feasible domains
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Av = Av, = Av, = 1). The variable Vi is not observed at the time of decision. The desired domain is specified as = In this context, Vi represents the drug intolerance (or allergy gene), while V2 represents diagnostic indicator. The structural equation for V3 implies that the symptom manifests primarily when both the intolerance (Vi) and the trigger (V2) are present. V4 represents the administration of potent drug. The target Y denotes patient recovery. The equation for Y captures critical medical contraindication: the drug (V4) effective for the general population (V₁ but harmful fatal to patients with the intolerance (V1 1). The objective is to determine the actionable variables to maximize patient recovery.
environmental trigger. V3 denotes symptom (e.g., rash), which serves
A.2 THE BERMUDA TASK
The BERMUDA case study is derived from real-world scenario involving the management of net coral ecosystem calcification in Bermuda, where environmental variables are recorded (Aglietti et al.) 2020
The sequence of variables in this task are listed as follows:
Light: bottom light levels;
T Tem: bottom temperature:
Sal: sea surface salinity;
DIC: seawater dissolved inorganic carbon;
T T'A: seawater total alkalinity;
DAi seawater saturation with respect aragonite;
Nut: PCI of NH4, NiO2 + NiO3, SiO4;
Chla: sea surface chlorophyll-a;
pHywi seawater pH;
:oos: seawater Poos:
NEC: net ecosystem calcification.
The underlying structure governing these variables adopted from Courtney al. (2017) and is illustrated in Figure 4 Consistent with previous studies Aglietti et al. (2020); Qin et al. (2025) the structural equations were obtained by performing linear regression on the 50 observations provided by Andersson & Bates (2018), and there are five actionable variables including DIC, T T.A Ω Nut, and Chla. Other variables are not observed at the time decision. To ensure compatibility with our method, we discretize the continuous variables by dividing their value ranges into six equal-width bins. Accordingly, the feasible domains for alterations are set to 2-5,, -15,, -0.5, 0.5, 1.5, 2.5). The desired domain is specified as S [1, The objective is to determine which variables to alter order to achieve net ecosystem calcification (NEC) within the desired range, thereby promoting coral reef health and resilience.
Light
DIC
PcO.
aA
Tem
TA
EC
pH sw
Chla
Sal
Nut
Figure 4: The underlying structure of the BERMUDA task.
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B PROOF OF THEOREM 1
Theorem 1. Let X and Y be two endogenous variables in an SEM. The following statements hold:
1. X E An)(Y) + x(X,)) # and p(X,Y) #0
+ X Anc(Y);
2. T(X,Y) #0
+ p(XY))
and
p(X,Y) # + T(X,)) # 0:
3. T(X,Y) #
+ p(X,Y) 2 and X Anc(Y) +> p(X,Y) 2 0.
Proof. We prove each statement separately by constructing counterexample. Statement (a): X E Anc(Y) + p(x, Y) + O.
To show that X E Anc(Y) does not imply p(x, Y) & O, it suffices to provide where variable an ancestor of another, yet its influence power on the latter zero.
Consider the following SEM over the sequence of variables (V1, Y):
V1 N1, Y = V Ny + (1 - (1--rr)
where N1 ~ (0.m(. Ny ~ (0rn().5). V1 is actionable with Av; = {0, 1}, and the desired domain for Y is S = {1}.
In the SEM, we have
V1 Anc(Y),
and
P(V,,Y) max,, EAv, P(Y 1 _ 01) (e)rr())P(( P(Y 1 01)
max Edv P(Y 1 _ 01) P(Y =1
max 0.5) 0.5
0.
Thus, an ancestral relation in the SEM does not imply non-zero influence power.
Statement (b): T(X, + 0 + P(X,) # O.
To show that 7(X,Y) 전 does not imply p(X, Y) * 0, it suffices to provide a case where a variable has non-zero average causal effect another, yet its influence power the latter is zero. Consider the following SEM over the sequence variables (V1, V2, Y):
Vi == N1, V/2 Vi N₂ + (1- (1-N)),
= V/2 + (1- V22) - (1-Ny)
where N1. N2, Ny ~ (0.99. V1 and 1/2 are actionable with Av, Av, = 10, and the desired domain for Y is (1). This SEM corresponds Example the main text.
lid
In the SEM, we have
TV.. P(Y 1|V, 1) P 1 0)
0.82 0.18
0.64,
and
P(V,,Y) maxenddy P(Y 1 _ v1) EE 1|V, v1)
max max P(Y 1|V, a U2, V1 v1)
-Ee~>((V) max P(Y 1 _ U2, V1 v1) eyEAv
-max(0.900.)
0.
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Thus, non-zero average causal effect in the SEM does not imply non-zero influence power. We also note that Statement (b) implies Statement (a), as T(X, Y) * 0 implies E Anc(Y).
Statement (c): p(X, Y 0 + T T(X Y) # O.
To show that p(X, Y) * 0 does not imply TXX, Y) * O, it suffices to provide a case where a variable has non-zero influence power another, yet its average causal effect on the latter is zero.
Consider the following SEM over the sequence of variables (V1, V22 Y):
V1 = N1,
V22 (1 - 1/1) N2.
Y = Vi Vy,
where N1. N2, Ny iid Bern(0.5). Vi and 1/2 are actionable with Av. Av. {0, 1). and the
desired domain for Y is {1}. This SEM corresponds to Example 3 the main text.
In the SEM, we have
TV,, = P(Y 1|1 1) - P(Y = 1 _ 0)
0.82 0.18
= 0.64,
and
P(V, max,leAv, P(Y = 1|V, = v1) Be,, P(Y 1|V vi)
max P(Y 1|V2 = =
diEAv; DEED
-Bni-((V)) uyEAv
P(Y = 1 - 022 V13 21
m0, 0.50 - 0.25
0.25.
Thus, non-zero influence power does not imply non-zero average causal effect in the SEM. Statement (d): p(X,Y) # + E Anc(Y).
To show that piX,Y) Y) * 0 does not imply X E Anc(Y', it suffices to provide a case where variable has non-zero influence power on another, yet it is not an ancestor of the latter.
Consider the following SEM over the sequence of variables (V1, V2, V3. V4, Y):
Vi = No.
1/2 = N2.
V/ == Vi V/2 (1-Ns),
V4 = No.
Y = V44 (1- Vi) (1- V1) Ny,
where N, ~ (0rn((.5), N2, N3. N4 sed (ern( ~ Bern(O.4), V2, V3, and V44 are actionable with Av, Av, Av. (0, 1). and the desired domain for Y is S {1). This SEM corresponds to Example in the main text.
~
In the SEM, we have
V₂ & Anc(Y),
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and
p(V2, Y) max P(Y = 1 02) E P(Vs) P(Y 1/12 v2)
max max e2
max BEE
P(Y - 1|V3 v3 1/2 U2),
P(a/(( P(Y = 1 = U3, V22 _ 02)
P(Vs) max
max P(Y = 1113 _ U3, V/2 = = UyEAvs
E B P(Y 11V3 vz 1/2 v2)
max
Pg P(Y = 1 = 03 1/2 _ U22)
"1
P(y Bog P(Y = 1 = U3 1/2 = U22
E. e2ED1 13 PV-
max EED
P(Y 1 14 V/3 rg 1/2 4 122)
max ME
P(Y 1 V/ = U3 V/2 = U2)
0.68 - 0.518
0.1622.
Thus, non-zero influence power does not imply an ancestral relation in the SEM. We also note that Statement (d) implies Statement (c), as X & Anc(Y) implies rXX, Y) = 0.
Statement (e): X E Anc(Y) + (X,Y) 0.
To show that X E Anc(Y) does not imply p(X, 2 0, it suffices to provide a case where a variable an ancestor of another, yet influence power on the latter negative.
Consider the following SEM over the sequence of variables (V1, 1/2 Y):
V1 NN.
1/2 = (1 V/1) N2.
Y (V1 V2) -N.,
where N1 ~ Bern(0.5), N2, Ny sid domain for Y is S (1).
~ Bern(0.8), V/2 actionable with Av; = -0, 1}. and the desired
In the SEM, we have
V/ E Anc(Y),
and
Y)
P(Y 1|V2 = v2) Eee P(Y =1122 =2) maX P(Y 1|V2 - v2) -P(Y= 11)
max(0.4,0.44 0.72
-0.32.
Thus, ancestral relation in the SEM does not imply non-negative influence power.
Statement (f): T(X,Y) * 0 + ((XY)) 2 0.
To show that (XXYY) * does not imply P(X, 0, it suffices to provide a case where variable has a non-zero average causal effect on another, yet its influence power on the latter is negative.
Consider the following SEM over the sequence of variables (V1, V2, Vs. Y):
Vi N1.
1/2 = Vi N2 + (1-1) . (1-N)),
V/3 =
+ V
(1- 12) Na),
Y = V/3 (1-1(V)) Vi) +1)+(1 1/) -N,
19

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where N1 ~ Bern(0.5), N2. N3 dd (0rn( Ny ~ O.rn. V2 and V3 are actionable with Av. = {0, 1 and Av; = {0, 1), and the desired domain for Y is S {1}. This SEM corresponds to Example in the main text.
In the SEM, have
T(V2, = P(Y 1 - 1) P(Y 1 _ 0)
-0.99 0.41
= 0.08,
and
2(V2, Y) maX P(Y 1| = 02) B P(Y 1|12 =2)
max max P(Y 1 U3 1/2 122
eyEAv, vyEd
-BVs)((s)) eyEAvs
max P(Y 1 U3 V22 = 022)
[0.5 0.5) 0.65
= 00.15.
Thus, a non-zero average causal effect in the SEM does not imply non-negative influence power. We also note that Statement (f) implies Statement (e), as r(X,Y) & 0 implies X E Ancc(Y).
C PROOF OF PROPOSITION 1
Proposition 1. Assume causal sufficiency, i.e., the joint distribution P(V, Y) is induced by an acyclic
SEM M with independent noises. Also assume positivity, i.e., P(V.[AA)) > 0 for all S S on the support of P. Then, the expression in Equation (TT consistent with the joint probability dictated by the SEM under the alteration of A. Furthermore, the expression in Equation (12) is consistent with the AUF probability dictated by the SEM under the alteration of and the observation of O.
Proof. Recall from Equation 11p. the joint distribution conditioned on the alteration set À is expressed as P(VIÅ) = IIv.A o(V)) IIvEvv P(V|VV,.... V2-1). As the sequence is topologi- cally consistent with the underlying SEM, and the SEM assumed to be acyclic, the value of each variable Vi depends solely its parents PA,, Consequently, P(V] V/-1) stituting this back into the product shows that P(VIÁ) IIv 8(V,) II VEVIA P(V][PA,). By invoking the manipulation theorem (i.e., Theorem 3.6 in Spirtes al (2000). we have that P(V Á) exactly the probability of V under the alteration of A. Moreover, quantity P(Y E SJÅ, O) in Equation 12 is fully determined by P(V| A), and therefore Equation 12) indeed gives to the true AUF probability dictated by the underlying SEM.
P(V]IPA,). Sub-
20


Reviews:
[{"review_id": "w0MoOLVeXW", "reviewer": "Reviewer_eoMz", "summary": "This paper proposes a framework for the “avoiding undesired future” (AUF) problem, which aims to identify which variables should be changed to prevent a predicted negative outcome. The authors introduce a metric, called influence power, to measure how useful each variable is to achieving this goal. They claim that influence power differs from traditional causal measures such as the average causal effect,  since variables with strong causal effects may have little or even a negative influence. In contrast, weakly causal or non-causal variables may still have a positive impact. The paper also introduces a Monte Carlo Tree Search (MCTS) method to estimate influence power using observational data.", "questions": "* Can the authors clarify how their \"influence power\" notion and overall framework relate to the above-suggested approaches?\n\n* Given the computational intensity and large number of Monte Carlo simulations required to achieve meaningful results, can the authors clarify how their method scales in practice?", "limitations": "", "rating": 4, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper tackles a relevant and important conceptual problem: moving from passive prediction to proactive intervention to avoid undesired outcomes. The motivation is to develop a principled approach to determine which variables to alter. I liked the connection between causal reasoning and utility-based decision theory, specifically through the principle of maximum expected utility and a Bellman-style recursive definition.", "weaknesses": "* The notion of \"influence power\" should be compared and contrasted with similar intervention-based approaches: for example, actual causes [1] (smallest set of variables that can be altered to change an outcome), counterfactual explanations [2] (set of interventions that optimise a counterfactual outcome), or the paper [3] on agent incentives (which also uses utility to evaluate interventions, and discusses a related notion of value of control).\n\n* The experimental section is limited to three toy models (trader, farmer, and doctor). The baseline comparisons focus on simple strategies (altering the highest probability variables or altering all variables); it'd be good to compare against a selection of the above-recommended approaches.\n\n\n[1] Halpern, Joseph Y. \"A modification of the Halpern-Pearl definition of causality.\" arXiv preprint arXiv:1505.00162 (2015).\n\n[2] Tsirtsis, Stratis, Abir De, and Manuel Rodriguez. \"Counterfactual explanations in sequential decision making under uncertainty.\" Advances in Neural Information Processing Systems 34 (2021): 30127-30139. \n\n[3] Everitt, Tom, Ryan Carey, Eric D. Langlois, Pedro A. Ortega, and Shane Legg. \"Agent incentives: A causal perspective.\" In Proceedings of the AAAI conference on artificial intelligence, vol. 35, no. 13, pp. 11487-11495. 2021.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Z2cFgRjz5C", "reviewer": "Reviewer_KrbU", "summary": "The paper introduces *Influence Power*, a metric aimed at quantifying how altering a variable affects the probability that an outcome variable falls within a predefined desired region ($P(Y\\in S)$).\n\nThe authors build on a Bellman-style recursive formulation and define Influence Power as the difference between the optimal success probability under intervention and the expected success under natural observation. They then propose a Monte-Carlo tree search (MCTS) approach to approximate influence power using observational data.\n\nExperiments on small synthetic SCMs suggest that the proposed MCTS approach outperforms trivial baselines in increasing $P(Y\\in S)$.", "questions": "- Are you solving a *counterfactual* question (\"what if we changed X in this observed world?\") or a *decision-theoretic* question (\"which variable should we alter next to maximize expected success\")?\n\n- Are all variables observed before Y is realized?\n\n- Clarify the formal problem and specify all assumptions clearly.\n\n- Improve the experimental design. \n\nIf you choose to use ACE, unroll the SCM over time to make it fair. Consider also adding the following:\n1. Add tasks with non-binary variables, nonlinear or multiplicative mechanisms, or denser graphs.\n2. Add a baseline that only observes.\n3. Include sensitivity analysis for $\\alpha$, order errors, and sample size.\n4. Report runtimes and sensitivity tests\n\n- How stable is influence estimation across multiple Monte Carlo seeds or data resamples?\n\n- Do you think that, given the exponential growth of the search tree in MCTS, the method is inherently limited to a small number of alterable variables (which is likely partly why the experiments use small toy models)? Perhaps you have future work in mind on sampling or pruning strategies to improve scalability?\n\n- Notes on improving notation (minor):\n1. The notation $(V_i^a=v_i) / (V_i^o=v_i)$ is non-standard. The paper introduces both this notation and the standard *do*-notation (lines 87-88). Ideally, introduce only one and use it consistently.\n2. Keep observed context (x) explicit in all conditional expressions.", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 1, "contribution": 2, "strengths": "- Influence power is a novel metric that combines ideas of potential intervention effect and value-of-information analysis.\n- The recursive Bellman-inspired formulation is a creative way to connect causal reasoning with sequential decision making. If done right, this could be impactful.\n- Examples 1-4 clearly illustrate the core intuitions (but their extended form could be moved to the Appendix).", "weaknesses": "- Conceptual confusion: counterfactual vs. interventional reasoning\n\nThe AUF problem is framed as counterfactual (\"given that the world will look like X, what if we changed Z?\"), but the paper only computes interventional and observational probabilities. This gap violates the Causal Hierarchy Theorem [1]: interventional data alone cannot answer counterfactual queries. For example, in lines 11-12: \"When a model predicts an undesired outcome, it is often crucial to determine what we can change to avoid it.\". Posed this way, it is theoretically impossible to solve the AUF problem (at least exactly). The formulation in the paper, though, is closer to a sequential decision process (or a planning problem) than to counterfactual inference. The authors likely need to just write their problem more formally to make this clear. See more points below around assumptions and the concept of time.\n\n[1] Bareinboim et al. \"On Pearl's Hierarchy and the Foundations of Causal Inference\", 2020\n\n\n- Ambiguous use of time and ordering\n\nThe paper implicitly assumes a temporal sequence of variables $(V_1,\\dots,V_d,Y)$ but **SCMs are static**.\nPhrases such as \"subsequent variables\" or \"before Y is finalized\" suggest temporal evolution, yet this is never formalized (no explicit time, transition, or policy definition).\nThe authors effectively simulate time by imposing a topological order, conflating causal order with temporal decision order (see below for discussion of ACE suitability to this setting). To resolve this ambiguity, the authors need to formalize their problem within a framework that supports time (Example unclear question: Are all variables observed before Y is realized? I would guess yes, but it's very unclear). \n\n\n- Unclear problem definition and assumptions\n\nKey assumptions are not clearly stated:\n\n1. Causal sufficiency (no hidden confounders) seems assumed but contradicted by examples involving an unobserved $U$. For instance, in example 4 the agent must choose $X$ without access to $U$, even though $U$ influences both $X$ and $Y$. From the decision-maker's perspective, $U$ acts as a hidden confounder, creating an inconsistency with the assumption of causal sufficiency (the time dimension is what makes this so complicated).\n2. Is the causal order taken as known? (I assume yes after reading Proposition 1)\n3. Positivity/overlap conditions for feasible alterations are unstated.\n4. The predictor $h(x)$ is mentioned in preliminaries but never used formally.\n\nThe absence of assumptions makes the formal objective of the AUF problem ambiguous.\n\n\n- Misinterpretation of causal relationships\n\nSeveral claims conflict with standard causal semantics. Some examples:\n\n1. *Example 2:* the paper concludes that altering $X$ is \"counterproductive\" despite a positive ACE. In standard SCMs, this is impossible. The claim arises only because intervening on $X$ destroys information useful for later decisions (N.B. SCMs don't support time and order of decisions).\n2. *Example 4:* a non-ancestor variable $W$ is said to have positive influence on $Y$. In causal terms, $do(W)$ cannot affect $Y$ at all. The \"influence\" is because of the effect of $W$ on how informative later observations are, not on the data-generating process itself.\n\nHence, \"influence\" in this paper measures more like *policy utility* rather than *causal effect* (which is also why comparing to ACE is not very suitable, see next point).\n\n\n- Over-reliance on ACE comparisons and why ACE is unsuitable for the current formulation.\n\nThe paper repeatedly contrasts Influence Power with ACE.\nACE is defined for **static SCMs**.\nComparing it to a metric that implicitly models information propagation over time is conceptually inconsistent. If the SCM were explicitly **unrolled over time** (with one variable per time step) and ACE were computed in that temporal formulation (e.g., changing $W_0$ and measuring its effect on $Y_t$), the comparison would then be meaningful, and $W_0$ would indeed be a causal ancestor of $Y_t$, even though it's not a causal ancestor of $Y_0$.\n\nAdditionally, ACE measures the *expected* change in $Y$ when a binary variable flips from 0 to 1. It is not informative when the goal is to increase the probability that $Y$ falls within an arbitrary region (even in a counterfactual setting, such reasoning is on an individual level, while ACE is population level) or when variables are non-binary. This comparison adds little insight and occasionally misleads (e.g., claiming that a variable with negligible ACE may still be \"highly influential\").\n\n- Toy experiments and limited validation\n\n1. The experiments involve only tiny binary SCMs, no non-linear or continuous settings are tested.\n2. The baselines (\"max-one\", \"max-all\") are trivial (and arguably not suited for this problem without SCM unrolling), so improvement isn't that unsurprising. There should also be a comparison to a baseline that only observes.\n3. No runtime analysis or sensitivity tests ($\\alpha$ parameter, wrong topology, data size). For example, MCTS becomes expensive as the number of alterable variables grows, since the tree expands exponentially.\n4. The claim that \"a rough approximation suffices\" lacks evidence.\n\nFurthermore, the experiments evaluate success in expectation over the entire data-generating process rather than at the level of individual contexts.\nIn each task, the authors compute ($P(Y\\in S)$) under different alteration strategies averaged across all exogenous realizations. This measures population-level effects but not the per-instance decision problem implied by the AUF formulation (\"given this particular observation, what should be altered to avoid the undesired future?\"). Consequently, the experiments do not validate whether influence power improves decision-making for specific predicted outcomes. They only show population-level improvements. This is misaligned with what AUF is framed as, a context-specific (conditional) decision problem.\n\n- The paper includes no reproducibility statement or supplementary material. Although not strictly required, this limits my ability to verify the reported experimental results.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "GCbt6Wh9Yh", "reviewer": "Reviewer_NVBs", "summary": "The authors present a novel approach to the AUF (avoiding undesirable future) problem\nwhich improves on SOTA (one-alteration and all-alteration). Their approach can also capture negative\ninfluence, ie changes which are strictly bad with respect to a given target. They use a Monte Carlo\nalgorithm to make estimating their measure more efficient (than brute force). They evaluate their\nmethod on three standard(?) problems and show that their measure allows them to compute\ninterventions which increase the chances of the model reaching the desired target region.", "questions": "(numbers are line numbers)\n\n089 how much is nearly negligible?\n103 it is assumed that all variables are alterable? How strong is this assumption?\n- it feels like a very strong assumption to me, as any variable on the causal path (or off of it)\n  can be directly manipulated. This certainly does not reflect real world limitations.\n\n104 \"desired region\" feels a little vague. S is a subset of the possible values of Y?\n107 \"as much as possible\" is also rather vague. Is there an implicit threshold or ratio here?\n\nI don't think that eq 2 implies that all variables *must* be altered, only that they can be altered.\nAfter all, the equation simply states that we must go over all variables and set their values, but\nit does not state that the values must be **different** from observed. So the criticism on l.139 does not hold.\n\n133 do not use contractions\n\n155 I don't like the way this is presented. k = d comes out of nowhere, and the MEP and AUF\nequations appear to be identical at this point. It's only later that it becomes (slightly) more\nobvious that this is deliberate.\n\n158 eq 3 is not well formatted. =o, unlike =a, has not been previously defined. Presumably it means\nsetting V_k to its actual value in the context. But wait, we get the definition on l.164. This is\nbackwards and very hard to read. Why not use <- for alteration, and = for observation?\n\n173 Def 1: clearly related to average causal effect. Is it a generalisation?\n\n211 causality need not be transitive. See Halpern for a detailed discussion.\nIt is also not that surprising that a alterable variable be non-influential. After all, the variable\nmay cause an exact value of Y in S, but not have sufficient influence to move out of S.\n\n200 notation for \\tau is really buried in the text here.\n\n226 this is a slightly forced example, as X and Z are essentially independent, they are both on the\ncausal path, but X is not a cause. However, it's good to see that IP can detect this.\n\n245 here we have the problem of talking about negligible ACE, as 0.08 is surely very low. Is this\nnegligible (see comment above)?\n\n262 again, not a surprising result. And the influence (271) is small.\n\n274 small note, but it should read \"considering altering\", not \"considering to alter\"\n\n278 \"despite it not being a causal ancestor\"\n\nExample 4. I do not understand your explanation. It is not intuitive (to me) and seems quite\nan important point. This needs clarification.\n\nEq (9) is the \\delta embedded in the equation standard notation for something?\nWhat is A hat? The MC approximation of A at any point?\n\n420 \"repeat experiment _with_ ten times\"\n\n421 I looked at Appendix A. I'm slightly disappointed by how small/simple the examples are. I wonder\nhow well the MC sampling approach would work on bigger/more difficult models. Moreover, they are\nlargely identical to your examples. I think you could just merge this together, name the examples\nappropriately, highlight differences if they exist, and remove the appendix. It would make it much\neasier to read.\n\n438 \"demosntrating\"\n\nFigure 2 is very small and quite difficult to read. Please replot or make larger for camera ready.\n\nTable 1 does not show how much work is required for your MC method. Your results are already better\nat T=10. I'd like an idea of how much comparable work is performed by Max-One and Max-All.\nI'm assuming (because it's not in the comment at least) that these are mean values. While I'm not in\nfavour of unnecessary statistical analysis, some might be appropriate here. Perhaps a box plot might\nbe more revealing here, as the standard deviations are quite large for all the tools, and this is\nsomeone obscured by presenting the data as a table.\n\nThe results seems to follow a logaritmic curve... why is this? Is this a limitation of the the MC\nmethod? After all, it should be possible, if all variables in V are alterable, to always guarantee Y\nin S? Or is this a result of the fact that you cannot alter variables before some point d?\n\nConclusion section. \"intriguing possibility...\" but I do not understand why this happens, and your\nintuition above is not (to me) intuitive. I could perhaps see the sharing of information via mutual\ninformation at play, but this is non-directional. Is this not instead revealing a limitation in\n(Pearl's conception of) SCMs based on probabilities? It's probably always possible to construct\nthese pathological examples (because they are divorced from real data producing systems) but do they\nreally tell us anything interesting, other than to look at for these potential errors?\n\n_Questions_:\n  * distance of alterations from outcome Y. Are they usually proximate or far away?\n  * are interventions consecutive or disjoint in general?\n  * are there any patterns/implications in distance from Y?\n\n  * frequency of achieving S in your simulations?\n      + S is always limited to one outcome I think, so the entire thing is boolean. Will this scale\n    to non-boolean settings.\n  * size of alterations vs frequency of S?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "I quite like this paper, I'd give it a weak accept. It's far from perfect, the novelty (while\nthere) is not earth-shattering and most of the worked\nexamples are little more than toys, but I enjoyed reading it.", "weaknesses": "I quite like this paper, I'd give it a weak accept. It's far from perfect, the novelty (while\nthere) is not earth-shattering and most of the worked\nexamples are little more than toys, but I enjoyed reading it. See the questions below.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "4layzQcGw6", "reviewer": "Reviewer_rkWZ", "summary": "The paper introduces the notion of influence power, a measure of the potential\nimportance of a variable for preventing a system from leading to an undesirable\noutcome. This notion is related to, but is shown to differ, to that of average\ncausal effects. The paper also introduces a UCT-based Monte-Carlo Tree Search\nestimator (from noiseless observational data). Theoretical support is provided\nfor the consistency of the estimator in the unconfounded case, as well as\nempirical support on three toy tasks with noiseless data.", "questions": "I'd appreciate if the authors could comment on the points I've marked as [**Q**] above.", "limitations": "", "rating": 8, "confidence": 2, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "**Originality**: to the best of my knowledge, the contribution is novel. It\nis definitely related to work in utility maximization and possibly algorithmic\nrecourse, but it is also different enough -- as clarified by the authors.\n\n[**Q**] Could you please clarify the points of difference with works in causal\nalgorithmic recourse? E.g., the works by Karimi and colleagues.  I'm sure this\nwould help the readers.\n\n**Quality**:\n\n**Clarity**: The text is well written and easy to follow.  All key definitions\nand arguments are accompanied by illustrative examples, which help a lot.\n\n**Significance**: I am not the most well versed in algorithmic decision making,\nbut I think this paper provides a useful contribution mixing elements from causality and decision making, and it could open the door to further research.\n\nI'm curious to know what the other reviewers think.", "weaknesses": "Focus on constructive and actionable insights on how the work could improve\ntowards its stated goals.\n\n**Clarity**: No major complaints on my end, but I did notice a handful of small\nlinguistic idiosyncrasies, such as:\n\n- Title: I'm not a native English speaker, but the sentence \"avoiding\n  undersired future\" sounds off to me. What about \"Avoiding undesirable future\n  events\"? This is how it's written in the abstract and it works just fine.\n\n- \"alterable variable\" also sounds off to me -- what about \"actionable variable\"?\n\n- Section 2, Notation: I don't think the notation {\\cal M}_{V_i} used\n  elsewhere in the text, it can probably be removed. Especially because it\n  looks incorrect: I suspect it should be {\\cal M}_{v_i} instead (lower case,\n  constant).\n\n- Section 2, Problem Definition: no need to repeat that \\Delta_{V_i} is the\n  alterable domain. This was already introduced previously.\n\n- Section 3.2: \"an alterable variable are worth\" -> is worth.\n\n- Section 3.3: \"Also, a variable such as...\" - why \"also\"? I'd drop it.\n\n- Above Example 4: \"it is not a causal\" -> \"not being a causal\".\n\nA minor issue is it's not immediately obvious to see the difference between\nthe two main terms in the main equation (at the end of p 3): one contains an\nintervention, the other an observation, but they are marked as \"a\" and \"o\",\nwhich are quite difficult to recognize as different. Perhaps use color to\nhelp the reader? The equation is much easier to understand once one spots this\notherwise tiny difference ;-)\n\n**Significance**: [**Q**] It'd be good to report the runtime difference between the\nthree algorithms (the baselines and the MCTS estimator). The results indicate\nthe estimator outperforms the baselines, which is good, but without\nunderstanding the its computational cost it's difficult to gauge whether the\nbenefits are worth it. This is my only major complaing with the work.\n\n[**Q**] It'd also be good to clarify from the get go - as soon as the introduction -\nthat Proposition 1 works only for unconfounded models (\"independent background\nnoises\"), which can be a strong assumption in practice, for clarity.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)