Intervention Generalization: A View from Factor Graph Models
Authors: Gecia Bravo-Hermsdorff
David Watson
Jialin Yu
Jakob Zeitler
Ricardo Silva
Venue: NeurIPS 2023 poster (2023)
Keywords: Causality
experimental design
TL;DR: We provide conditions and learning algorithms by which we can infer the outcome of an unseen choice of experimental condition given a database of past experiments.

Abstract: One of the goals of causal inference is to generalize from past experiments and observational data to novel conditions. While it is in principle possible to eventually learn a mapping from a novel experimental condition to an outcome of interest, provided a sufficient variety of experiments is available in the training data, coping with a large combinatorial space of possible interventions is hard. Under a typical sparse experimental design, this mapping is ill-posed without relying on heavy regularization or prior distributions. Such assumptions may or may not be reliable, and can be hard to defend or test. In this paper, we take a close look at how to warrant a leap from past experiments to novel conditions based on minimal assumptions about the factorization of the distribution of the manipulated system, communicated in the well-understood language of factor graph models. A postulated interventional factor model (IFM) may not always be informative, but it conveniently abstracts away a need for explicitly modeling unmeasured confounding and feedback mechanisms, leading to directly testable claims. Given an IFM and datasets from a collection of experimental regimes, we derive conditions for identifiability of the expected outcomes of new regimes never observed in these training data. We implement our framework using several efficient algorithms, and apply them on a range of semi-synthetic experiments.

Intervention Generalization: A View from Factor Graph Models
Gecia Bravo-Hermsdorff" David Watson Jialin Yu*
Jakob Zeitler Ricardo Silva"
Abstract
One of the goals of causal inference is to generalize from past experiments and observational data to novel conditions. While is in principle possible to eventually learn mapping from novel experimental condition to an outcome of interest, provided sufficient variety experiments is available in the training data, coping with a large combinatorial space of possible interventions is hard. Under typical sparse experimental design, this mapping is ill-posed without relying on heavy regularization or prior distributions. Such assumptions may or may not be reliable, and can be hard to defend or test. In this paper, we take a close look at how warrant leap from past experiments to novel conditions based minimal assumptions about the factorization of the distribution of the manipulated system, communicated in the well-understode language of factor graph models. A pos- tulated interventional factor model (IFM) may not always be informative, but it conveniently abstracts away need for explicitly modeling unmeasured confound- ing and feedback mechanisms, leading to directly testable claims. Given an IFM and datasets from collection of experimental regimes, we derive conditions for identifiability of the expected outcomes of new regimes never observed in these training data. We implement our framework using several efficient algorithms, and apply them on a range of semi-synthetic experiments.
Introduction
Causal inference is fundamental problem in many sciences, such clinical medicine [8, 73, 69] and molecular biology [66, 41, 29]. For example, causal inference can be used to identify the effects of chemical compounds cell types [73] determine the underlying mechanisms of disease [52].
One particular challenge causal inference is generalization - the ability to extrapolate knowledge gained from past experiments and observational data to previously unseen scenarios. Consider laboratory that has performed several gene knockouts and recorded subsequent outcomes. Do they have sufficient information to predict how the system will behave under some combination(s) of knockouts? Conducting all possible experiments in this setting would be prohibitively expensive and time consuming. A supervised learning method could, in principle, map vector representation of the design to outcome variables of interest. However, past experimental conditions may be too sparsely distributed in the set of all possible assignments, and such direct supervised mapping would require leaps of faith about how assignment decisions interact with the outcome, even under the guise of formal assumptions such linearity.
In this paper, we propose a novel approach to the task intervention generalization, i.e., predicting the effect of unseen treatment regimes. We rely on little more than a postulated factorization of the
Department of Statistical Science, University College London.
Department Informatics, King's College London. Department of Computer Science, University College London.
37th Conference on Neural Information Processing Systems (NcurIPS 2023).

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XIT
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Figure 1: (a) A dynamic process where only the final step is (simultaneouslyy observed. Intervention variables are represented as white squares, random variables as circles, and dashed lines indicate hidden variables. (b) A bird's eye unstructured view of the sampling process, where an intervention vector dictates a the distribution obtained at an agreed-upon time frame T.
distribution describing how intervention variables interact with a random process X, and possible target outcome Y.
Related setups appear in the causal modeling literature soft interventions [18], including its use in methods such as causal bandits [45] and causal Bayesian optimization [4] (see Section 6). However, these methods rely on directed acyelic graphs (DAGs), which may be hard to justify in many applications. Moreover, in the realistic situation where the target of a real-world intervention is a set of variables [26] and not only the "child" variable within DAG factor, the parent-child distinction is blurred and previous identification results do not apply. Without claiming that our proposal is appropriate for every application, model causal structure as a set of soft constraints, together with their pulative (arbitrary but local) modifications by external actions. Our interventional factor model (IFM) leads to provable intervention generalization via a factor graph decomposition which, when informative, can be tested without further assumptions beyond basic relations of conditional independence. IFMs are fully agnostic with regards to cycles or hidden variables, in the spirit of Dawid [23]'s approach to causal inference, where the key ingredient boils down to statements of conditional independence among random and intervention variables.
suggest the following take on intervention generalization:
Our primary contributions are as follows. (I) We introduce the interventional factor model (IFM), which aims to solve intervention generalization problems using only claims about which interventions interact with which observable random variables. (2) We establish necessary and sufficient conditions for the identifiability of treatment effects within the IFM framework.
(3) We adapt existing results from conformal inference our setting, providing distribution ffreeeno--eeeeeeee predictive intervals for identifiable causal effects with guaranteed finite sample coverage. (4) We implement our model using efficient algorithms, and apply them range of semi-synthetic experiments.
Problem Statement
Data assumptions. Fig. ((a) illustrates generative process common many applications and particularly suitable to our framework: perturbation, here represented by set of interventional variables a, is applied and a dynamic feedback loop takes place. Often in these applications (e.g. experiments in cell biology restrictive, and the time-resolution may boil down to a single snapshot, as illustrated by Fig. 1bb). We assume that data is given as samples of random vector X collected cross-sectionally well-defined time-point (not necessarily at equilibrium) under a well-defined intervention vector a. The importance of establishing a clear sampling time in the context of graphical causal models is discussed by [21]. The process illustrated in Fig. ((a) may suggest no particular Markovian structure the time the snapshot is taken. However, in practice, it is possible to model the black-box sampling process represented in Fig. ((b) in terms of an energy function representing soft constraints [48]. These could be the result of particular equilibrium processes [47], including deterministic differential equations [40], or
[66] and social science [59]), the sampling of this process is highly
verifiable approximations [59].
The role of a causal model is to describe how intervention locally changes the energy function. In the context of causal DAG models, sometimes this takes the guise of "soft interventions" and other variations that can be called "interventions structure" [43, 55] edge interventions [70]. Changes of structural coefficients in possibly cyclic models have also been considered [37]. The model family propose takes this to the most abstract level, modeling energy functions via recombinations of

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Figure 2: Examples of distinct causal graphical models, expressing different factorization assump- tions. Random variables are represented as circles, intervention variables as white squares, and factors as black squares. (a) A directed acyclic graph (DAG) with explicit intervention variables. (b) The corresponding interventional factor model (IFM) for (a). (c) A Markov random field (MRF) with interventional variables, thus, forming chain graph.
local interventional effects without acyclicity constraints, such constraints should be taken for granted [22] and are at best a crude approximation for some problems at hand (e.g.. [66]).
Background and notation. The notion of an intervention variable in causal inference encodes an action that modifies the distribution of a system of random variables. This notion sometimes brought up explicitly in graphical formulations of causal models [60, 72, 23]. To formalize it, let a denote vector of intervention variables (also known as regime indicators), with each o; taking values in finite set (0, 1,2.... N; 1}. A fully specified vector of intervention variables characterizes a regime or an environment (we use these terms interchangeably). Graphically, we represent intervention variables using white squares and random variables using circles (Fig. 2). We use subscripts to index individual (random or interventional) variables and superscripts to index regimes. For instance, X denotes the random variable X, under regime o², while of denotes the ith intervention variable of the jth environment. (Note that the order of the variables is arbitrary.) We occasionally require parenthetical superscripts to index samples, so a X, under regime o².
(k) denotes the kith sample of
For causal DAGs, Pearl's do operator [60] denotes whether a particular random variable X; is manipulated to have a given value (regardless of the values of its parents). As example, consider a binary variable X; E (True, False). We can use a (categorical) intervention variable a; to index the two "do-interventional" regimes: 0; denotes "do(X; True)", and Ø; 2 denotes =
We reserve o; = 0 to denote the choice of "no manipulation", " also known as the "obseevattonal"" regime. However, one need not think of this o, setting fundamentally different than the others; indeed, it is convenient to treat all regimes as choices of data generating process.* 4 Moreover, there is need for intervention variables correspond deterministic settings of random variables; they may in principle describe any well-defined change in distribution, such as stochastic conditional interventions [23, 17]. As stochastic example, d; 3 could correspond to randomly choosing do(X; = True) 30% of the time and do(X, = F False) 70% of the time. As a conditional example, σ; 4 could mean "if parents of X; satisfy given condition, do(X; True), otherwise do nothing". When X; can take more values, the options for interventional regimes become even more varied. The main point to remember is that o; 1, 2, ..... should be treated as distinct categorical options, and we reserve o; 0 to denote the "observational" case of "no manipulation".
As discussed in the previous section, the probability density/mass function p(z; o)) can be the result of a feedback process that does not naturally fit a DAG representation. Indeed, a growing literature in causal inference carefully considers how DAGs may give rise equilibrium distributions (e.g., [47, 12, 11]), or marginals of continuous-time processes (e.g., [57]). However, they come with considerable added complexity of assumptions ensure identifiability. In this work, abstract away all low-level details about how an equilibrium distribution comes to be, and instead require solely a model for how distribution p(x) factorizes as function of a. These assumptions naturally formulated as a factor graph model [44] augmented with intervention variables, which we call an interventional factor model (IFM).
'In general, there can be infinitely many choices for interventional settings. However, for the identifiability results in the next section, we cover the finite case only, as it removes the need for smoothness assumptions on the effect of intervention levels.
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Problem statement. We are given a space E of possible values for an intervention vector of dimension d, making | - IT i N, For range of training regimes Etrain E, are given col- Dr, with dataset Di collected under environment/regime o¹ E Errain. The goal is to learn p o²)) and plo")) := E|Y for all test regimes o² E Σtest = Σ Etrain.
lection of datasets Di, D2
In each training dataset, we measure sample of post-treatment i..d. draws of some m-dimensional random vector X (possibly with m & d, as there is no reason to always assume a one-to-one mapping between intervention and random variables), and optionally an extra outcome variable Y. The data generating process p(x; od) is unknown. We assume Y -. a X for simplicity", which holds automatically in cases where Y is known deterministic summary of X. We are also given a factorization of p(z;o),
p(250) & II [M(ossooR) Vo E E k=1
where S& S [m]] and F6 - [d] define the causal structure. The (positive) functions fx( are unknown. The model p(y x) can also be unknown, depending on the problem.
As illustrated in Figs. 2(a-b), such an intervention factor model (IFM) can also encode (a re- laxation of) the structural constraints implied by DAG assumptions. In particular, note the one-to-one correspondence between the DAG factorization in Fig. 2(a) and the factors in the IFM in Fig. 2(b): f1(21:01) == p(z1; 11). f2 Z2i 02) == p(æ2 02), and fa 22, Z3i 0T)) := p(23 I1, I2; 03). This explicitly represents that the conditional distribution of æ1 given all other variables fully factorizes in a: i.e., p(21 22, F3; σ) x (111;0) (1);:)))2(( f2(11, 32; fs #2, 23; dz). Such factorization is not enforced by usual parameterizations of Markov random fields (i.e., graphical models with only undirected edges) or chain graphs (graphical models with acyclic directed edges and undirected edges) [25] such as the example shown in Fig. 2(c).
Scope and limitations. The factorization in Eq. (1) may come from different sources, c.g., from knowledge about physical connections (it is typically the case that one is able to postulate which variables are directly or only indirectly affected by an intervention), or as the result of structure learning methods (e.g., [1]). For structure learning, faithfulness- like assumptions [72] are required, as conditional independencies discovered under configurations Etrain can only be extrapolated to Etest by assuming that independencies observed over particular values of can be generalized across all regimes. We do not commit to any particular structure learning technique, and refer to the literature on eliciting and learning graphical structure for variety of methods [42]. In Appendix A, we provide a general guide on structure elicitation and learning, and a primer on causal modeling and reasoning based solely on abstract conditional independence statements, without a priori commitment particular f'ailly of graphical models [23].
Even then, the structural knowledge expressed by the factorization in Eq. (1) may be uninformative. Depending on the nature of Eurain and Evest, it may be the case that we cannot generalize from training to test environments, and p(z; a²) is unidentifiable for all o* E Exest- However, all methods for causal inference rely on a trade-off between assumptions and informativeness. For example, unmeasured confounding may imply no independence constraints, but modeling unmeasured confounding is challenging, even more so for equilibrium data without observable dynamics. If we can get away with pure factorization constraints implied by an array of experimental conditions and domain knowledge, we should embrace this opportunity. This is what is done, for instance, in the literature on causal bandits and causal Bayesian optimization [45, 49, 4, 75], which leverage similar assumptions to decide what to do next. However, we are not proposing a method for bandits, Bayesian optimization, or active learning. The task of estimating p(x; d²) and u(aa)) for a novel regime is relevant in itself.
3 Interventional Factor Model: Identification
We define our identification problem as follows: given the population distributions P(Errain) == :p(;;
(p(x;;o") p(æ; .p(......p:: o')) for the training regimes a E Exrain and knowledge of the fac- torization assumptions as given by Eq. (I), can we identify a given p(x; corresponding to some unobserved test regime o* E Ytest?
'Otherwise, we can just define Y := Xm. and use the identification results in Section 3 to check whether Y be predicted from

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Figure 3: (a) An interventional factor model (IFM) with three (binary) intervention variables. (b) The junction tree of the a-graph associated with the IFM in (a): intervention variables are arranged as a hypergraph, where the hypervertices represent the sets of intervention variables that share factor and the edge represents the overlap between the two sets of intervention variables. (c) A table displaying the assignments for regimes in Etrsin. From this Merain and the assumptions in (a), possible to generalize to the two missing regimes, i.e., 01 d2 03 = 1, and 01 = 03 1, 0.
This identification problem is not analyzed setups such as causal Bayesian optimization [4, 75], which build on DAGs with interventions targeting single variables. Without this analysis, it is unclear to what extent the learning might be attributed to artifacts due to the choice of prior distribution or regularization, particularly for a sparsely populated Etrain As extreme case that is not uncommon in practice, we have binary intervention variables and never observe more than one a, set to 1 within the same experiment, it is unclear why we should expect non-linear model to provide information about pairs of assignments o, = aj = using an off-the-shelf prior or regularizer. Although eventually& dense enough process of exploration will enrich the database of experiments, this process may be slow and less suitable to situations where the goal is not just to maximize some expected reward but to provide a more extensive picture of the dose-response relationships in system. While the smoothness of p(z; a) as a function of a is a relevant domain-specinf information, it complicates identifiability criteria without further assumptions. In what follows, assume smoothness conditions, meaning that for some pair p(z; ") p(z; a²), with a * b, these probability density/mass functions are allowed to be arbitrarily different. Such setting is particularly suitable for situations where interventions do take categorical levels, with limited to no magnitude information.
preliminaries. Before introducing the main result of this section, let start by considering the toy example displayed in Fig. 3: Fig. 3(a) shows the graphical model (how X could be factorized is not relevant here), and Fig. 3(c) shows the assignments for the training regimes Strai (all intervention variables are binary). This training set lacks the experimental assigmment σ - 02 σ3 1.6 However, this regime is implied by the model and Eerain To this, consider the factorization p(z; o) x (1 an /2); 03), it implies:
Identification:
p (=: (1,1,0)
((=:(1,))
p [x; (0, 1, 0))
(1 (0,
where we used (1, 1) etc. to represent the o assignments. Because both (1, and (0, 1, 0) are in Etrain. the ratio is identifiable up to multiplicative constant. Moreover, multiplying and dividing the result by fz (1, we get:
p fa:: 1,), p p(a; (0, 1))
fi (1,1)) (z(2; [1 (0. 1)) (z,
p(:: (1,
p 1))
from which, given that (0, 1, 1) is also in Eusins we can derive p(z; (1,1,1)).
3.1 Message Passing Formulation
The steps in this reasoning can be visualized in Fig. 3(b). The leftmost hypervertex represents (Ø1, 02) and suggests 0j can be isolated from d3. The first ratio p : (1, 1, ()))//((;; (0, 1,))) considers three roles: 0i can be isolated (it is set a "baseline" of 0 in the denominator) and 03 is yet to be considered (it is set to the baseline value of 0 in the numerator). This ratio sets the stage for the next step where (a2, 073) is "absorbed" in the construction the model evaluated at (1, 1, 1). The entries in Errain were chosen so that we see all four combinations of (01, o2) and all four combinations of
"ss well the assignment 01 03 1,02 0, which can be recovered by an analogous argument.
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(02, 0z), while avoiding the requirement of seeing all eight combinations of (01. How to generalize this idea is the challenge. Next, we present our first proposed solution.
Definitions. Before we proceed, we need a few definitions. First, recall that we represent the baseline (or "observational") regime as 01 = d2
od = This describes data captured under a default protocol, e.g.. transcriptomic study in which no genes are knocked down. Let i/z denote the set of all environments of Σ such that of 0 i E Z. For instance, for the example in Fig. 3, set Z = {2} implies that 20 = { (0, 0, 0), (0, 1, 0)J. Finally, let g be the intervention vector given by o; of i E Z, and 0 otherwise.
what follows, we also use of the concepts of graph decompositions, decomposable graphs and junction trees, commonly applied graphical models [46, 20j. In Appendix B, we review these concepts." ? An IFM I with intervention vertices di...., od has an associated a grapp denoted by GA(I) defined as an undirected graph with vertices di...., Fd. and where edge o, oj is present if and only if o; and o, are simultancously present in at least one common factor f& in I. For example, the a-graph of the IFM represented in Fig. 3(a) is 02 03- This is a decomposable graph with vertex partition A = (o1). ). B = {02] and C = {Ø3} As this o-graph is an undirected decomposable graph, it has junction tree, which we depict in Fig. 3(b).
Identification: message passing formulation. Let I be an IFM representing a model for the m-dimensional random vector X under the d-dimensional intervention vector a E E. Model I has unknown factor parameters but a known factorization p(æ; a) x II [u( (F,))
Theorem 3.1 Assume that the o-graph (I) is decomposable. Given set Etrain = fo!..... S E and known distributions p(a;; ot),.....,(............. p(),,")), the following conditions are sufficient to identiyy any pla;o* ))), o² E EE:
i) all distributions indexed by o² o'. o', have the same support; and ti) [Pp)] E for all Fk in the factorization If
The algorithm for computing p(z;a* works as follows. Construct a junction tree T for I, choose an arbitrary vertex in T to be the root, and direct T accordingly. V2 hypervertex in T, let Dy be the union of all intervention variables contained in the descendants of V2 in T. Let BN be the intersection of the invervention variables contained in Vi with the intervention variables contained in the parent Va(aa of Vs in T.
We define a message from a non-root vertex Vg to its parent Vx(N) as
p(1; o (P((+)))) m pl:; a (Be(+)]))
(2)
with the update equation
plæ; Rl (P(+))) x p(a; (Fr
II
my:
(3)
VirEch((k)
where the product over ch)),,, the children of Vy in T, is defined to be ch()) = 0.
(All proofs in Appendix B.) In particular, if no factor contains more than one intervention variable, the corresponding o- o-graph will be fully disconnected. This happens, for example, for an IFM derived from a DAG without hidden variables and where each intervention has one child, as in Fig.
3.2 Algebraic Formulation
If a o-graph is not decomposable, the usual trick of triangulation prior to clique extraction can be used [46, 20), at the cost of creating cliques which are larger than the original factors. Alternatively, and generalization of Eqs. (2) and (3), we consider transformations of the distributions in IP (P(ttrmm)
Quick summary: a junction tree is formed by turning the cliques of undirected graph into the (hy- per)vertices of the tree. Essential the definition, junetion tree must have running intersection property: given an intersection S : H. H, of any two hypervertices H., H, all hypervertices in the unique path in T between H. and H, must contain S. This property captures the notion that satisfying local agreements ("equality of properties" of subsets of elements between two adjacent hypervertices in the tree) should imply global agreements. A decomposable graph is simply graph whose cliques can be arranged junction tree.

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Figure 4: Example of how to infer the target distribution p(;; (1,1,)) from the other seven possible
regimes given the postulated factorization p(z;o) x fi (1; f2) (x; 02 fa) 01, Jg). We use f as a shorthand notation for [u(2; 0h₁ = 0, Tbe = b), e.g. (l (2: (1, 1,) x ʃ 12 f3 For PR-transformation (f1 12 ) x (ff f10 J3 ) X ..... x (fl f ) f f2 up to mul- 92 46 0, 0r = 1. (In the table, this can be seen by going through each fx column and adding up the corresponding exponents Gj when there is tick in j; the sum must agree with the corresponding entry in the final row). A solution is (1 q4 96 47 1, 92 gs q3 = 11 corresponding to p(z; o²) (p(z; a )p(z; o p)(;; o )p(; a (p(2; p(x: o )p(=;;'))
tiplicative constant independent of z, need to find solution satisfying: 91 + 95
by products and ratios. We define a PR-transformation of a density set {p(2; "(o;,"""",,,;;;,)")) as any formula [[; p(2; o'),, for collection Q1.... q of real numbers.
Theorem 3.2 Let br denote a particular value of Fur and let Dy be the domain of o Given collection P(Etrain) == {p(x; p(z; } and a postulated model factorization p(a; o) x Ih- (s( oF.)) ). a sufficient and almost- metnostteeeryheree necessary condition for a given p(æ; o²) to be identifiable by PR-rransformations of IP( Erraia) is that there exists some solution to the system
Vk E (1. Vor, E Db,
Σ
qi = 1 =b),,
(4)
where 1(.) is the indicator function returning or O if its argument is true or false, respectively.
The solution to the system gives the PR-transformation. An example is shown Fig. The main idea is that, for a fixed Ir any factor fie ( TP.) can algebraically be interpreted as an arbitrary symbol indexed by ar, say "f"""" This means that any density p(a; a²) is (proportional to) "squarefree" monomial m² in those symbols (i.e, products with exponents 0 or 1). We need to manipulate a set of monomials (the unnormalized densities in IP (PE into another monomial m (the unnormalized p(x;o")). More generally, the possible set functions g(.). fa()) such that g(m", {mm ...., ....,,'') mn') h([me, ((mm"....,ni) mm')), with g(-) invertible in mi, suggest that g(-) and fu(-) must the product function (monomials are closed under products, but not other analytical manipulations), although we stop short of stating formally the conditions required for PR-cransformations to be complete in the space of all possible functions of (PCussin) This would be stronger claim than Theorem 3.2, which shows that Eq. (4) is everyheeverywhere complete in the space of PR transformations.
The message passing scheme and the algebraic method provide complementary views, with the former giving andie--and-ooquer perspective that identifies subsystems that can be estimated without requiring changes from the baseline treatment everywhere else. The algebraic method is more general, but suggests no hierarchy of simpler problems. These results show that the factorization over X is unimportant for identifiability, which may be surprising. Appendix C discusses the consequences of this finding, along with a discussion about requirements on the size of Σ Etrain-
4 Learning Algorithms
The identification results give a license to choose any estimation method we want if identification established - while we must make the assumption of the model factorizing according to a postulated IFM, we are not required explicitly use a likelihood function. Theorems 3.1 and 3.2 provide ways of constructing target distribution p(a; o*) from products of ratios of densities from PP (Pttrann). This
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suggests a plug-in in approach for estimation: estimate products of ratios using density ratio estimation methods and multiply results. However, in practice, we found that fitting a likelihood function directly often works better than estimating the product of density ratios, even for intractable likelihoods. We now discuss three strategies for estimating some : of interest.
Deep energy-based models and direct regression. The most direct learning algorithm is to first maximize the sum of log-likelihoods L(0;; D ..... D') := -i=1 Ej= log po o"), where n. is the number of samples in the ith regime. The parameterization of the model is indexed by a vector 0, which defines p(-) as logpo(<;; a) == Ei= &0 (Irs,) ) + constant. Here, Φo(() is differentiable black-box function, which in our experiments is a MLP (aka multilayer perceptron, or feedforward neural network). Parameter vector OK.OF, is the collection of weights and biases of the MLP, different instance for each factor k and combination of values in F In principle, making the parameters smooth functions of dF, is possible, but in the interest of simplifying the presentation, we instead use look-up table for completely independent parameters as indexed by the possible values of Fi Parameter set 0 is the union of all Σ JEE, 18y1 MLP parameter sets. As maximizing log- ligellklii is generally intractable, in our implementation we apply pseudo-likelihhn with discretization of each variable X in a grid. The level of discretization does not affect the number of parameters. as we take their numerical value as is, renormalizing over the pre-defined grid. Score matching [38], noise contrastive estimation [32] or other variants (e.g., [71]) could be used; our pipeline is agnostic to this choice, with further details in Appendix G. We then estimate [x(( = F[[ ær] using an off-the-shelff method, which in our case is another MLP. For given a², we sample from the corresponding p(æ; a²) with Gibbs sampling and average the results of the regression estimate fy(2) to obtain an estimate fi(a").
Inverse probability weighting (IPW). A more direct method is to reweight each training sample by the target distribution p(z; to generate p(aa*t == ーー= density ratio (")/())""")(((()) o²), rescaled such that > w" = 1. There are several direct methods for density ratio estimation [53] that could be combined using the messuges/product-ratie of the previous section, but found that it was stabler just take the density ratio of the fitted models using deep energy-based learning, just like in the previous algorithm. Once estimators ((o+ , (1)) are obtained, we combine them by the usual inverse variance weighting rule, (((0") == Σi= r')4,100 / Ei=r"', where r² == 1/0", and e² == >"=>>>>> (1)0))) method requires neither model for (x) nor Markov chain Monte Carlo. However, it may behave more erratically than the direct method described above, particularly under strong shifts in distribution.
w())" where wf))" is the
ya. This
Covariate shift regression. Finally, a third approach for estimating p(o*) is to combine models for fx)) with density ratios, learning a customized (x) for each test regime separately. A this is very slow and did not appear to be advantageous compared to the direct method, we defer a more complete description to Appendix D.
Predictive Coverage. Even when treatment effects are identifiable within the IFM framework, the uncertainty of resulting estimates can vary widely depending on the training data and learning algorithm. Building recent work in conformal inference [78, 51, 76], we derive the following finite sample coverage guarantee for potential outcomes, which requires no extra assumptions beyond those stated above.
Theorem 4.1 (Predictive Coverage.) Assume the identifiability conditions of Theorems 3.1 or 3.2 hold. Fix a target level a E (0,1), 1), and let I1, In be a random partition of observed regimes intro training and test sets of size n/2. Fit a model şa using data from I1 and compute conformity scores s( (i) - [(aa)) using data from Iz. For some new test environment a², compute the normalized likelihood ratio w( (a*) x p") g*)), rescaled to sum to n/2. Let f(a") be the qth smallest value in the reweighted empirical distribution Σ; (ar) 8()())), where d denotes the Dirac delta function and = [nn/2 + [))] Then for any new sample n + 1, we
have:
((x+(m+1)) E pa(a*) + 7(0")) 2 21-a. a.
Moreover, if weighted conformity scores have a continuous joint distribution, then the upper bound on this probability is a + 1//n22 1).
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5 Experiments
We run a number of semi-synthetic experiments to evaluate the performance of the IFM approach on a range of intervention generalization tasks. In this section, we summarize our experimental set-up and main results. The code for reproducing all results and figures is available online"; in Appendix E, we provide detailed description of the datasets and models; and in Appendix F we present further analysis and results.
Datasets. Our experiments are based on the following two biomolecular datasets: i) Sachs [66]: a cellular signaling network with 11 nodes representing phosphorylated proteins and phospho- lipids, several of which were perturbed with targeted reagents stimulate or inhibit expression. There are binary intervention variables. Errain has 5 regimes: a baseline, with all Ø; = 0, and 4 "single-interventoon" regimes, each with different single o; 1. Etest consist of the remaining 11 (= 2 5) unseen regimes. ii) DREAM [30]: Simulated data based on a known E. coli regulatory sub-network with 10 nodes. There are 10 binary intervention variables, and (similarly) Eerain has baseline regime, with all 0; = O, and 10 single-intervention regimes, with single choices of o; = 1.
Etest consists of 45 (10 9/2 45) unseen regimes defined by all pairs 0; - oj =
Oracular simulators. The first step to test intervention generalization is to build a set of proper test beds (i.e., simulators that serve as causal effect oracles for any E E), motivated by expert knowledge about the underlying system dynamics. Neither the original Sachs data nor the DREAM simulator used provide joint intervention data required in evaluation. Thus, for each domain, we trained two ground truth simulators: i) Causal-DAG: a DAG model following the DAG structure and data provided by the original Sachs et al. and DREAM sources (DAGs shown in Fig. 10(a)) and Fig. (a), respectively). Given the DAG, we fit a model where each conditional distribution is a heteroskedastic Gaussian with mean and variance parameterized by MLPs (with 10 hidden units) of the respective parents. ii) Causal-IFMs: the corresponding IFM is obtain by direct projection of the postulated DAG factors (as done in, e.g., Fig. 2(b)). The likelihood is neural energy model (Section 4) with MLPs with 15 hidden units defining potential functions. After fitting these models, we compute by Monte Carlo simulation their implied ground truths for every choice of a². The outcomes Y are then generated under 100 different structural equations of the form tanh(X (A X) with random independent normal weights À and e (O, vyy) À and By are scaled such that the ground truth variance of AT X, Var(A X), is sampled uniformly at random from the interval [0.6, 0.8). and set By == Var(X ( XX..
Compared models. We implement three variants of our proposed IFM model, corresponding to the three learning algorithmss described in Section 4: i) IFM1 uses deep energy- -baed models and direct regression; ii) IFM2 uses an IPW estimator; and iii) IFM3 relies on covariate shift regression. We compare these models to the following benchmarks: i) Black-box: We apply an off- fff-the-shelt algorithm (XGBoost [14]) to learn a direct mapping from o to Y without using X. This model does not exploit any structural assumptions. ii) DAG: We estimate the structural equations in an acyclic topological ordering that is consistent with the data generating process. This represents a strong baseline that exploits ground truth knowledge of the underlying causal graph. The likelihood is defined by conditional Gaussian models with mean and variances parameterized as MLPs, matching exactly one of the simulators described below.
Results. We evaluate model performance based on the proportional root mean squared error (pRMSE), defined as the average of the squared difference between the ground truth Y and estimated Y. with each entry further divided by the ground truth variance of the corresponding Y. Results are visualized in Fig. 5. We additionally run a series of one-sided binomial tests to determine whether models significantly outperform the black box baseline, and compare the Spearman's 's rank correlation [80] between expected and observed outcomes for all test regimes. Unsurprisingly, DAGs do best when the ground truth is a Causal- -DAG, while IFM methods do better when the data generator is a Causal-IFM. Still, some IFMs robust both ground truth models, with IFMI (the deep energy model) doing especially well on the DREAM dataset, and IFM2 (the IPW estimator) excelling on the Sachs data.
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Oubaat DRCAU BMG
Oraai DREAM star:
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Eror (PRMSS))
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500
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Figure 5: Experimental results on range of intervention generalization tasks, see text for details.
In particular, for the DREAM datasets, IFMI (scaled) errors remain stable, while the DAG has major loss of performance when a non-DAG ground truth is presented. However, IFM2 underperform on DREAM datasets, which might be because the knockdowns regimes in those datasets induce dramatic shifts in distribution overlap among regimes. (We omit results for IFMB (covariate shift regression) on these datasets, the method does not converge in reasonable time.) Though the black-box method (XGBoost algorithm and no structural assumption) sometimes struggles to extrapolate, displaying a long-tailed distribution of errors, it actually does surprisingly well in some experiments, especially on the DREAM dataset. In fact, the black-box method is essentially indistinguishable from linear regression in this benchmark (results not shown), which is not surprising given the sparsity of the training data. However, the non-additivity of the a effect on Y is more prominent in the Sachs datasets, making difficult generalize without structural assumptions.
6 Related Work
factor graph interpretation of Pearl's do-calculus framework is described in [79]. but without addressing the problem of identifiability. More generally, several authors have exploit structural assumptions to predict the effects of unseen treatments. Much of this research falls under the frame- work of transportability [61, 7]. where the goal is to identify causal estimands from combination of observational and experimental data collected under different regimes. If only atomic interventions are considered, the do-caleulus is sound and complete for this task [50]. The o-calculus introduced by [28, 17] extends these transportabilitty results to soft interventions [18, 19]. However, these methods are less clearly defined when an intervention affects several variables simultaneously, and while DAGs with additive eπrors have some identifiability [67] and estimation [75] advantages, acyclic error additivity may be an appropriate assumption in some domains. Under further parametric assumptions, causal effects can be imputed with matrix completion techniques [73, 3] or more generic supervised learning approaches
[31], but these methods often require some data to be collected for all regimes in Dest. Finally. [2] is the closest related work in terms of goals, factorizing the function space of each [XX;;; a) directly as function of a.
Another strand of related research pertains to online learning settings. For instance, several works have shown that causal information boost convergence rates in multi-armed bandit problems when dependencies are present between arms [45, 49, 24], even when these structures must themselves be adaptively learned [9]. This suggests promising direction for future work, where identification strategies based on the IFM framework are used to prioritize the search for optimal treatments. Indeed, combining samples across multiple regimes can be an effective strategy for causal discovery. as illustrated by recent advances in invariant causal prediction [63, 33, 64, 81], and it can also help with domain adaptation and covariate shift [54, 6, 13, 15).
7 Conclusion
We introduced the IFM framework for solving intervention generalization tasks. Results from simulations calibrated by real-world data show that our method successfully predicts outcomes for novel treatments, providing practitioners with new methods for conducting synthetic experiments. Future work includes: i) integrating the approach with experimental design, Bayesian optimization and bandit learning; iï) variations that include pre-treatment variables and generalizations across heterogeneous subpopulations, inspired by complementary matrix factorization methods such as [2]; tii) tackling sequential treatments; and iv) diagnostics of cross-regime overlap issues [36, 58).
10

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Acknowledgments and Disclosure of Funding
We thank the anonymous reviewers and Mathias Drton for useful discussions. GBH was supported by the ONR grant 62909-19-11 19-1-299,,,, and JY was supported by the EPSRC grant EP/W024330/1, RS was partially supported by both grants. JZ was supported by UKRI grant EP/S021566/1.
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Reviews:
[{"review_id": "Q9oTtT6Exn", "reviewer": "Reviewer_T46G", "summary": "The paper introduce Interventional Factor Models (IFM), a graphical model that encodes assumptions about a data-generating process and the interventions that can be performed on top of it. The model explicitly includes intervention variables to impose a factorization over the distributions generated by the model under any interventional regime.\n\nBeing a variable Y a function of the other observable variables X (or at least independent of the interventional regimes given X), the problem setting aims to learn E[Y] under a particular interventional regime of interest, given a subset of all possible interventional distributions and a corresponding IFM. For this task, the paper presents identification criteria and learning/estimation algorithms to find the parameter of interest.\n\nThe authors present the result of experiments on semi-synthetic data to evaluate the performance of their approach.\n\nI have some questions related to the scope of the results in the paper. I leave them in the comments section and would like to hear from the authors. I'm willing to update my scores based on the response.", "questions": "- When Thm. 3.2 says no, is it the case that there is no way to identify the causal effect of interest from the given model and distributions, even if it is not in the form of a PR-transformation?\n- If not, does it mean that Thm. 3.2. is necessary only under PR-transformations?", "limitations": "I believe the discussion on limitations and societal impacts is sufficient.", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "- The paper presents a novel graphical model that can be used to generalize known experimental settings to new unseen settings of interest.\n- Relevant setup, definitions, and results needed to understand the result are included.\n- The writing is clear enough to understand the setting and main contributions of the paper.\n- The paper presents sound results for the identification of causal effects and algorithms to estimate it, based on seen experimental settings and an IFM.", "weaknesses": "- The introduction and contributions claim that Thm. 3.2 is necessary, in general. However, the statement talks about PR-transformations in particular. It is not clear to me if it is proven that there are no other identification routes than PR-transformations.\n- It is mentioned in the paper that the IFM graph could be elicited from domain experts. This is something that sounds natural to me in the case of DAG, but not so intuitive in this case. I believe the reader could benefit from a more explicit example where variables have associated meaning and one could make sense of the assumptions.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you very much for the appreciation and excellent questions! We found the clarification questions about the generality of identifiability to be particularly useful for readers.\n\n**On Theorem 3.2:** Indeed we don't make any claims of completeness outside of the class of PR transformations. We do conjecture though that PR transformations include all \"practical\" ways of finding a mapping from distributions in $\\Sigma_{train}$, which we can explain as follows. If we fix $x$ and think of each factor $f_k(x; \\sigma)$ as a symbol indexed by $\\sigma$ (as in the caption of Figure 3), then each unnormalized density in each regime is a monomial over those symbols. A symbolic mapping from the set of monomials encoded by $\\Sigma_{train}$ to a target monomial of interest (i.e., a test set unnormalized density) boils down to ratios and products, as monomials are closed \"only\" under multiplication (we didn't elaborate that the exponents in the PR transformation will be integers, as this is not necessary to assume in order to solve it). We will clarify in the  manuscript to which extent PR transformations are general enough and adjust the abstract accordingly.\n\n**IFM elicitation:** similar to stages in causal discovery algorithms that start from an undirected network (like the PC algorithm), we can construct an undirected network that starts fully connected, and repeatedly ask an independence oracle whether two variables $V_i$ and $V_j$ are conditionally independent given all other variables. Special attention should be paid about the meaning of \"conditional independence\" when speaking of two intervention variables, see e.g. Section 7 of Constantinou and Dawid, \"Extended conditional independence and applications in causal inference\" (Annals of Statistics, 2017). This is how Figure 1c comes to be if we query an independence constraint oracle that follows the independencies entailed by the DAG in Figure 1a. The cliques in this graph (ignoring the directionality) can conservatively be translated to factors in a factor graph, as in Figure 1b. \n\nThe independence constraint oracle can be substituted by independence constraint/interaction tests (including the knowledge that we know that $\\sigma_i$s are regime indicators and hence their sampling distribution is chosen by design). Despite relying on interventional data, faithfulness-like assumptions still play a role e.g. pairwise independence leading to joint independence. Moreover, as we observe only a limited number of values of $\\Sigma$, assumptions about the lack of contextual independencies also play a role (e.g., if $X_i$ doesn't change in distribution when $\\sigma_j$ changes from 0 to 1 while keeping all other $\\sigma$ variables at 0, we may want to assume that this implies that $X_i$ and $\\sigma_j$ should not be in any factor, regardless of the configuration of $\\sigma$).\n\nComing from a more domain-specific theory, such structures also emerge from models of equilibrium. Consider this example that can be found in [8], Section 3.1.1, where $f$ denote differential equations at equilibrium, $X$ denotes observed random variables, $U$ denotes (mutually independent) latent variables and $I$ denotes an intervention indicator:\n\n$$f_I: X_I - U_I = 0$$\n$$f_D: U_1(X_I - X_O) = 0$$\n$$f_P: U_2(gU_3X_D - X_P) = 0$$\n$$f_O: U_4(U_5I_KX_P - X_O) = 0$$\n\nAfter marginalizing the $U$ variables, what we get is an IFM (this is not the whole story, though, as other non-graphical constraints may take place given the particular equations. [8] discusses $U_I$ being marginally independent of $I_K$ on top of the above). In general, energy-based models are to be interpreted as conjunctions of soft constraints, the factor graph is one implication of a system of stochastic differential equations, and interventions denote change to particular constraints. A SDE model may have *other* assumptions on top of the factorization, and parameters which carry particular meaningful interpretations, but this fits well with our claims that the IFM is a \"minimalist\" family of models in terms of structural assumptions - a reasonably conservative direction to follow particularly when the dynamics of many natural phenomena cannot be (currently) measured at individual level, as it's the case of much of cell biology data. [33] has further elaborations on some of these ideas."}, {"review_id": "HDhZ3KuBIF", "reviewer": "Reviewer_ybr1", "summary": "This paper maps from available observational and experimental datasets to unseen interventional distributions given the factorization of the joint distribution of the intervened system. They utilize an interventional factor model equipped with factor graphs to provide necessary and sufficient conditions for causal effect identifiability. Finally, they provide some practical algorithms to estimate the final outcomes.", "questions": "I would request the authors to provide explanations for the previously mentioned main weaknesses and answer the following questions:\n* For the example in line 74, why is the DAG $\\sigma \\rightarrow X$, $\\sigma \\rightarrow  Y  \\leftarrow X$, not an option?\n* Are the authors refuting the utility of theorem 3.1 and 3.2 and adopting only the new approaches provided in section 4? \n* How is this paper dealing with cycles and confounders? The authors should mention that more explicitly.", "limitations": "If the variables have more than two states or we have more variables, the number of interventional datasets will also be significantly high. In real-life applications, collecting these many interventional datasets are not feasible or possible. The authors should discuss this challenge in detail.", "rating": 5, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The authors provided good examples of different concepts in the main paper and in the appendix. It's a different approach from approaches that use causal graphs but is quite interesting. The paper has a nice flow in the writing and is easy to read. The explanations and detailed examples provided in the appendix are worth appreciating, and they are quite helpful for readers.", "weaknesses": "Main weakness:\n\n* Some concepts such as junction tree, hypervertex, and message passing algorithms should have been defined with short examples in the main paper since they are used in the main paper theorems.\n\n* Multiple approaches have been described in sections 3 and 4 and but most of them were not explained with enough details and intuition.\n\n* The main contribution seems a little unclear. The authors first discussed two approaches in section 3. Later in section 4, they mentioned that in practice using deep energy-based models works better. They suggested employing a differentiable black box to learn parameters for each factor and estimate E[Y| x]. Using a black box is not completely novel. \nIt appears that the approaches in section 3 are not very useful and thus the authors are proposing three more methods that work better in practice. I would request the authors to clarify the mentioned issues.\n\n* In the experiment section, the authors completely ignore the approaches they discussed in Theorem 3.1 or Theorem 3.2. They considered the deep learning approach as the best version and compared the benchmarks with that.\n\nMinor comments:\n* Some concepts in sections 1 and 2 are used without proper definitions and examples. Readers would need to know those background knowledge beforehand to go with the flow of the paper.\n* Line 119: Unmeasured confounders are used without any definitions or proper examples.\n* Line 207:  Theorem 3.2 seems less intuitive. A proof sketch in the main paper would be appreciated.\n* Line 227: The approach “Deep energy-based models and direct regression” should be described in more detail since this approach worked better than other approaches.\n* Conformity scores are not defined in the main paper although has been used in theorem 4.1.\n\n\nI have read the author's rebuttal. The authors resolved some of my concerns. But I am not confident enough to increase the scores.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for commenting that our paper is \"quite interesting\" and that it \"has a nice flow in the writing and is easy to read\". Much appreciated!\n\nIn the following, we address your comments and questions.\n\n**\"Some concepts... should have been defined... .\":** We were struggling for space and we decided that concepts which are present in textbooks would be referred to without explicit definition in the main body of the text, but we fully agree that this isn't ideal. As a possible camera-ready allows for another page, we will take this opportunity to use it for modifications like that if given the opportunity.\n\n**\"...details and intuition.\":** We tried to present a minimal example of the junction tree approach in lines 151-159. We will flesh it out a bit more, and point more explicitly to Appendix A where a more complex example is given, assuming the opportunity to provide an accepted camera-ready version. Likewise, we will explicit write further intermediate steps into the explanation of the algebraic method in the caption of Figure 3. It can be explained as follows.\n\nEssentially, we need to find a vector $\\{q_i\\}$ such that $p(x; \\sigma) \\propto \\prod_{k = 1}^t p(x; \\sigma^i)^{q_i}$, or show that there is no solution. With $x$ fixed, this is done by treating each factor $f_k(x_{S_k}, \\sigma_{F_k})$ as a symbol in an algebraic system. This means that each density, up to a multiplicative constant, is a monomial in those symbols. \n\nFor instance, in the example in Figure 3, we use the symbol $f_1^{00}$ to denote the factor $f_1(x; \\sigma_1 = 0, \\sigma_2 = 0)$, while e.g. $f_3^{10}$ denotes $f_3(x; \\sigma_1 = 1, \\sigma_3 = 0)$ and so on. The monomial corresponding to $p(x; \\sigma = (1, 1, 1))$ is $f_1^{11}f_2^{11}f_3^{11}$.\n\nNow, a PR transformation from the set of 7 training densities in Figure 3 (all configurations of binary $\\{\\sigma_1, \\sigma_2, \\sigma_3\\}$ except for the test configuration $\\sigma_1 = \\sigma_2 = \\sigma_3 = 1$) is of the form $$(f_1^{00}f_2^{00}f_3^{00})^{q_1} \\times (f_1^{01}f_2^{10}f_3^{00})^{q_2} \\times \\dots \\times (f_1^{10}f_2^{01}f_3^{11})^{q_7}.$$ \n\nFor an algebraic identity between $f_1^{11}f_2^{11}f_3^{11}$ and the above to hold (up to a multiplicative factor), it is necessary that $q_1, q_2, \\dots, q_7$ are such that, in the resulting multiplication, the resulting exponents for $f_1^{11}$, $f_2^{11}$, and $f_3^{11}$ are 1, and all others are zero. For instance, $f_1^{00}$ appears in the regimes in lines 1 and 5 in the table of Figure 3. Hence, we must have $q_1 + q_5 = 0$. Symbol $f_1^{01}$ appears in lines 2 and 6, and therefore we must have $q_2 + q_6 = 0$. This goes on for all symbols (columns in the table), ending with $f_3^{11}$, which only appears in the $7^{th}$ training density, making $q_7 = 1$. This gives a system with one solution, the one shown in the caption. The admittedly ugly Eq. (4) is a compact way of describing this system of equations.\n\n**\"It appears that the approaches in section 3 are not very useful...\":**. It is clear that we should have been more explicit in the transition between these sections. Section 3 is about identification methods, and those don't necessarily need to translate directly into estimation methods. By analogy, think of the formulas implied by the do-calculus or G-computation, and how it's not necessarily the case they are used in estimation methods by plugging-in a corresponding estimate of a density. \n\nThat is, the proofs in Section 3 are constructive, but this doesn't imply they should lead to a one-to-one correspondence with a learning algorithm (for instance, G-estimation looks very different from what a direct approach based on G-computation may suggest). As long as we know that we can get $\\Sigma_{test}$ by products of ratios of densities in $\\Sigma_{train}$, it doesn't matter whether we can identify each factor as long as all densities follow the common factorization that shares factors, so a purely likelihood-based approach suffices (we did try density ratio estimation methods, but they worked poorly out-of-the-box and we decided to omit further discussion on them). We will clarify this in a final manuscript by a more fleshed out bridge between Sections 3 and 4. All problems in Section 5 can be identified based on the fact that it satisfies the conditions of Theorem 1, as there is no more than one $\\sigma$ per factor and $\\Sigma_{train}$ spans all necessary elementwise support for each $\\sigma$ variable (a scenario we mention in Section 3). We will comment on this explicitly in a revised manuscript.\n\n**\"dealing with cycles and confounders?\":** We do not model confounders/cycles explicitly. As the only information that matters is the Markov blanket, the presence of an edge may be due to confounders or cycles or \"directed dependence\", and for better or worse that's all treated on the same footing. For instance, if the generative model happens to be a DAG with one latent variable $U$ and edges\n$$X_1 \\leftarrow U, X_1 \\leftarrow \\sigma_1;$$\n$$X_2 \\leftarrow U, X_2 \\leftarrow X_1, X_2 \\leftarrow \\sigma_2;$$\n$$X_3 \\leftarrow X_2, X_3 \\leftarrow \\sigma_3,$$\n\nthen one factorization of the above after marginalizing $U$ is defined by two factors, $f_1(x_1, x_2; \\sigma_1, \\sigma_2)$, and $f_2(x_2, x_3; \\sigma_3).$ The presence of $U$ makes identification harder due to the induced interaction of $\\sigma_1$ and $\\sigma_2$, but the implied factorization by itself doesn't introduce structural constraints not found in the original DAG. With more confounders, we may get a fully uninformative model, see the **Scope and Limitations** section.\n\n**\"Why isn't [this DAG] an option?\":** It may certainly be the case that a system has edges from $\\sigma$ to $Y$. We don't cover this case. A case where $Y$ may be shielded from $X$ is when $Y$ is a longer-term phenotype that is a predictable result from *which* values $X$ attain at some point in equilibrium, without requiring to know *how* ($\\sigma$) those measurements came to be.\n"}, {"review_id": "4FJHHB1dny", "reviewer": "Reviewer_4dmo", "summary": "This paper consider how to use data from past interventions to allow it to generalize to new unseen interventions. This is an important practical problem to consider, as running additional experiments is often costly/infeasible. In order to tackle this problem, the authors consider a graphical models approach, specifically they consider an interventional factor graph model (IFM). Using posit an IFM factorization of the the density $p(x; \\sigma)$. Then they provide the sufficient conditions for identification, and provide a message passing algorithm to do so. Next, they discuss multiple approaches to estimate the density based on ML models, as as well via IPW methods. They also consider a covariate shift regression approach. Further, they provide a conformal inference approach to establish coverage. Lastly, they perform a number of experiments to establish the empirical efficacy of their method. ", "questions": "I have made some suggestions in the weaknesses section. I list some other questions here. \n\nDeep-energy based models:  Does fitting the parameter vector $\\theta_{k, \\sigma_{F_{k}}}$ require knowledge of the set of  variables in $F_k$? I am confused by this, and if it does require, how do we determine these variables in practice. ", "limitations": "Yes, they have. ", "rating": 7, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "The paper tackles a very important problem, how can we use past interventions to generalize to new interventions. They propose an innovative IFM model and message passing algorithm to establish identification for this problem. Viewing this problem under this lens is an interesting one, and potentially of practical use. Further, I appreciate the authors providing examples so that its easier to understand their identification argument. I also appreciated the authors providing multiple methods for estimating the density discussed earlier. I believe providing multiple approaches is often of great practical use since no one algorithm often works in all scenarios.  In terms of empirical evaluation, I think its interesting that the authors used semi-synthetic data. I believe this is good practice, and should be followed more often.  ", "weaknesses": "Presentation: Presentation of both the regression and coverage algorithm is confusing, and seems to require a lot of additional knowledge on behalf of the reader. I do not fully understand how these algorithms proceed. For example, the deep-energy based models, lines 237-239 are very unclear. Making it clear what exactly is being fit would be very useful. More generally, being clear and rigorous regarding these things will go a long way in making the paper clearer. \n\nEmpirical Evaluation of Coverage: I did not see any simulations to this effect.\n\nEmpirical Evaluation: It is not clear to me what exactly X is in these datasets. Once again, being clear and rigorous about these details will enhance understanding, and give the reader a chance to appreciate the empirical evaluation. Further, even after reading the appendix, I do not understand how the outcomes were generated. Could the authors please clarify empirical details in the rebuttal? \n\nComparison to related work: The comparison to [2] is incorrect. The authors claim that a series of works including that of [2] requite data to be collected for all regimes in $\\Sigma_{\\text{test}}$. This is not the case. For example, the experimental design section of [2] shows that this isn't the case. ", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for all of your feedback, for the kind words on this being a \"very important problem\" and \"innovative\" in this solution! Moreover, thanks for asking clarification questions that will definitely improve on the presentation of the paper.\n\n**Method clarification:** to fit a deep energy model, proceed as follows:\n\n1. Each log-factor $\\phi_{\\theta_{k, \\sigma_{F_k}}}(X_{S_k})$ is given by a MLP with a chosen number of hidden units. It is a different MLP for possible value of $\\sigma_{F_k}$. For instance, in our empirical studies, there is at most one $\\sigma_i$ variable associated with each factor. If $\\sigma_{F_k}$ is non-empty, then we have two MLPs for factor $k$, which are $\\phi_{\\theta_{k, \\sigma_{F_k} = 0 }}(X_{S_k})$ and $\\phi_{\\theta_{k, \\sigma_{F_k} = 1}}(X_{S_k})$. Set $\\theta_{k, \\sigma_{F_k} = 0}$ are the weights and biases of that corresponding MLP/local regime.\n\n2. We fit all parameters $\\theta_{k, \\sigma_{F_k}}$ by pseudo-log-likelihood. This means maximizing the sum of the univariate conditional log-likelihoods over all datasets $\\mathcal D^i$ as\n$$\\sum_{i = 1}^t \\sum_{j = 1}^{n_t} \\sum_{q = 1}^p \\log p_\\theta(x^{ij}_q|x^{ij}_w; \\sigma^i),$$\nwhere $x^{ij}_q$ is the $q$-th column of the $j$-th data point of $\\mathcal D^i$, and $x^{ij}_w$ are the remaining columns of the same row of the same dataset. Each entry in the sum above can be given as the negative energy function minus the corresponding log normalizing constant, which requires summing over the possible values of $X_q$ only *(apologies for not writing the equation explicitly, but after a long fight with apparent OpenReview's Markdown bugs, it seemed impossible to write the log-sum-exp and sub/superscripts we wanted to write. We will write it explicitly in the final manuscript)*.\n\nThe above is then optimized by Adam, a gradient-based algorithm, and each MLP has a single hidden layer with a hyperbolic tangent activation function and a linear output layer.\n\nWe fit a model for $f(X) := \\mathbb{E}[Y|X]$ using a supervised learning method. In our experiments, another MLP. At test time, we evaluate any $\\mathbb{E}[Y; \\sigma^\\star]$ by Monte Carlo, that is, we generate a large sample from the learned $p(x; \\sigma)$ by Gibbs sampling, then average $f(X)$ over this sample. \n\nSimulations of coverage are very expensive (it requires generating at least dozens of datasets from the same model, and fitting a deep-energy based model to each to get a single evaluation). We have begun work on this, but experiments are ongoing and will not be ready ahead of the rebuttal period deadline. Our final manuscript will include coverage results either in the appendix or main text, if space allows. \n\nWithout prior knowledge of what pseudo-likelihood means, we agree that it is tedious to get this from the Julia code provided in the supplement, and we will tweak the supplementary material to add the above. As we say in the main text, we have no reason to be attached to pseudo-likelihood, and one should feel free to use other methods for energy-based learning such as score matching, contrastive divergence, etc.\n\n**What exactly $X$ is in these dataset:** In the datasets, the intermediate variables $X$ refer to concentrations of metabolites in cells, such as lipids or, in the case of DREAM, (simulated) gene expression. The Jupyter notebook and the original references provide a description of the variables; see also Appendix D. \n\n**Outcome models:** they are artificially generated as follows for each of the 100 simulated problems in each of the two studies. Basically, for each of the 100 cases studies we artificially generate a vector $\\lambda_{true}$ and simulate outcome data as $$Y = \\tanh(\\lambda_{true}^{\\mathsf T}X) + \\epsilon_y.$$\n\nThe scale of $\\lambda_{true}$ and the variance of $\\epsilon_y$ are set so that $var(\\lambda_{true}^{\\mathsf T}X)$ under the baseline regime is about 1.5 to four times $var(\\epsilon_y)$, with the ratio chosen uniformly at random from $[1.5, 4]$. For each regime $\\sigma$, we then numerically compute the ground truth $\\mathbb{E}[Y; \\sigma]$ by Monte Carlo.\nFor more details, we refer reviewers to the section \"Synthetic Ground Truth and Training Data Generation\" in the Jupyter notebook `starting_demo.ipynb`, where this process is described more explicitly without requiring much code reading.\n\n**\"...knowledge of the set of variables in $F_k$...\":** Yes, it does. How to get it? Similar to stages in causal discovery algorithms that start from an undirected network (like the PC algorithm), we can construct an undirected network that starts fully connected, and repeatedly ask an independence oracle (human expert or statistical test) whether two variables $V_i$ and $V_j$ are conditionally independent given all other variables. Special attention should be paid about the meaning of \"conditional independence\" when speaking of two intervention variables, see e.g. Section 7 of Constantinou and Dawid, \"Extended conditional independence and applications in causal inference\" (Annals of Statistics, 2017). This is how Figure 1c comes to be if we query an independence constraint oracle that follows the independencies entailed by the DAG in Figure 1a. The cliques in this graph (ignoring the directionality) can conservatively be translated to factors in a factor graph, as in Figure 1b. (There are more details about this in the response to T46G).\n\n**Comparison to [2]:** excellent point, we intended to say that about [23]. Approach [2] is very interesting but with widely different assumptions (and data requirements), more akin to sparse ANOVA in Fourier space if we were to give a very simplified description of it. We attempted to fit our very-sparse-design data with their code, but we did not succeed in getting meaningful stable results -- to be fair, these are not cases where [2] is meant to be used anyway. We will amend the text to clarify this point.\n"}, {"review_id": "n5PpRFmJZy", "reviewer": "Reviewer_t5cb", "summary": "The authors present the interventional factor model, a more general formalization used to predict the effect of treatment on an outcome in unseen regimes. It is more general than existing formalizations since it does not assume causal graphs to be directed acyclic graphs (DAGs). The authors show in this new formalization what are the conditions to ensure identifiability of treatment effects. Finally, the authors propose several methods that can estimate treatment effects and show their effectiveness with semi-synthetic experiments.", "questions": "Minor typos or style suggestions:\n- line 39: \"we submit\" => \"we argue\" or \"we claim\" \n- line 46: use a colon instead of a full stop\n- line 59: why use the aleph symbol, could another common letter be used instead?\n- line 204: \"Equ.\" => \"Eq.\". It is more common and \"Eq.\" is used for all the other references to equations.\n- line 350: citation 2 is repeated", "limitations": "Yes the limitation have been addressed, and the societal impact is not really applicable.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The article is well written. It proposes a really general formalization using factor graphs that is original. It encompasses identification results, several algorithms, and experiments.", "weaknesses": "The contribution is limited in the sense that identification results for DAGs are already well established (do-calculus and $\\sigma$-calculus). This work address the more general case where the graph is not necessarily a DAG. It supposes that the graph is known, but this assumption is strong in practice since it is challenging to know these kinds of general graphs both from the expert knowledge and structure learning perspective (more than DAGs).\n\nReal-world applications could surely help motivate the use of this formalization. The present semi-synthetic experiments are interesting, however the simple black-box baseline method is overall performing really well, undermining the use of the more involved proposed methods.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Many thanks for your comments, and by finding our article \"well written\" and \"original\"! \n\n**On identification/elicitation/learning:** we are not claiming that such a family is universally superior to DAGs, but there is no shortage of domains where energy-based formulations are more natural than directed representations, such as in data composed of snapshots of an equilibrium distribution (as in many cell biology applications). In particular, a set of equilibrium equations\n$$f_k(X_{S_k}, \\sigma_{F_k}, U_k) = 0,$$ \n\nwhere $f_k$ is a stochastic differential equation and $U_k$ is a set of hidden variables, is an IFM with hidden variables and partially deterministic factors. A full-blown SDE is likely to be challenging to specify (even more so if no dynamics are observable), but the factorization implied by it can still be used to set up an IFM. We have amended the text to motivate our method with practical use cases of this form.\n\nRegarding learning/elicitation of energy-based model structures, if an expert or structure learning algorithm is able to answer queries about Markov blankets, then that's already sufficient. A minimal procedure is as follows: start with a fully connected undirected graph among all variables. For all pairs of variables $(V_i, V_j)$, ask: \"is $V_i$ conditionally independent of $V_j$ given all other variables? If yes, then remove the $(V_i, V_j)$ edge\". When done, transform each clique in this graph into a factor in a IFM. That's it, but bear in mind that the definition of \"conditional independence\" for two regime indicators $\\sigma_i$ and $\\sigma_j$ is non-standard; see Section 7 of Constantinou and Dawid, \"Extended conditional independence and applications in causal inference\" (Annals of Statistics, 2017). For structure learning methods, we do need to bear in mind that only a limited set $\\Sigma_{train}$ of interventional configurations are available, and it's possible that independencies within other ranges found in $\\Sigma$ won't hold. Where data is lacking for sufficient levels of $\\sigma_i$ variables, knowledge of physical/spatial structure can play a role, suggesting which direct interactions between interventions and random variables can be safely discarded or not. After all, without this knowledge, not even $do$ operations could be linked to real data.\n\n**Black-box baseline method:** The black-box algorithm did perform competitively in a fair number of experiments. This is just a matter of fact that we encountered after running them. There is no expectation that it will work in general (it's not meant to), and we did not want to perform dataset-selection of any kind.\n\n**Notation:** Thank you very much for the detailed comments about style and presentation, we will implement them (we are fond of $\\aleph$ for denoting cardinality, but if this causes confusion with its more fundamental uses in set theory, we have no problem changing it).\n"}, {"review_id": "8o0Ze1m23D", "reviewer": "Reviewer_EjuP", "summary": "This work proposes the use of factor models as a graphical causal model to generalize from past experiments. The authors introduce factor models, describe their relative merit then describe how a factorization can be derived for a given intervention, and give approaches for estimation using deep energy based modeling and weighting approaches. Experimental results are shown which provide a comparison of the performance of the proposed approach to other approaches (DAGs and black box estimation)", "questions": "* Is the set of identifiable estimands comparable to other graphical causal models like ADMGs? \n\n* For these estimation approaches (and more generally) is it possible to provide a sense of convergence/consistency of the causal parameters?\n\n* Given the experimental results where there does not seem to be a clearly preferable approach in all settings, how should someone decide when it is appropriate/necessary to employ an IFM?\n\n* Can you provide a discussion of the current proposal with \"Causality with Gates\" by Winn (AISTATS 2012). I can see that there are significant differences between these two texts, but given that they are both concerned with the use of factor graphs for causal inference I think it should be discussed.", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "* This is a very interesting idea, and I generally agree with the authors' central claims around the opportunities provided by using factor models for causal inference. \n\n* The addition of conformal inference for uncertainty intervals here is a nice, and elegant addition to the paper. \n\n* The authors do a nice job of providing a thorough empirical evaluation of the proposed approach", "weaknesses": "One of the weaknesses of factor models is that it is more difficult to perform inference than in DAGs where there is a simple factorization that can be exploited. Factor models are also less immediately interpretable than DAG models. While this isn't necessary a problem in itself, it would be useful if there was a more plain discussion about the tradeoffs involved in this representation. \n\nI also found the presentation to be a little difficult to follow. There are a number of missing discussions that would be useful to contextualize the proposed approach in the broader literature (e.g., ADMGs, segregated graphs, gated factor graphs). It's also a little unclear to me whether there is a sound and complete identification algorithm here.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for finding our idea to be \"very interesting\", implemented with \"elegant additions\" on uncertainty quantification and a \"through empirical evaluation\"! In what follows, we address your questions.\n\n**Trade-offs in the representation:** one of the main defining features of DAGs is the natural presence of marginal independencies between intervention variables and random variables (a random variable $X_i$ is independent of an intervention variable $\\sigma_j$ when its distribution doesn't change as we choose different levels for $\\sigma_j$. See [17] for further discussion about what \"independence\" means in the context of non-random intervention variables). That \"the future doesn't cause the past\" is one of the sources of this type of independence. \n\nIn contrast, no marginal independencies of this type are structurally encoded in a (connected) IFM. This is more natural in a scenario where a design vector $\\sigma$ is set at the initial stage in a process, and the system is runs towards some equilibrium. No element $\\sigma_i$ in the intervention vector is decided based on outcomes caused by the other $\\sigma_j$ in this vector. This is what happens in many datasets such as Sachs et al. Although traditionally DAGs have been applied to it, the nature of the data sampling mechanism, and the feedback generative mechanism itself, puts such a construction in dispute. References [8] and [32] have a discussion on such issues, where directed components are still used - but even there, we believe there is much to appreciate about the relative simplicity of an IFM, leading to both more tractable theoretical and practical analyses when integrating data from multiple jointly-intervened design vectors. \n\nDAGs are also natural where there is a sequential plan, i.e., instances of closed-loop control where an action is chosen based on the outcomes of previous actions. Even in this case, cross-sectional IFMs could be combined in a sequence, as an instance of chain graph modelling where a design vector is jointly decided at different stages, a next equilibrium distribution is achieved, and a consecutive round of decisions is taken based on the previous equilibrium. So, we envision that a natural extension is a chain-graph combination of directed components and IFMs. Data for that is not as easy to find as for a single-shot IFM, hence our focus here. \n\n**Soundness and completeness:** an IFM is definitely not meant to capture possible identification conditions that rely of marginal independencies, which can be exploited by DAG-based models. Likewise, the closest DAG relaxation of an IFM model will fail to capture relevant constraints (e.g., a bivariate model given by a copula $c(x_1, x_2)$ and marginals $f_1(x_1; \\sigma_1)$, $f_2(x_2; \\sigma_2)$). We also define a broad family of transformations (PR transformations), which we conjecture informally as encompassing all symbolic mappings from training to test regimes. Within PR transformations, we provide a sound and (measure one) complete method to show identifiability.\n\n**When to decide on the use of an IFM:** following the above discussion, it is clear that the choice of family matters. Our DAG-based simulations were designed to make the DAG alternative competitor look as good as possible, matching the exact additive-error conditionally Gaussian family. In practice, off-the-shelf goodness-of-fit techniques, even if using the baseline regime only, should provide diagnostic tools for this choice. Even plotting simulations from the fitted model $p(x|\\sigma^0)$, compared against the training data, already provides strong indications of what each model can accomplish.\n\n**Estimation:** in the sense of Fisher (pointwise) consistency, learning densities with the postulated factorization, and supervised methods for mapping $X$ to $Y$, will guarantee consistency for any $\\mathbb{E}[Y; \\sigma]$ identifiable from the conditions in Section 3. This is true even if not all individual factors in the factorization are identifiable (up to a multiplicative constant): the constructive identification results bind training densities to test densities; having the required factorization and support is what matters.  We agree it would be interesting to have other inferential tools for the causal parameters, e.g., rates of convergence. It is not our goal to address it here but we hope this fosters further work in the community, particularly considering that the causal functionals of interest, $\\mathbb{E}[Y; \\sigma^\\star]$, are relatively simple.\n\n**Winn (2012):** many thanks for the reference! It is an early example of how factor graphs can be used to encode dependencies between variables in a causal graph and well worth discussing in our paper. We highlight that our focus is on explicit modeling of non-atomic interventions and identifiability of out-of-distribution regimes, while Winn (2012) sets the stage on how do-calculus applied to DAGs can be translated to a type of factor graph, although it must rely also on context-specific independencies (via the \"gates\" variation of a factor graph model). An excellent pointer to be added to our references.\n"}, {"review_id": "2ztmqqGXGY", "reviewer": "Reviewer_2Na9", "summary": "This paper studies the problem of generalization in causal inference. In particular, it extends the factor graph to interventional factor graph (IFM). It shows when can such model be identified as well as proving practical algorithm for learning. The setting assumes knowing the factorization, this type of structural knowledge has its advantages and limitations as well. ", "questions": "1. Because the term junction tree is explicitly mentioned in theorem 3.1, it would be good to have a brief definition of it?\n2. Can you elaborate a bit more on the algebraic formulation of this problem? This section is a bit harder to digest.\n3. I am a little confused about the usefulness of theorem 3.1 and 3.2. As authors mentioned in the first paragraph of Section 4, they do not use it to product of density ratios but rather use deep energy-based model or IPW. Are the identifiability results in 3.1 and 3.2 necessary for these methods to work?", "limitations": "Not applicable", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper is well motivated with good theoretical results and empirical experiments. ", "weaknesses": "1. This paper does not provide too much real-world example of the interventional factor model. The simulation is also semi-synthetic. I am a little concerned about the applicability of such model.\n2. There seems to be a disconnect between the identifiability results and learning algorithms (see the questions section).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thanks for commenting that our paper is \"well motivated with good theoretical results and empirical experiments\"!\n\n**Real-world applicability:** coming from a more domain-specific theory, such structures also emerge from models of equilibrium. Consider this example that can be found in [8], Section 3.1.1, where $f$ denote differential equations at equilibrium, $X$ denotes observed random variables, $U$ denotes (mutually independent) latent variables and $I$ denotes an intervention indicator:\n\n$$f_I: X_I - U_I = 0$$\n$$f_D: U_1(X_I - X_O) = 0$$\n$$f_P: U_2(gU_3X_D - X_P) = 0$$\n$$f_O: U_4(U_5I_KX_P - X_O) = 0$$\n\nAfter marginalizing the $U$ variables, what we get is an IFM (this is not the whole story, though, as other non-graphical constraints may take place given the particular equations. [8] discusses $U_I$ being marginally independent of $I_K$ on top of the above). In general, energy-based models are to be interpreted as conjunctions of soft constraints, the factor graph is one implication of a system of stochastic differential equations, and interventions denote change to particular constraints. A SDE model may have *other* assumptions on top of the factorization, and parameters which carry particular meaningful interpretations, but this fits well with our claims that the IFM is a \"minimalist\" family of models in terms of structural assumptions - a reasonably conservative direction to follow particularly when the dynamics of many natural phenomena cannot be (currently) measured at individual level, as it's the case of much of cell biology data. [33] has further elaborations on some of these ideas.\n\n**Usefulness of Theorems 3.1 and 3.2:** it is clear that we should have been more explicit in the transition between these sections. Section 3 is about identification methods, and those don't necessarily need to translate directly into estimation methods. By analogy, think of the formulas implied by the do-calculus or G-computation, and how it's not necessarily the case they are used in estimation methods by plugging-in a corresponding estimate of a density. \n\nThat is, the proofs in Section 3 are constructive, but this doesn't imply they should lead to a one-to-one correspondence with a learning algorithm (for instance, G-estimation looks very different from what a direct approach based on G-computation may suggest). As long as we know that we can get $\\Sigma_{test}$ by products of ratios of densities in $\\Sigma_{train}$, it doesn't matter whether we can identify each factor as long as all densities follow the common factorization that shares factors, so a purely likelihood-based approach suffices (we did try density ratio estimation methods, but they worked poorly out-of-the-box and we decided to omit further discussion on them). We will clarify this in a final manuscript by a more fleshed out bridge between Sections 3 and 4. All problems in Section 5 can be identified based on the fact that it satisfies the conditions of Theorem 1, as there is no more than one $\\sigma$ per factor and $\\Sigma_{train}$ spans all necessary elementwise support for each $\\sigma$ variable (a scenario we mention in Section 3). We will comment on this explicitly in a revised manuscript.\n\n**Algebraic formulation:** essentially, we need to find a vector $\\{q_i\\}$ such that $p(x; \\sigma) \\propto \\prod_{k = 1}^t p(x; \\sigma^i)^{q_i}$, or show that there is no solution. With $x$ fixed, this is done by treating each factor $f_k(x_{S_k}, \\sigma_{F_k})$ as a symbol in an algebraic system. This means that each density, up to a multiplicative constant, is a monomial in those symbols. For instance, in the example in Figure 3, we use the symbol $f_1^{00}$ to denote the factor $f_1(x; \\sigma_1 = 0, \\sigma_2 = 0)$, while e.g. $f_3^{10}$ denotes $f_3(x; \\sigma_1 = 1, \\sigma_3 = 0)$ and so on. So the monomial corresponding to $p(x; \\sigma = (1, 1, 1))$ is $f_1^{11}f_2^{11}f_3^{11}$ and so on.\n\nNow, a PR transformation from the set of 7 training densities in Figure 3 (all configurations of binary $\\{\\sigma_1, \\sigma_2, \\sigma_3\\}$ except for the test configuration $\\sigma_1 = \\sigma_2 = \\sigma_3 = 1$) is of the form $$(f_1^{00}f_2^{00}f_3^{00})^{q_1} \\times (f_1^{01}f_2^{10}f_3^{00})^{q_2} \\times \\dots \\times (f_1^{10}f_2^{01}f_3^{11})^{q_7}.$$ \n\nFor an algebraic identity between $f_1^{11}f_2^{11}f_3^{11}$ and the above to hold (up to a multiplicative factor), it is necessary that $q_1, q_2, \\dots, q_7$ are such that, in the resulting multiplication, the resulting exponents for $f_1^{11}$, $f_2^{11}$, and $f_3^{11}$ are 1, and all others are zero. For instance, $f_1^{00}$ appears in the regimes in lines 1 and 5 in the table of Figure 3. Hence, we must have $q_1 + q_5 = 0$. Symbol $f_1^{01}$ appears in lines 2 and 6, and therefore we must have $q_2 + q_6 = 0$. This goes on for all symbols (columns in the table), ending with $f_3^{11}$, which only appears in the final density, making $q_7 = 1$. This gives a system with one solution, the one shown in the caption. The admittedly ugly Eq. (4) is a compact way of describing this system of equations.\n\n**Brief definitions:** provided with the opportunity to write a camera-ready version of this paper, we will have one extra page and we will be able to move selected and summarized definitions from the appendix into the main body, including that of a junction tree."}]

Author rebuttal:
Thank you all for such detailed and helpful reviews! Getting the time and attention of no fewer than six experts is no small privilege. 

Although there is some overlap among questions, it's not too substantive. As there are many reviewers, we think it will be most convenient for them if we make our rebuttals self-contained, so there will be some degree of repetition in the replies.

Again, many thanks for your help!

Meta-review: The authors propose using factor graph graphical models to build a bridge between observational and interventional distributions. Pearl does this with SCMs and CBNs. Authors argue there are several practical settings where acyclicity assumption is not warranted, and fitting the SCM framework requires strong assumptions in such cases. They suggest using a collection of "intervention nodes" to represent any possible intervention.

The core techniques in the paper are not new. But I think using factor graphs to represent different interventions, and showing that one can use this probabilistic model to extrapolate across interventional distributions is a nice modeling tradeoff, in between a causally sufficient graph where every intervention can be estimated vs. no structure between interventional distributions where nothing can be done.

My main comments are below:

- As also specified by many reviewers, authors should make a more serious effort in comparing with the DAG-based modeling. For example, DAGs and undirected graphs can model distributions that cannot be modeled by each other (under faithfulness at least). There is no discussion on what kind of distributions can be modeled by factor models vs. DAGs in a natural way. The only DAG comparison is given in Figure 1 on a non-representative DAG where the DAG imposes no conditional independence.

- There is no discussion on the number of intervention requirements. Clearly one will need more interventions than a DAG-based model, but by how much? Demonstrative examples comparing ADMGs (causal graphs with latents) by possibly pulling some examples from the transportability literature of Bareinboim and Pearl would be nice to add to showcase the tradeoff in terms of how many interventions are required to transport across the two modeling frameworks.

I am still in favor of acceptance. But I strongly encourage the authors to address the above concerns and the reviewer feedback to correctly position their paper, and ensure their contribution will be conveyed more clearly to a general audience.
Decision: Accept (poster)