Identifying and Estimating Causal Effects under Weak Overlap by Generative Prognostic Model
Year: 2021

Abstract:
As an important problem of causal inference, we discuss the identification and 1 estimation of treatment effects (TEs) under weak overlap, i.e., subjects with certain 2 features all belong to a single treatment group. We use a latent variable to model 3 a prognostic score (PGS), which is widely used in biostatistics and sufficient for 4 TEs, i.e., we build a generative prognostic model. We prove that the latent variable 5 recovers a PGS, and the model identifies individualized treatment effects. The 6 model is then learned as the Intact-VAE, a new type of variational autoencoder 7 (VAE). We derive counterfactual generalization bounds which motivate represen8 tation balanced for treatment groups conditioned on individualized features. The 9 proposed method is compared with recent methods using (semi-)synthetic datasets. 10

text: As an important problem of causal inference, we discuss the identification and1 estimation of treatment effects (TEs) under weak overlap, i.e., subjects with certain2 features all belong to a single treatment group. We use a latent variable to model3 a prognostic score (PGS), which is widely used in biostatistics and sufficient for4 TEs, i.e., we build a generative prognostic model. We prove that the latent variable5 recovers a PGS, and the model identifies individualized treatment effects. The6 model is then learned as the Intact-VAE, a new type of variational autoencoder7 (VAE). We derive counterfactual generalization bounds which motivate represen-8 tation balanced for treatment groups conditioned on individualized features. The9 proposed method is compared with recent methods using (semi-)synthetic datasets.10
1 Introduction11
Causal inference [21, 34], i.e, inferring causal effects of interventions, is a fundamental problem. In12 this work, we focus on treatment effects (TEs), such as effects of public policies or a new drug, based13 on a set of observations consisting of binary labels for treatment / control (non-treated), outcome, and14 other covariates (e.g, patients’ personal records). The fundamental difficulty of causal inference is15 that we never observe counterfactual outcomes, which would have been if we had made the other16 decision (treatment or control). While the ideal protocol for causal inference is randomized controlled17 trials (RCTs), they often have ethical and practical issues, or suffer from prohibitive costs. Thus,18 causal inference from observational data is indispensable. It introduces other challenges, however.19 The most crucial one is confounding: there may be variables (called confounders) that causally affect20 both the treatment and the outcome, and spurious correlation follows.21
A large majority of works, including this work, rely on the unconfoundedness, which means that22 appropriate covariates are collected so that the confounding can be controlled by conditioning on23 covariates. That is, all the confounders are in essence observed. This is still challenging, due to24 systematic imbalance (difference) of the distributions of the covariates between the treatment and25 control groups, introducing bias in estimation. Among classical ways of dealing with imbalance26 are matching and re-weighting [44, 35]. Machine learning methods are also exploited; there are27 semi-parametric methods, e.g, [48, TMLE], which have better finite sample performance, and also28 non-parametric, tree-based methods, e.g., [49, Causal Forests (CF)]. Notably, starting from [23], there29 is a recent rise of interest in learning representation of covariates, which is independent of treatment30 groups, i.e., balanced representation learning (BRL) .31
The most serious form of imbalance is that sample points with certain values of covariate are all32 belong to a single treatment group, which is called weak overlap of the covariate. Causal effects are33 not directly estimable at non-overlapped covariate values. There are lines of work that give robustness34 to weak overlap [3], trim non-overlapped sample points [52], or study convergence rate depending35 on overlap [19]. Weak overlap is particularly relevant to machine learning methods exploiting rich36 covariates, because, with higher-dimensional covariates, overlap is harder to satisfy and verify [10].37
Submitted to 35th Conference on Neural Information Processing Systems (NeurIPS 2021). Do not distribute.
Our approach to the weak overlap issue is based on the prognostic score (PGS) [14], which is among38 the important concepts of sufficient scores. While the most well-known score is the propensity score39 (PPS) [36], PGSs have also long been known to improve methods using PPS [37, 5], and interests last40 in biostatistics [45, 2]. Prognostic modeling can benefit more from predictive systems and exploit41 richer literature than propensity modeling, particularly in Medicine and Health. A comparative study42 in [13] shows PGS-based methods perform better, or as well as, PPS methods. Thus, it is promising43 to combine the predictive powers of prognostic modeling and machine learning.44
To solve the inverse problem of recovering PGSs, our method exploits also the recent advance45 of identifiable representation, particularly of VAE [26, iVAE]. Identification means parameters of46 interest (for us, representation function and causal effects) are uniquely determined and given by47 true observational distribution. Identification logically precedes estimation and inference. Without48 identification there is no hope of a consistent estimator, and a model would fail silently; it may fit49 perfectly but return an estimator that converges to the wrong one or does not converge [29, particularly50 Sec. 8]. Identification is even more important for causal inference, because, unlike usual (non-causal)51 model misspecification, causal assumptions are often unverifiable through observables [50]. Thus, it52 is critical to specify theoretical conditions for identification, and then the applicability of methods53 can be judged by knowledge of an application domain.54
In this work, we study identification (Sec. 3) and estimation (Sec. 4) of TEs under weak overlap. We55 particularly discuss individualized treatment effects, conditioned on the covariates. Code and proofs56 are in Supplementary Materials. The main contributions of this paper are:57 1) theory of TE identification under weak overlap of covariates, using PGS and identifiable model;58 2) counterfactual generalization bounds on TE error, which motivates our conditional BRL;59 3) a new regularized VAE to estimate TEs, with connections to identification and balancing;60 4) experimental comparison to state-of-the-art methods on (semi-)synthetic datasets.61
2 Setup and motivation62
2.1 Counterfactuals, treatment effects, and identification63
Following [21], we introduce potential outcomes (POs, or counterfactual outcomes) y(t) ∈ Rd, t ∈64 {0, 1}. y(t) is the outcome that would have been observed, if treatment value t = t had been applied.65 Formally, this is the consistency of counterfactuals: y = y(t) if t = t, or simply y = y(t). We see66 y(t) as the hidden variables that give factual y under factual assignment t = t. The fundamental67 problem of causal inference is that, for a unit under research, we could observe only one of y(0) or68 y(1), corresponding the treatment value applied. That is, “factual” refers to y or t that is in principle69 observable in data, or statistical entities (e.g, estimators) built on them. We also observe relevant70 covariate(s) x ∈ X ⊆ Rm, which is associated with individuals, with distribution D ∼ p(x,y, t).71 Note, we use Roman fonts for random variables (e.g., t) and italic for realization (e.g., t).72
The expected PO is denoted by µt(x) = E(y(t)|x = x), conditioned on x = x. The estimands in73 this work are the Conditional Average TE (CATE) and Average TE (ATE), defined respectively by74
τ(x) = µ1(x)− µ0(x), ν = E(τ(x)). (1)
CATE is an individual-level, personalized, treatment effect, given highly discriminative covariate.75
Standard results [38][16, Ch. 3] give sufficient conditions for identification under general setting.76 They are Exchangeability: y(t) |= t|x, and Overlap: p(t|x) > 0 for any x ∈ X . Both are required for77 t ∈ {0, 1}. When t appears in a statement without quantification, we always mean “for both t”. Often,78 Consistency is also listed, but, as above, it is better known as well-definedness of counterfactuals.79 Exchangeability means, just as in RCTs but additionally given x, that there is no correlation between80 factual treatment t and counterfactual outcomes y(t). Overlap means that the supports of p(x|t = 0)81 and p(x|t = 1) should be the same, and this ensures it is valid to condition on any (x, t).82 We relax overlapped covariate in Sec. 3.2, to allow some non-overlapped values x, i.e., covariate x is83 weakly overlapped. In Sec. 2.2, we introduce a condition which gives exchangeability (and PGSs). In84 this paper, we also discuss overlap of variables other than x (e.g. PGSs), and we give a definition for85 any random variable v with support V as following.86 Definition 1. Overlap of v means p(t|v) > 0 for all t ∈ {0, 1},v ∈ V . We also say v is overlapped.87 If the condition is violated at some value v, then v is non-overlapped and v is weakly overlapped.88
2.2 Prognostic scores89
Our method is motivated by PGSs [14], adapted as Pt-score and P-score in Definition 2 in this paper,90 related to balancing scores b(x), which is defined by t |= x|b(x) [36]. The PPS p(t = 1|x) is a special91 case of this. Both are sufficient scores for identification; PGSs are sufficient statistics of outcome92 predictors and b(x) is for the treatment (see Appendix for details).93 Definition 2. A Pt-score (PtS) is two functions Pt(x) (t = 0, 1) such that y(t) |= x|Pt(x). A PtS is94 called a P-score (PS) if P0 = P1.95
Note that, a PtS is by definition two functions, thus overlapped Pt(x) means both P0(x) and P1(x)96 are overlapped. Why PtS (PGS)? PtS is more applicable than balancing score b(x) under weak97 overlap. Overlapped b(x) implies overlapped x, which in turn implies overlapped PGS [10]. Lower-98 dimensional than x, PtS is likely more overlapped than x, and, moreover, there is evidence that PtS99 maximizes overlap among all sufficient scores for ATE [9].100
Below is a direct corollary of Proposition 5 in [14]. Both of PtS and PS give CATE, but, as we will101 see, PS is better as a conditionally balanced representation, since Pt(x) |= t|x only when P0 = P1.102 Proposition 1 (CATE by PtS). If Pt is a PtS, then CATE can be given by103
µt(x) = E(E(y(t)|Pt,x)) = E(E(y|Pt(x), t = t)) = ∫ p(y|Pt = Pt(x), t)ydy (2)
With the knowledge of Pt, we choose one of P0,P1 corresponding to the counterfactual outcome of104 interest. This ability of counterfactual assignment resolves the problem of non-overlap at x.105
PtSs exist under general settings when y(t) follows an additive noise model (ANM).106
(G1) (ANM) the data generating process (DGP) for y is y = f∗(M(x), t) + e where f∗,M are107 functions and e is a zero-mean exogenous (external) noise.108
The DGP defines y(t) by setting t = t in the equation. And it also specifies how other variables109 causally affect y. For example, x affects y through M, so M(x) is the effect modifier [14], which is110 often components of x affecting y directly. Note (G1) also implies exchangeability given x, through111 y(t) |= t|M(x). ANMs are also commonly used in nonparametric regression methods for TEs [6].112 Under (G1), 1) Pt := f∗t (M(x)) = µt(x)1 is a PtS2 but not PS, 2) M is a PS (x is a trivial PS), and113 3) P := (µ0(x), µ1(x)) is a PS. We use the same symbol to denote a PtS and the random variable114 defined by it, when appropriate.115
3 Identification under generative prognostic model116
In Sec. 3.1, we introduce our generative prognostic model and VAE based117 on p(y, z|x, t) and prove identifiability of our model. In Sec. 3.2, we prove118 identification of CATEs, one of our main contributions. The theoretical119 analysis involves only our generative model (i.e., prior and decoder), but120 not encoder, because model identifiability is a property of model, and121 causal identification is about DGP and model. The encoder is involved122 in estimation which is studied in Sec. 4.123
3.1 Intact-VAE: model, architecture, and identifiability124
Generative models are useful to solve the inverse problem of recovering125 Pt-score. Our goal is to build a model that can be learned by VAE from126 observational data to obtain a PtS, or more ideally PS, via the latent127 variable z. That is, a generative prognostic model.128
With the above goal, the generative model of our VAE is built as129
p(y, z|x, t) = p(y|z, t)p(z|x, t). (3) The first factor is our decoder which models p(y|Pt, t) in (2), and the130
second factor is our conditional prior which models Pt(x). Conditioning on x in the joint model131
1We often write t of function argument in subscript, which indicates possible counterfactual assignment. 2µt is the most common PGS, to the extent that some call it the PGS (e.g, [39, 9, 47]), even without ANMs.
p(y, z|x, t) reflects that our estimand is CATE given x. Modeling the score by a conditional distri-132 bution rather than a deterministic function is more flexible. We parameterize our model by ANM133 outcome and factorized Gaussian prior as134
pf (y|z, t) = p (y − ft(z)); pλ(z|x, t) ∼ N (z;ht(x),diag(kt(x))) (4) where θ = (f ,h,k) are functional parameters and is a noise. λ(x) := diag(k−1t (x))(h(x),− 12 ) T135 is the natural parameter of the Gaussian prior, and we also use it as a shorthand for both h,k.136
The ELBO of our model can be derived from standard variational lower bound137 log p(y|x, t) ≥ log p(y|x, t)−DKL(q(z|x,y, t)‖p(z|x,y, t))
= Ez∼q log p(y|z, t)−DKL(q(z|x,y, t)‖p(z|x, t)). (5)
Our encoder q, which conditions on all the observables, is standard, and we will see its importance138 later. We name this architecture Intact-VAE (Identifiable treatment-conditional VAE).139
We naturally have an identifiable conditional VAE (CVAE), as the name suggests. Note that (3) has a140 similar factorization with the generative model of iVAE [26], that is p(y, z|x) = p(y|z)p(z|x); the141 first factor does not depend on x. Further, since we have the conditioning on t in both the factors of142 (3), our VAE architecture is a combination of iVAE and CVAE [42, 28], with t as the conditioning143 variable. See Figure 1 for the comparison in terms of graphical models. The core idea of iVAE is144 reflected in our model identifiability (Lemma 1 below). See Appendix for the basics of VAEs.145
The following conditions on the model are used in theoretical analysis.146
(M1) i) ft is injective, ii) ft is differentiable, and iii) n := dim(z) = d(= dim(y)).147 Lemma 1 (Model identifiability). Given model (3) and (4) under (M1) i) and ii), for t = t, assume148
(D1) (Linear independence of λ) there exist 2n+1 points x0, ...,x2n ∈ X such that the 2n-square149 matrix Lt := [γt,1, ...,γt,2n] is invertible, where γt,k := λt(xk)− λt(x0).150
Then, given t = t, the family is identifiable up to an equivalence class. That is, if pθ(y|x, t = t) =151 pθ′(y|x, t = t), we have the relation between parameters: for any yt in the image of ft,152
f−1t (yt) = diag(a)f ′ t −1 (yt) + b =: A(f ′t −1
(yt)) (6) where diag(a) is an invertible n-diagonal matrix and b is a n-vector, both depend on λt.153
The conditions are inherited from iVAE. (D1) holds easily in practice, if the components of λt(x)154 are linearly independent; if (D1) fails, then the support of λt(x) is in a (2n− 1)-dimensional space.155 The essence of the result is f ′t = ft ◦ At, that is, ft can be identified (learned) up to an affine156 transformation defined by λt. This is achieved by combining the techniques from [26] and [43], and157 essentially the same results can be proved for other exponential family priors [43]. In this paper,158 symbol ′ (prime) always indicates another parameter (variable, etc.).159
3.2 Nonparametric identifications under weakly-overlapped covariate160
In this subsection, we give two identification results based on (partial) recovery of PS or PtS,161 respectively. Since PtSs are functions of x, the recovery is achieved by a noiseless prior, that is,162 k(x) = 0; the prior zλ,t ∼ pλ(z|x, t = t) degenerates to deterministic function ht(x).163 PtSs with dimensionality lower than or equal to y are essential to work for weak overlap of x, to the164 extent that, from now on, we simply say PSs / PtSs when referring to this kind of low-dimensional165 PSs / PtSs, unless particularly indicated. (M1) iii), i.e. n = d, is not restrictive because µt is a PtS of166 the same dimension as y under (G1). Also, in practice, n = d means that we seek a low-dimensional167 representation of x. In fact, to make the dimensionality explicit in (G1), we introduce an alternative168 (G1’) which includes (G1) with Pt = µt and jt is identity.169
(G1’) (Low-dimensional PtS) Under (G1), µt(x) = jt(Pt(x)) for some Pt and injective jt.170 We use (G1’) afterwards. Clearly, Pt in (G1’) is a PtS, and injectivity and n = d ensure n =171 dim(y) ≥ dim(Pt). Similarly, the next (G2) reduces unverifiable n ≥ dim(P) to n = d, for PS.172
(G2) (Low-dimensional PS) Under (G1), µt(x) = jt(P(x)) for some P and injective jt.173
(G2) means that CATEs are given by µ0 and an invertible function i := j1 ◦ j−10 . See Appendix for174 more discussion and a (closely related) real world example. In Sec. 4.1, we argue that there often175 exist equivalent PSs under (G1’), at least approximately.176
With (G1’) or (G2), overlapped x can be relaxed to overlapped P(t)S plus the following.177
(M2) (Score partition preserving) For any x,x′ ∈ X , Pt(x) = Pt(x′) =⇒ ht(x) = ht(x′).178 Note that (M2) is in fact required for optimal h, in the sense specified in Proposition 1 and Theorem 1179 below. The intuition is that, Pt maps non-overlapped x to an overlapped value, and ht preserves this180 property, through learning. In fact, (M2) is trivially satisfied if Pt and ht are linear, and this is still181 challenging and considered by many works [32, 9], or some with linear outcome models [11, 39].182
Our first identification, Proposition 2, relies on (G2) and our generative model, without model183 identifiability (so differentiable ft is not needed). This is a nonparametric3 identification under shape184 restriction [7], because f ,h are functional parameters, and injectivity is monotonicity if jt is on R.185 Proposition 2 (Identification with PS). Given (G2) and model (3) and (4) under (M1) i) and iii),186 and (M3) (PS matching) h0(x) = h1(x) and k(x) = 0. Then, if Epθ (y|x, t) = E(y|x, t), we have4187
1) (Recovery of PS) zλ,t = ht(x) = v(P(x)) on overlapped x,188 where v : P → Rn is an injective function and P := {P(x)|overlapped x}189 2) (Identification) if P in (G2) is overlapped, and (M2) is satisfied, then µt(x) = µ̂t(x)190 for any t ∈ {0, 1},x ∈ X , where µ̂t(x) := Epλ(z|x,t)Epf (y|z, t) = ft(ht(x)).191
The essence is, i) the true DGP is identified up to an invertible mapping v, so that ft = jt ◦ v−1 and192 ht = v ◦ Pt, and ii) Pt is recovered up to v and y(t) |= x|Pt is preserved, with same v for both t.193 PS is preferred since it satisfies overlap more easily and (M2) than PtS which refers to two functions.194 However, the existence of low-dimensional PS is uncertain in practice when our knowledge of the195 DGP is limited. Thus, we need Theorem 1 to work under PtS which generally exists.196 Theorem 1 (Identification with PtS). Given the DGP (G1’) and model (3)&(4) under (M1) and197 (M3’) (Noise matching) p = pe and k(x) = kk′(x), k → 0, assume (D1) and198
(D2) (Spontaneous balance) There exist 2n + 1 points x0, ...,x2n ∈ X , 2n-square matrix C,199 and 2n-vector d, such that L−10 L1 = C and β0 − C−Tβ1 = d/k for optimal λt (see200 below), where Lt is defined in (D1), βt := (αt(x1)−αt(x0), ..., αt(x2n)−αt(x0))T , and201 αt(x;λt) is the log-partition function of the prior in (4).202
Then, if pθ(y|x, t) = p(y|x, t), conclusions 1) and 2) in Proposition 2 hold with P replaced with Pt203 in (G1’), and the domain of v becomes P := P0 ∪ P1,Pt := {Pt(x)|overlapped x}.204
Theorem 1 also achieves the two essential points, but in different and complementary ways. Proposi-205 tion 2 starts from the prior by P0 = P1 and setting h0 = h1. Conversely, Theorem 1 starts from the206 decoder with p = pe and strengthens model identifiability (6) by (D2). (D2) restricts the discrepancy207 between λ0,λ1 on 2n+ 1 points of x, thus is relatively easy to satisfy with high-dimensional x.208
We see more reasons to prefer PS. In general, to identify the mean function µt(x), a regression is209 enough, and p = pe is unnecessary as in Proposition 2. Also, (D2) is trivial if we have PS and set210 λ0 = λ1. See Appendix for more on the complementarity between the two identifications.211

heading: 4 Estimation by β-Intact-VAE212
text: 4.1 Prior as PS, posterior as PtS, and β as regularization strength213
In this subsection, we discuss our focus on balanced PtSs and give an estimation method to realize it.214
In learning the Intact VAE with data, we assume that there is a PtS and the decomposition of (G1’)215 holds. Such a decomposition is not unique in general, however. Among possible PtSs, we wish to216 learn a balanced PtS, which is close to PS. This is based on the observations in Sec. 3.2: we saw217 that existence of PS is preferable in identifying the true DGP up to equivalent expression. Here, we218 introduce the notion of balanced PtS in a non-rigorous way: a PtS Pt is called balanced if the value of219 some measure for the conditional independence Pt(x) |= t|x is small. The idea is common in practice.220 For example, in a real-world nutrition study, [20] reduces 11 covariates to a 1-dimensional linear PS.221
3Some references (e.g, [16]) only refer to identification without models as “nonparametric”. However, with recent advances, we have identification under nonparametric models [29], also the case in this paper.
4Same results hold without “under (G1)” in (G2) (or (G1’) for Theorem 1), except that zλ,t is not necessarily a PS (or PtS for Theorem 1). This is because (G1) is required to ensure µt is a PtS.
Assuming that there is a balanced PtS, we consider two ways for estimating it with Intact VAE. One222 is to exploit a prior that does not depend on t. Namely, we set λ0 = λ1 =: λ in (4). The other is223 to introduce a hyperparameter β in the ELBO as in β-VAE [17]. More specifically, with the prior224 pλ(z|x), we use factorized Gaussian for the decoder and encoder:225 pf ,g(y|z, t) ∼ N (y;ft(z),diag(gt(z))); qφ(z|x,y, t) ∼ N (z; rt(x,y),diag(st(x,y))). (7)
The modified ELBO with β, up to additive constant, is derived as226
ED{−βDKL(qφ(z|x,y, t)‖pλ(z|x))− Ez∼q[(y − ft(z))2/2g2t (z)]− Ez∼q log |gt(z)|}. (8) Here, for convenience, we omit the summation (also in Lf in Sec. 4.2), as if y was univariate. The227 approximate posterior (or encoder) qφ(z|x,y, t) depends on t, which can realize a PtS. With β, we228 control the trade-off between the first and second term: the former is the divergence of the posterior229 from the balanced prior, and the latter is the reconstruction of the outcome. By choosing β in an230 appropriate way, such as by validation, the ELBO can find a solution that explains the outcome while231 keeping the balancedness of the posterior. Also note that, the parameters g and k, which models the232 outcome noise and expresses uncertainty of the prior, respectively, are both learned by the ELBO.233 This deviates from the theoretical conditions in Sec. 3.2, but is more practical and gives better results234 in experiments. See Appendix for much more on ideas and connections behind the ELBO.235
Once the encoder qφ is learned5, the estimate of the expected POs is given by236
µ̂t̂(x) = Eq(z|x=x)ft̂(z) = ED|x∼p(y,t|x)Ezft̂(z)qφ(z|x,y, t), t̂ ∈ {0, 1}, (9) where q(z|x) := Ep(y,t|x)qφ(z|x,y, t) is the aggregated posterior and D ∼ p(x,y, t). In estimation,237 we consider the case where x is observed in the data, and the sample of (y, t) are taken from the238 data given x = x (when x is not in the data, we replace qφ with pλ in (9), see Appendix for details239 and results). Note that t̂ in (9) indicates counterfactual assignment which may not be the same as240 factual t = t in the data. That is, we set t = t̂ in the decoder. The assignment is not applied to the241 encoder, but it is learned from factual x,y (see also Sec. 4.2, the explanation for CF,t). The overall242 algorithm steps are i) we train VAE by (8), ii) infer CATE τ̂(x) = µ̂1(x)− µ̂0(x) by (9).243
4.2 Conditional balanced representation learning244
We formally justify our ELBO (8) from the viewpoint of BRL. Usually, particularly in ATE estimation,245 balance means covariate balance, i.e., x |= t [44]. Influenced by this, most BRL methods learn balanced246 covariate representation z such that z |= t [40, 31] and usually z is a function of x. From Sec. 4.1, we247 understand that larger β in ELBO (8) encourages z |= t|x which is given by the prior, corresponding248 to Pt(x) |= t|x for a balanced PtS. Here, we show that, this conditional balance of representation z249 is natural for CATE estimation, and CATE error due to bad recovery of jt in (G1’) is controlled by250 ELBO (8). In Appendix, we detail novel implications of our bounds, compared to those in [40, 31].251
Using (9) to estimate CATE, τ̂f (z) = f1(z)− f0(z) is marginalized on q(z|x). The bounds below252 motivate both prior and posterior balancing. Let us first consider errors defined by the aggregated253 prior p(z|x) := ∑ t p(t|x)pt(z|x) (denote pt(z|x) := pλ(z|x, t = t)), and q(z|x) afterwards. We254 introduce the objective we bound. The true CATE, given covariate x = x or score z, is255
τ(x) = m1(P1(x))−m0(P0(x)); τm(z) = m1(z)−m0(z) (10) where mt(z) := E(y(t)|Pt = z) and Pt is a balanced PtS in (G1’). Accordingly, given x, the error256 of prior CATE, with or without knowing Pt, is naturally defined as257
∗,pf (x) := Ep(z|x)(τ̂f (z)− τ(x)) 2; pf (x) := Ep(z|x)(τ̂f (z)− τm(z)) 2. (11)
We bound pf instead of ∗,p f because the error between τ(x) and τm(z) is small if balanced Pt is258 recovered (then z ≈ P0(x) ≈ P1(x) in (10), see Appendix for details). Instead, we consider the error259 between τ̂f and τm below. We define the risks of outcome regression, into which p is decomposed.260 Definition 3 (PO Risks). The expected loss of PO at (z, t), factual risk, and counterfactual risk are261
Lf (z, t) := g−2t Ep(y(t)|Pt=z)(y(t)− ft(z)) 2 = gt(z) −2 ∫ (y − ft(z))2p(y(t) = y|Pt = z)dy; pF,t(x) := Ept(z|x)Lf (z, t); p CF,t(x) := Ep1−t(z|x)Lf (z, t) = ∫ Lf (z, t)p1−t(z|x)dz.
5As usual, we expect variational inference and optimization procedure are (near) optimal, i.e., Consistency of VAE (see Appendix for formal statement). Consistent estimation using the prior is a direct corollary of consistent VAE. Under Gaussian models, it is possible to prove consistency of posterior estimation, as shown in [4].
With y(t) involved, Lf is a PO error of f weighted by g. Factual and counterfactual counterparts are262 defined accordingly, w.r.t factual pt learned from data. Note, in F,t, unit u = (x,y, t) involves in263 the learning of pt(z|x) (and qt(z|x) afterwards), and also in Lf (z, t) since y(t) = y for the unit. In264 CF,t, however, y(t) 6= y′ = y(1− t) for u′ = (x,y′, 1− t) (particularly relevant for the posterior).265 Thus, the regression error (second) term in ELBO (8) controls pF,t via factual data. On the other266 hand, pCF,t is not estimable due to unobservable y(1− t), but is bounded as below.267 Lemma 2 (Counterfactual risk bound). Assume |Lf (z, t)| ≤M , we have268
pCF (x) ≤ ∑ t p(1− t|x) p F,t(x) +MDp(x) (12)
where pCF (x) := ∑ t p(1− t|x) p CF,t(x), and Dp(x) := ∑ t √ DKL(pt‖p1−t)/2.269
pCF (x) is bounded by p F,t plus MDp(x), which measures the imbalance between pt(z|x) and is270 symmetric for t. We can implicitly control pCF by making Dp small. Again, this means PS is preferred271 as a conditional balanced representation, and justifies our balanced prior pλ(z|x). Moreover, the272 results (including Theorem 2 below) also hold for posterior estimation, that is, replace pt(z|x) with273 qt(z|x) := q(z|x, t = t) = Ep(y|x,t)qφ(z|x,y, t) , the results and proofs hold as it was. This implies274 that the imbalance between qt should also be controlled. Correspondingly, the symmetric KL term in275 ELBO (8) balances qt(z|x) by encouraging z |= t|x for the posterior.276 Theorem 2 in turn bounds pf , by decomposing it to p F,t, p CF,t, and Vpy.277
Theorem 2 (Generalization bound). Assume |Lf (z, t)| ≤M and |gt(z)| ≤ G, then,278 f (x) ≤ 2(G2( F,0(x) + F,1(x)) +MD(x)− Vy(x)|p (13)
where Vpy(x) := Ep(z|x) ∑ t Ep(y(t)|Pt=z)(y(t)−mt(z))2, and “|p” collects all superscripts q.279
The new term Vpy(x) reflects the intrinsic variance in the DGP and can not be controlled, and it is280 negative because pf is defined by mean functions ft and mt, not y(t). The other two terms, as we281 indicated, is estimated by our ELBO.282
Estimating G,M is nontrivial. Instead, similarly to [40], we rely on β in the ELBO to weight the283 two terms in (13). We do not need two hyperparameters since G is implicitly controlled by the third284 term in ELBO (8), which is a norm constraint. As in matching methods, β is a trade-off between285 conditional balance of learned PtS (affected by ft) and precision / effective sample size of outcome286 regression, and can be seen as the probabilistic counterpart of [47, 25].287
Finally, we note the bounds do not directly address non-overlap; in Lemma 2, when p(1− t|x) = 0,288 pF,1−t in the r.h.s is unbounded since p1−t(z|x) can not be learned from data. However, as we argued289 in Sec. 3.2, with more balanced Pt recovered as representation, overlap is more easily satisfied.290
5 Related work291
Weak overlap. Under (respective versions of) weak overlap, [32] estimates ATE by reducing292 covariates to a linear PGS, [11] estimates homogeneous (constant) TE under partial linear outcome293 model, and [9] studies identification of ATE by a general class of scores, given (linear) PPS and PGS.294 In machine learning, current focus is on finding overlap regions [33, 8], or indicating possible failure295 under weak overlap [22], but not remedies. An exception is [24] which provides bounds without296 overlap. [40, 31] are in line of [24] and have similar bounds to ours, without relating to overlap. Our297 method is the first in machine learning that gives identification without overlap.298
Prognostic scores are recently combined with machine learning, mainly in biostatistics. For example,299 [39] trains a flexible PGS and fits a linear regression on the PGS among others, for constant TEs, and300 [12] models PGS in its Bayesian regression tree for CATE. More related, [20] estimates CATE by301 reducing covariates to a linear score that is a joint PPS and PGS, and [47] uses SVM to minimize302 the worst-case bias due to PGS imbalance. However, in machine learning, few methods consider303 PGSs; [55, 15] learn outcome predictors, without connection to PGS, while [24] conceptually, but304 not formally, connects BRL to PGS. Our work follows the recent boom in biostatistics and is the first305 to formally connect generative learning, PGS, and BRL (see below on BRL) for TE estimation.306
Identifiable representation. Recently, independent component analysis (ICA) and representation307 learning, both ill-posed inverse problems, meet together to give nonlinear ICA and identifiable308
representation, e.g., using VAEs [26], and energy models [27]. The results are exploited in causal309 discovery [51] and out-of-distribution generalization [46]. This work is the first to explore identifiable310 representations in TE identification.311
BRL and related methods amount to a major direction. Early BRL methods are BLR/BNN [23]312 and TARnet/CFR [40]. Adding to this, [53] also exploits the local similarity of between data points.313 [41] uses similar architecture to TARnet, considering the importance of treatment probability. There314 are also methods using GAN [54, GANITE] and Gaussian process [1]. Our method shares the idea of315 BRL, and further extends to conditional balancing more suitable for CATE.316
Our work hopefully lays conceptual and theoretical foundations of VAE methods for TEs (e.g.,317 [30, 31]), under unconfoundedness. Also, monotonicity, which is injectivity on R, is important in318 causal inference, and some works consider it together with overlap [24, 56]. See Appendix for details.319
6 Experiments320 We compare the proposed method with existing methods on three types of datasets. Here we present321 two experiments, and the rest one, on the Pokec social network dataset, can be found in Appendix. As322 in previous works [40, 30], we report the absolute error of ATE ATE := |ED(y(1)−y(0))−ED τ̂(x)|,323 and, as a surrogate of CATE, the empirical PEHE [18] PEHE := ED((y(1)− y(0))− τ̂(x))2.324 Unless otherwise indicated, for each function f ,h,k, r, s in (4)(7), we use a multilayer perceptron325 (MLP) that has 3*200 hidden units with ReLU activation, and λ = (h,k) depends only on x. We fix326 g(x) = 1 because the datasets have fixed noise scale, and results with learned g on synthetic dataset327 with dependent noise is in Appendix. The Adam optimizer with initial learning rate 10−4 and batch328 size 100 is employed. All experiments use early-stopping of training by evaluating the ELBO on a329 validation set, and results are reported on a testing set. Each set of running on synthetic dataset (a line330 in the figure) is within 1 hour on an 8-CPU machine, and it is within a day for IHDP. More details on331 hyper-parameters and settings are given in each experiment and Appendix.332
6.1 Synthetic dataset333
We generate synthetic datasets following (14). Both x,w are factorized Gaussians. µ,σ are randomly334 sampled. The functions h,k, l are linear. Outcome models f0, f1 are built by NNs with invertible335 activations. y is univariate, dim(x) = 30, and dim(w) ranges from 1 to 5. w is a PS, but is not336 low-dimensional when dim(w) > 1. We control overlap by ω which multiplies the logit value, and337 have 5 different overlap levels from strong overlap to very weak overlap. See Appendix for details.338
x ∼ N (µ,σ);w|x ∼ N (h(x),k(x)); t|x ∼ Bern(Logi(ωl(x))); y|w, t ∼ N (ft(w), 1). (14)
With the same (dim(w), ω), we evaluate our method and339 CFR on 10 random DGPs, with different sets of functions340 f,h,k, l in (14). For each DGP, we sample 1500 data341 points, and split them into 3 equal sets for training, val-342 idation, and testing. We show our results for different343 hyperparameter β. For CFR, we try different balancing344 parameters and present the best results (see Appendix for345 details). We report PEHE , see Appendix for ATE results.346
In each panel of Figure 2, we adjust one of ω,dim(w)347 respectively, with the other fixed to the lowest. In the left348 panel, both our method and CFR are quite robust to overlap349 level, supporting respective theories ([24] gives bounds for350 CFR under weak overlap). Too large β seems to worsen351 the results, possibly because we already have an apparent352
PS (w with dim(w) = 1) and large β incurs sub-optimal ELBO with no gain.353
In the right panel, when dim(w) > 1, ft in (14) is non-injecitve and learning of PtS is necessary.354 Thus, larger β has a negative effect, and particularly, β = 1 is significantly better than β = 3. The355 drop of error for dim(w) > 3 is due to the randomness of f in (14). In Sec. 2.2, we saw that the356 2-dimsensional PS P := (µ0(x), µ1(x)) always exists under ANMs. Thus, when dim(w) > 2, our357 method tries to recover that P, and generally performs not worse than under dim(w) = 2, but still358 not better than under dim(w) = 1.359
Our method is much robuster against different DGPs than360 CFR (see error bars), though it is worse than CFR when361 dim(w) > 1. This is unsurprising because our model362 has 1-dimensional z, while CFR uses 200-dimensional363 representation. Thus, the results already show the power of364 identification and recovery of scores (see Figure 3 also). In365 fact, we observed that our method outperforms or matches366 CFR with higher-dimensional z (see Appendix). Thus, we367
believe the performance gap with dim(z) = 1 is due to the capacity of NNs in Intact-VAE.368
When dim(w) = 1, there are no better PSs than w, because ft is invertible and no information369 can be dropped from w. Thus, as shown in Figure 3, our method learns z as an approximate affine370 transformation of the true w, showing identification. For comparison, we run [30, CEVAE] which371 is also based on VAE but without identification, and it shows much lower quality of recovery. As372 expected, both recovery and estimation are better with balanced prior pλ(z|x), and we can see an373 example of bad recovery using pλ(z|x, t) in Appendix. More latent plots can also be found there.374
6.2 IHDP benchmark dataset375
This experiment shows our conditional balancing matches state-of-the-art BRL methods, and does not376 overfit to PEHE. The IHDP dataset [18] is widely used to evaluate machine learning based methods,377 e.g. [40, 41]. It is also used in [24] which considers weak overlap, because the covariates are weakly378 overlapped due to their correlation to the artificial treatment assignment. Finally, there is a linear PS379 (linear combination of the covariates). See Appendix for details.380
Note, most of covariates are binary, so the support of the PS is often on small and separated intervals381 and is possibly discrete. Thus, Gaussian latent z is misspecified. We use multivariate z in model382 to address this, similarly to [30]. We set β = 1 since it works well on synthetic dataset with weak383 overlap. To see our balancing property clearly, we modify our method and add two components for384 unconditional balancing from [40] (see Appendix), and compare this modified version to the original.385
As shown in Table 1, Intact-VAE outperforms or matches the state-of-the-art methods. Particularly,386 our method has the best ATE estimation, and is slightly worse than CFR for PEHE. This is possibly387 due to the fitting capacity (recall Sec. 6.1), and also we do not tune β. Notably, our method388 outperforms other generative models (CEVAE and GANITE) by large margins. The modified389 version is slightly improved, but we should note that the improvement for ATE is barely significant.390 This indicates overfitting to PEHE. In fact, PEHE estimates the marginalized error E (x) where391 (x) = (τ(x)− τ̂(x))2, and, compared with ATE , it focuses on values x with high probability and392 / or large (x). The balancing in [40] is based on bounding E (x), and thus tends to overly focus on393 the above values of x, resulting in sub-optimal estimation of CATE and even of ATE. This tendency394 is more apparent with sub-optimal hyperparameter for the unconditional balancing (see Appendix).395
7 Conclusion396
In this work, we proposed a method for CATE estimation, under weak overlap. Our method exploits397 identifiable VAE, a recent advance in generative models, and is fully motivated and theoretically398 justified by causal considerations: identification, PGS, and balancing. We show that VAEs are399 suitable for principled causal inference thanks to its probabilistic nature, if not compromised by ad400 hoc heuristics. We believe it is possible to extend the bounds in Sec. 4.2 to weak overlap, just as401 [24] extends [40] to weak overlap, and leave this for future. Experiments show evidence that the402 injectivity of f in our model is possibly unnecessary because dim(z) > dim(y) often gives better403 results. Theoretical study of this is an interesting future direction. To avoid potential negative societal404 impact (e.g, bad prescriptions), practitioners should judge the conditions of the proposed method by405 their domain expertise, and careful trials are always recommended.406
References407 [1] Ahmed M Alaa and Mihaela van der Schaar. Bayesian inference of individualized treatment408 effects using multi-task gaussian processes. In Advances in Neural Information Processing409 Systems, pages 3424–3432, 2017.410
[2] Joseph Antonelli, Matthew Cefalu, Nathan Palmer, and Denis Agniel. Doubly robust matching411 estimators for high dimensional confounding adjustment. Biometrics, 74(4):1171–1179, 2018.412
[3] Timothy B. Armstrong and Michal Kolesár. Finite-sample optimal estimation and inference on413 average treatment effects under unconfoundedness. arXiv preprint arXiv:1712.04594v5, 2021.414
[4] Stéphane Bonhomme and Martin Weidner. Posterior average effects. arXiv preprint415 arXiv:1906.06360v5, 2021.416
[5] M Alan Brookhart, Sebastian Schneeweiss, Kenneth J Rothman, Robert J Glynn, Jerry Avorn,417 and Til Stürmer. Variable selection for propensity score models. American journal of epidemi-418 ology, 163(12):1149–1156, 2006.419
[6] Alberto Caron, Ioanna Manolopoulou, and Gianluca Baio. Estimating individual treatment420 effects using non-parametric regression models: a review. arXiv preprint arXiv:2009.06472,421 2020.422
[7] Denis Chetverikov, Andres Santos, and Azeem M Shaikh. The econometrics of shape restrictions.423 Annual Review of Economics, 10:31–63, 2018.424
[8] Wangzhi Dai and Collin M Stultz. Quantifying common support between multiple treatment425 groups using a contrastive-vae. In Machine Learning for Health, pages 41–52. PMLR, 2020.426
[9] Alexander D’Amour and Alexander Franks. Deconfounding scores: Feature representations for427 causal effect estimation with weak overlap. arXiv preprint arXiv:2104.05762, 2021.428
[10] Alexander D’Amour, Peng Ding, Avi Feller, Lihua Lei, and Jasjeet Sekhon. Overlap in429 observational studies with high-dimensional covariates. Journal of Econometrics, 2020.430
[11] Max H Farrell. Robust inference on average treatment effects with possibly more covariates431 than observations. Journal of Econometrics, 189(1):1–23, 2015.432
[12] P Richard Hahn, Jared S Murray, Carlos M Carvalho, et al. Bayesian regression tree models433 for causal inference: Regularization, confounding, and heterogeneous effects (with discussion).434 Bayesian Analysis, 15(3):965–1056, 2020.435
[13] David Hajage, Yann De Rycke, Guillaume Chauvet, and Florence Tubach. Estimation of436 conditional and marginal odds ratios using the prognostic score. Statistics in medicine, 36(4):687–437 716, 2017.438
[14] Ben B Hansen. The prognostic analogue of the propensity score. Biometrika, 95(2):481–488,439 2008.440
[15] Negar Hassanpour and Russell Greiner. Learning disentangled representations for counterfactual441 regression. In International Conference on Learning Representations, 2019.442
[16] Miguel A. Hernan and James M. Robins. Causal Inference: What If. CRC Press, 1st edition,443 2020.444
[17] Irina Higgins, Loïc Matthey, Arka Pal, Christopher Burgess, Xavier Glorot, Matthew Botvinick,445 Shakir Mohamed, and Alexander Lerchner. beta-vae: Learning basic visual concepts with a con-446 strained variational framework. In 5th International Conference on Learning Representations,447 2017.448
[18] Jennifer L Hill. Bayesian nonparametric modeling for causal inference. Journal of Computa-449 tional and Graphical Statistics, 20(1):217–240, 2011.450
[19] Han Hong, Michael P Leung, and Jessie Li. Inference on finite-population treatment effects451 under limited overlap. The Econometrics Journal, 23(1):32–47, 2020.452
[20] Ming-Yueh Huang and Kwun Chuen Gary Chan. Joint sufficient dimension reduction and453 estimation of conditional and average treatment effects. Biometrika, 104(3):583–596, 2017.454
[21] Guido W Imbens and Donald B Rubin. Causal inference in statistics, social, and biomedical455 sciences. Cambridge University Press, 2015.456
[22] Andrew Jesson, Sören Mindermann, Uri Shalit, and Yarin Gal. Identifying causal-effect457 inference failure with uncertainty-aware models. Advances in Neural Information Processing458 Systems, 33, 2020.459
[23] Fredrik Johansson, Uri Shalit, and David Sontag. Learning representations for counterfactual460 inference. In International conference on machine learning, pages 3020–3029, 2016.461
[24] Fredrik D Johansson, Uri Shalit, Nathan Kallus, and David Sontag. Generalization bounds and462 representation learning for estimation of potential outcomes and causal effects. arXiv preprint463 arXiv:2001.07426, 2020.464
[25] Nathan Kallus, Brenton Pennicooke, and Michele Santacatterina. More robust estimation of465 sample average treatment effects using kernel optimal matching in an observational study of466 spine surgical interventions. arXiv preprint arXiv:1811.04274, 2018.467
[26] Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. Variational au-468 toencoders and nonlinear ica: A unifying framework. In International Conference on Artificial469 Intelligence and Statistics, pages 2207–2217, 2020.470
[27] Ilyes Khemakhem, Ricardo Monti, Diederik Kingma, and Aapo Hyvarinen. Ice-beem: Iden-471 tifiable conditional energy-based deep models based on nonlinear ica. Advances in Neural472 Information Processing Systems, 33, 2020.473
[28] Durk P Kingma, Shakir Mohamed, Danilo Jimenez Rezende, and Max Welling. Semi-supervised474 learning with deep generative models. In Advances in neural information processing systems,475 pages 3581–3589, 2014.476
[29] Arthur Lewbel. The identification zoo: Meanings of identification in econometrics. Journal of477 Economic Literature, 57(4):835–903, 2019.478
[30] Christos Louizos, Uri Shalit, Joris M Mooij, David Sontag, Richard Zemel, and Max Welling.479 Causal effect inference with deep latent-variable models. In Advances in Neural Information480 Processing Systems, pages 6446–6456, 2017.481
[31] Danni Lu, Chenyang Tao, Junya Chen, Fan Li, Feng Guo, and Lawrence Carin. Reconsidering482 generative objectives for counterfactual reasoning. Advances in Neural Information Processing483 Systems, 33, 2020.484
[32] Wei Luo, Yeying Zhu, and Debashis Ghosh. On estimating regression-based causal effects485 using sufficient dimension reduction. Biometrika, 104(1):51–65, 2017.486
[33] Michael Oberst, Fredrik Johansson, Dennis Wei, Tian Gao, Gabriel Brat, David Sontag, and487 Kush Varshney. Characterization of overlap in observational studies. In International Conference488 on Artificial Intelligence and Statistics, pages 788–798. PMLR, 2020.489
[34] Judea Pearl. Causality: models, reasoning and inference. Cambridge University Press, 2009.490
[35] Paul R Rosenbaum. Modern algorithms for matching in observational studies. Annual Review491 of Statistics and Its Application, 7:143–176, 2020.492
[36] Paul R Rosenbaum and Donald B Rubin. The central role of the propensity score in observational493 studies for causal effects. Biometrika, 70(1):41–55, 1983.494
[37] Donald B Rubin. Estimating causal effects from large data sets using propensity scores. Annals495 of internal medicine, 127(8_Part_2):757–763, 1997.496
[38] Donald B Rubin. Causal inference using potential outcomes: Design, modeling, decisions.497 Journal of the American Statistical Association, 100(469):322–331, 2005.498
[39] Alejandro Schuler, David Walsh, Diana Hall, Jon Walsh, and Charles Fisher. Increasing the499 efficiency of randomized trial estimates via linear adjustment for a prognostic score. arXiv500 preprint arXiv:2012.09935, 2020.501
[40] Uri Shalit, Fredrik D Johansson, and David Sontag. Estimating individual treatment effect:502 generalization bounds and algorithms. In International Conference on Machine Learning, pages503 3076–3085. PMLR, 2017.504
[41] Claudia Shi, David Blei, and Victor Veitch. Adapting neural networks for the estimation of505 treatment effects. In Advances in Neural Information Processing Systems, pages 2507–2517,506 2019.507
[42] Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using508 deep conditional generative models. In Advances in neural information processing systems,509 pages 3483–3491, 2015.510
[43] Peter Sorrenson, Carsten Rother, and Ullrich Köthe. Disentanglement by nonlinear ica with511 general incompressible-flow networks (gin). In International Conference on Learning Repre-512 sentations, 2019.513
[44] Elizabeth A. Stuart. Matching Methods for Causal Inference: A Review and a Look Forward.514 Statistical Science, 25(1):1 – 21, 2010.515
[45] Elizabeth A Stuart, Brian K Lee, and Finbarr P Leacy. Prognostic score–based balance measures516 can be a useful diagnostic for propensity score methods in comparative effectiveness research.517 Journal of clinical epidemiology, 66(8):S84–S90, 2013.518
[46] Xinwei Sun, Botong Wu, Chang Liu, Xiangyu Zheng, Wei Chen, Tao Qin, and Tie-yan Liu.519 Latent causal invariant model. arXiv preprint arXiv:2011.02203, 2020.520
[47] Alexander Tarr and Kosuke Imai. Estimating average treatment effects with support vector521 machines. arXiv preprint arXiv:2102.11926, 2021.522
[48] Mark J Van der Laan and Sherri Rose. Targeted learning: causal inference for observational523 and experimental data. Springer Science & Business Media, 2011.524
[49] Stefan Wager and Susan Athey. Estimation and inference of heterogeneous treatment effects525 using random forests. Journal of the American Statistical Association, 113(523):1228–1242,526 2018.527
[50] Halbert White and Karim Chalak. Identification and identification failure for treatment effects528 using structural systems. Econometric Reviews, 32(3):273–317, 2013.529
[51] Pengzhou Wu and Kenji Fukumizu. Causal mosaic: Cause-effect inference via nonlinear ica530 and ensemble method. In International Conference on Artificial Intelligence and Statistics,531 pages 1157–1167. PMLR, 2020.532
[52] S Yang and P Ding. Asymptotic inference of causal effects with observational studies trimmed533 by the estimated propensity scores. Biometrika, 105(2):487–493, 03 2018.534
[53] Liuyi Yao, Sheng Li, Yaliang Li, Mengdi Huai, Jing Gao, and Aidong Zhang. Representa-535 tion learning for treatment effect estimation from observational data. In Advances in Neural536 Information Processing Systems, pages 2633–2643, 2018.537
[54] Jinsung Yoon, James Jordon, and Mihaela van der Schaar. GANITE: Estimation of individ-538 ualized treatment effects using generative adversarial nets. In International Conference on539 Learning Representations, 2018.540
[55] Weijia Zhang, Lin Liu, and Jiuyong Li. Treatment effect estimation with disentangled latent541 factors. arXiv preprint arXiv:2001.10652, 2020.542
[56] Yao Zhang, Alexis Bellot, and Mihaela Schaar. Learning overlapping representations for543 the estimation of individualized treatment effects. In International Conference on Artificial544 Intelligence and Statistics, pages 1005–1014. PMLR, 2020.545
Checklist546 1. For all authors...547 (a) Do the main claims made in the abstract and introduction accurately reflect the paper’s548 contributions and scope? [Yes]549 (b) Did you describe the limitations of your work? [Yes] Particularly in Conclusion.550 (c) Did you discuss any potential negative societal impacts of your work? [Yes] We men-551 tioned possible misuse of our method and suggestions for practitioners in Introduction552 and Conclusion.553 (d) Have you read the ethics review guidelines and ensured that your paper conforms to554 them? [Yes]555 2. If you are including theoretical results...556 (a) Did you state the full set of assumptions of all theoretical results? [Yes]557 (b) Did you include complete proofs of all theoretical results? [Yes] In Appendix.558 3. If you ran experiments...559 (a) Did you include the code, data, and instructions needed to reproduce the main experi-560 mental results (either in the supplemental material or as a URL)? [Yes] In supplemental561 material.562 (b) Did you specify all the training details (e.g., data splits, hyperparameters, how they563 were chosen)? [Yes] Possibly in Appendix.564 (c) Did you report error bars (e.g., with respect to the random seed after running experi-565 ments multiple times)? [Yes]566 (d) Did you include the total amount of compute and the type of resources used (e.g., type567 of GPUs, internal cluster, or cloud provider)? [Yes]568 4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...569 (a) If your work uses existing assets, did you cite the creators? [Yes]570 (b) Did you mention the license of the assets? [Yes] In code project.571 (c) Did you include any new assets either in the supplemental material or as a URL? [Yes]572 In supplemental material.573 (d) Did you discuss whether and how consent was obtained from people whose data you’re574 using/curating? [Yes] In Appendix. And we refer to the original paper where this is575 discussed in details.576 (e) Did you discuss whether the data you are using/curating contains personally identifiable577 information or offensive content? [Yes] Refer to the original paper. And we believe578 there are no privacy issues.579 5. If you used crowdsourcing or conducted research with human subjects...580 (a) Did you include the full text of instructions given to participants and screenshots, if581 applicable? [N/A]582 (b) Did you describe any potential participant risks, with links to Institutional Review583 Board (IRB) approvals, if applicable? [N/A]584 (c) Did you include the estimated hourly wage paid to participants and the total amount585 spent on participant compensation? [N/A]586

Review summaries:
The paper studies identification and estimation of treatment effects (TEs) under weak overlap of covariates (Definition 1, probability of treatment conditioned on covariate has different supports for some covariate, which is closer to the real world). For identification of TEs, the paper exploits the prognostic score (PGS) and the identiﬁable representation, and as a result, proposes a novel, VAE-based generative prognostic model. Prior studies have seen good practical performances of PGS in many settings and hence, the paper expects it to be a good choice and to outperform other candidates in joint use with additional techniques. The identifiable representation is essential to guarantee the identification of TEs. The estimation of the VAE model is via a $\\beta$-regularized-ELBO which helps to keep the posterior distribution of latent variables of VAE ''balanced'' (defined in line 219). Motivations for e.g. balanced representation are provided later in the theoretical analyses on the counterfactual risk bound and the generalization bound. The proposed method is compared with baselines on (semi-)synthetic datasets.

This work studies the identification and estimation of treatment effects (with special attention on the CATE) under "weak" overlap.  It proposes a theory of treatment effect identification when the overlap assumption does not hold for the entire domain of $\mathbf{x}$ using the prognostic score.  It offers counterfactual generalization bounds on treatment effect estimates. It develops the Intact-VAE as a regularized treatment effect estimation model that accounts for identification and balancing. It offers proof that their generative model p(y, x | x, t) is identifiable. It offers proof that CATE is identifiable via their generative model. The methods are validated via a comparison to existing methods using synthetic and semi-synthetic datasets.

This paper proposes a generative prognostic model for identifying and estimating causal effects under weak overlap. The proposed method uses a latent variable to model a prognostic score (PGS). The authors prove that the latent variable recovers a PGS, and the model identifies individualized treatment effects. The model is learnt as an intact-VAE. The authors provide a theory of treatment effect identification and counterfactual generalization bounds on the treatment effect error. The generalization bounds motivate conditional balanced representation learning (BRL). Then they propose a new regularized VAE to estimate treatment effects. In experiments, they compare the proposed method with some state-of-the-art benchmarks on some synthetic and semi-synthetic datasets.