Bounding the Effects of Continuous Treatments for Hidden Confounders
Abstract
Observational studies often seek to infer the causal effect of a treatment even though both the assigned treatment and the outcome depend on other confounding variables.
An effective strategy for dealing with confounders is to estimate a propensity model that corrects for the relationship between covariates and assigned treatment.
Unfortunately, the confounding variables themselves are not always observed, in which case we can only bound the propensity, and therefore bound the magnitude of causal effects.
In many important cases, like administering a dose of some medicine, the possible treatments belong to a continuum.
Sensitivity models, which are required to tie the true propensity to something that can be estimated, have been explored for binary treatments.
We propose one for continuous treatments .
We develop a framework to compute ignorance intervals on the partially identiﬁed dose-response curves, enabling us to quantify the susceptibility of an inference to hidden confounders.
We show with real-world observational studies that our approach can give non-trivial bounds on causal effects from continuous treatments in the presence of hidden confounders.
Introduction
The goal of causal inference is to separate the effect of one treatment variable from the inﬂuence of many related but irrelevant “confounding” variables.
Physical interventions accomplish this most effectively, but practical barriers often force researchers to rely on observational studies and clever statistics.
A plethora of methods operate on the basis of a learned propensity model for the assigned treatment conditioned on covariates, for instance to reweigh the sample and remove any visible biases.
Usually, the covariates are inadequate to account for all the hidden paths between the treatment and the outcome, and propensity-based approaches may struggle to discern the real effect.
The scientiﬁc community at large continues to vascillate on the health implications and ideal consumption levels of coffee [Atroszko, 2019], alcohol [Ystrom et al., 2022], and cheese [Godos et al., 2020], to name a few substances.
Most sensitivity analyses to hidden confounding require the treatment categories to be binary or at least discrete.
This weakens the empirical studies that would have been better speciﬁed by doseresponse curves [Calabrese and Baldwin, 2001] from a continuous treatment variable, for example.
Estimated dose responses are indeed vulnerable in the presence of hidden confounders.
A simulated example in Figure 1 demonstrates how a J-shaped treatment effect can appear ﬂipped in observational data due to confounding.
The phenomenon is an example of Simpson’s paradox [Simpson, 1951,
Yule, 1903].
1.1 Related works
[Figure]
There is growing interest in causal methodology for treatments (or exposures, interventions) that take on speciﬁc values within a continuum, especially in the ﬁelds of econometrics [e.g. Huang et al., 2021, Tübbicke, 2022], health sciences [Vegetabile et al., 2021], and machine learning [Ghassami et al., 2021, Colangelo and Lee, 2021, Kallus and Santacatterina, 2019].
So far, much scrutiny on partially identiﬁed potential outcomes has focused on the case of binary treatments, the simplest setting [e.g. Rosenbaum and Rubin, 1983,
Louizos et al., 2017, Lim et al., 2021].
A number of creative approaches were exhibited in the past few years to make strides in this binary setting.
Most of them relied on a sensitivity model for bounding the extent of possible unobserved confounding, to which downstream tasks may be adapted by noting which treatment effects degrade the quickest.
A number of quite recent attempts have been made to handle unobserved confounding with continuous treatments.
Some optimize the treatment-effect bounds through generative models [Padh et al., 2022,
Hu et al., 2021], rather than a sensitivity model.
Another employs instrumental variables [Kilbertus et al., 2020].
One with a sensitivity model was developed in parallel to the present work [Jesson et al., 2022].
Regarding binary treatments, the so-called Marginal Sensitivity Model (MSM) due to Tan [2006] continues to be studied extensively
[Zhao et al., 2019, Veitch and Zaveri, 2020, Yin et al., 2021].
Variations thereof include Rosenbaum’s earlier sensitivity model [2002] that enjoys ties to regression coefﬁcients [Yadlowsky et al., 2020].
Other groups have borrowed strategies from deep learning [Wu and Fukumizu, 2022] rather than opting for the MSM.
Another active line of work constructs bounds not due to ignorance on confounding but instead viewed from the lens of robustness [Guo et al., 2022,
Makar et al., 2020, Johansson et al., 2020].
The MSM is highly interpretable with its single free parameter, and applicable to a wide swath of models.
Other approaches require additional structure to the data-generating ( observed outcome, treatment, covariates ) process.
Proximal causal learning [Tchetgen et al., 2020, Mastouri et al., 2021] requires additional proxy structures.
Chen et al.
[2022] rely on multiple large dataset partitions.
1.2 Contributions
Our ﬁrst contribution is to propose a unique sensitivity model (§2.1) that extends the MSM to a treatment continuum.
Next, we derive general (§3) and specialized (§3.2) formulas.
We devise an efﬁcient algorithm (§C) to compute ignorance bounds over dose-response curves, following up with experiments on real (§4) datasets.
Potential outcomes
Causal inference is often cast in the nomenclature of potential outcomes, due to Rubin [1974].
The broad goal is to measure a treatment’s effect on an individual, marked by a set of covariates, while accounting for all the confounding between the covariates and the treatment variable.
Effects could manifest heterogenously across individuals.
In non-interventional settings, observed covariates may not entirely overlap across treatment regimens.
It is typical to estimate two models, (1) the outcome predictor and (2) a model for the propensity of treatment conditioned on the covariates.
The latter may help account for biases.
z x . .
t observedoutcome potentialoutcomes t r e a t m e n t p r
o p e n s i ty p r
o x
y c o v a r i a t e s
Y₀
Y₀.₅
Y ₁
Y t
The ﬁrst two assumptions involved in Rubin’s framework are that observations of outcome, assigned treatment, and covariates { Y ( i ) , T ( i ) ,
X ( i ) } are i.i.d draws from the same population and that all treatments have a chance to occur for each covariate vector: p T | X ( t | x ) > 0 (overlap/positivity) for all t, x ∈ [0 , 1] × X , speciﬁcally in our context of continuous treatments.
The third and most challenging of these fundamental assumptions is that of ignorability , or sufﬁciency.
Our study is concerned with the scenarios where that assumption is violated: when there exists a dependency, not blocked by the covariates, between the assigned treatment and true potential outcomes.
Let p ( y t | x ) denote the probability density function of potential outcome Y t =
y t from a treatment t ∈ [0 , 1] , given covariates X = x .
Formally, we have a violation of ignorability: { ( Y t ) t ∈T (cid:54)⊥⊥ T } | X.
It is only realistic to observe samples of Y | T
= t, X
= x with density p ( y t | t, x ) .
However, to account for possible hidden confounding, we also require a p ( y t | τ (cid:54) =
t, x ) for quantifying treatment effects of the general form E [ f ( Y t ) | X ] , involving the density
[Equation]
where p ( y t | τ, x ) is the distribution of potential outcomes conditioned on actual treatment T = τ ∈ [0 , 1] that may differ from the potential outcome’s index t .
Throughout this study, y t will indicate the value of the potential outcome at treatment t , and to disambiguate with assigned treatment τ will be used for events where T
= τ .
For instance, we may care about the counterfactual of a smoker’s ( τ = 1) health outcome had they not smoked ( y t =0 ) , where T = 0 signiﬁes no smoking and T =is “full” smoking.
We aim to develop some intuition before introducing the novelties.
On notation.
We will use the shorthand p ( · · · ) with lowercase variables whenever working with probability densities of the corresponding stochastic variables in uppercase.
In other words, p ( τ | x ) means d d τ P [ T ≤ τ | X = x ] , and p ( y t | τ, x ) means d d u P [ Y t ≤ u | T = τ, X = x ] (cid:12)(cid:12)(cid:12) u = y t .
Interpretation.
How would one interpret p ( y t | τ, x ) ?
The potential-outcomes vector ( Y t ) t ∈ [0 , 1] of inﬁnite dimensionality is intrinsic to each individual with true confounder Z , for which X is a noisy proxy.
By “true” confounder we refer to any set of variables that sufﬁce to block all backdoor paths between Y t and T .
The potential-outcomes vector would only change from knowledge of assigned treatment T
= τ if it betrayed additional information about Z , absent in X , that further informed any Y t .
We may express p ( y t | τ, x ) explicitly in terms of hypothetical true confounders as (cid:82) p ( y t | z ) p ( z | τ, x ) d z because z subsumes both x and τ .
This way, p ( y t | z ) is the true potential outcome and p ( z | τ, x ) acts as a ﬁlter for how parts of the true confounder mix together into the proxy x and the assigned treatment τ .
Propensities.
The probability density p ( τ | x ) is termed the nominal propensity .
A quantity often examined is the complete propensity , speciﬁcally referring to p ( τ | y t , x ) in our realm.
The complete propensity can differ from p ( τ | x ) because of hidden confounders.
In that instance, conditioning on potential outcome y t modulates the distribution.
Similarly, by connection through Bayes’ rule, conditioning the potential outcomes p ( y t | x ) on assigned treatment τ modulates those distributions.
Absent any unobserved confounding, p ( y t | τ, x ) =
p ( y t | x ) and Equation 1 trivializes.
See Figurefor a graphical illustration on the runaway inﬂuence of τ on the potential outcomes.
Sensitivity.
Explored by Kallus et al.
[2019] and Jesson et al.
[2021] among many others, the
Marginal Sensitivity Model (MSM) serves to bound the extent of (putative) hidden confounding in the regime of binary treatments T (cid:48) ∈ { 0 , 1 } .
Speciﬁcally, it couples the odds of treatment under the nominal propensity to the odds of treatment under complete propensity, limiting the discrepancy:
Deﬁnition 1 (The Marginal Sensitivity Model) .
For binary treatment t (cid:48) ∈ { 0 , 1 } and Γ ≥ 1 , the following restriction is placed on complete propensities: Γ − 1 ≤ (cid:104) p
[Table]
Restricting ourselves to binary treatments affords us a number of conveniences.
For instance, one probability value is sufﬁcient to describe the whole propensity landscape on a set of conditions, p (1 − t (cid:48) | · · · ) = 1 − p ( t (cid:48)
| · · · ) .
As we transfer to the separate context of t ∈ [0 , 1] , we must contend with inﬁnite treatments and inﬁnite potential outcomes.
2.1 Towards continuous sensitivity
We require a constraint on the quantity p ( τ | y t , x ) that is fundamentally unknowable across all the combinations of assigned treatments T = τ ∈ [0 , 1] and potential outcomes y t ∈ [0 , 1] .
As with the
MSM, our target is to associate p ( τ | y t , x ) to the knowable p ( τ | x ) .
In other words, we seek to constrain the knowledge conferred on propensity by a single potential outcome y t .
It is not necessary for the functions pertaining to ( y t ) t ∈ [0 , 1] to exhibit any degree of smoothness in t .
The potentialoutcome variables are treated as entries in an inﬁnitely long vector.
However, we do impose that the propensity probability densities p ( τ | . . . )
are at least once differentiable in τ .
What sort of analogue exists for the notion of “odds” in the MSM?
Contrast treatment τ versus τ + δ locally, for some inﬁnitesimal δ , at any part of the curve.
A translation of the MSM might appear as (cid:104) p ( τ + δ | x ) p ( τ | x ) (cid:105) − 1 (cid:104) p ( τ + δ | y t ,x ) p ( τ | y t ,x ) (cid:105) .
Let us peer into one of those ratios.
In logarithms,
[Equation]
Hence, we introduce the inﬁnitesimal MSM ( δ MSM), tying ∂ τ log p ( τ | y t , x ) to ∂ τ log p ( τ | x ) .
Deﬁnition 2 (The Inﬁnitesimal Marginal Sensitivity Model) .
For treatments in the closed unit interval, t ∈ [0 , 1] , and violation factor Γ ≥ 1 , the complete propensities must obey the following inequality:
[Equation]
We crafted the δ MSM with the intention of functionally mirroring the MSM—locally, on a treatment continuum.
Whereas Deﬁnition 2 is stated in logarithms, Deﬁnition 1 is not; the difference is merely cosmetic and hyperparameter Γ plays an equivalent role in both structures.
Nevertheless, the emergent properties are vastly different.
The framework
We list the core assumptions surrounding our problem.
Assumption 1 (Bounded Hidden Confounding) .
Invoking Deﬁnition 2, the violation of ignorability is constrained by a δ MSM with some Γ ≥ 1 .
Assumption 2 (Fully Observed Confounding at No Treatment) .
The utter lack of treatment is not informed by potential outcomes: p ( τ = 0 | y t , x ) = p ( τ = 0 | x ) for all t and y t .
Assumption 2 states that we look for sensitivity to hidden confounders outside the control group at T = 0 .
The restriction is reasonable in situations like the following: we seek to estimate the effect of a prescription drug, and some clinics prescribe different dosages.
Our T = 0 group would be individuals who have not received any such prescription, and T > 0 would place patients on a scale depending on prescription dosage.
We expect a dramatically lessened vulnerability to hidden confounders for the well-represented—in observed and unobserved attributes—control group.
From a technical perspective, Assumption 2 is necessary for our derivations, and should be interpreted as a blind spot in the sensitivity model rather than a requirement for the underlying process.
There is no additional constraint, besides the δ MSM itself, on how much the complete propensity function may ﬂuctuate around any T > 0 .
We motivate and validate this assumption in the real world with §4.
Next, we proceed with derivations.
The key to cracking open Equation 1 is to carve out a region inside the domain of integration where an approximation can be trusted.
This will extrapolate from the singular point τ = t where estimation is feasible.
3.1 Dealing with an unreliable approximation
We approximate p ( y t | τ, x ) around τ = t, where p ( y t | t, x ) = p ( y | t, x ) is learnable from data.
Suppose that p ( y t | τ, x ) is twice differentiable in τ .
Construct a Taylor expansion
[Equation]
Denote with ˜ p
( y t | τ, x ) an approximation of ﬁrst or second order as laid out above.
We will encounter that even ∂ τ p ( y t | τ, x ) | τ = t is intractable.
Thankfully, it can be bounded using the δ MSM machinery.
Let us quantify the reliability of this approximation by a set of weights 0 ≤ w t ( τ ) ≤ 1 , where typically (but need not necessarily) w t ( t ) = 1 .
Decompose the integral of Equation 1—
[Equation]
This separation into recoverable (A) and entirely unknown (B), demarcated by the weights, ensures that the inaccurate regimes of the approximation vanish (as w t ( τ ) → 0 away from t ) and are replaced with the ignorant quantity.
We simplify part B of Equation 3 ﬁrst, into p ( y t | x )[1 − (cid:82)0 w t ( τ ) p ( τ | y t , x ) d τ ] .
We witness already that p ( y t | x ) shall take the form of
[Equation]
How the approximation error of Equation 2 carries into Equation 4 depends on the peakedness of the weight function.
To proceed further demands reﬂecting on Assumptions 1 & 2, as we do in §A.
A note on ensemble uncertainty.
One should quantify empirical uncertainties [Jesson et al., 2020] alongside sensitivity to hidden confounding.
In our experiments we learn both the predictor and the propensity model as ensembles from bootstrapped resampled [Lo, 1987] data.
Then ˜ p ( y t | x ) can also be resampled for conﬁdence intervals via its component ensembles.
3.2 Tractable weight combinations
In addition to developing the general framework above, we derive analytical forms for a speciﬁc paramametrization to the weighting function and propensity distribution.
Here, we look to the Beta function and its associated probability density for a natural solution.
Suppose that
[Equation]
We designed the reliability weights to mirror the propensity’s form by rescaling a Beta density.
We assert that w t ( τ ) peaks at τ = t , and that w t ( τ ) = 1 .
We ﬁnd that c t (cid:44) t rt (1 − t ) r (1 − t ) , even though the solution is irrelevant for our purposes.
The mode is ﬁxed: ( a t − 1) / ( a t + b t − 2) =
t .
See §B for details on the solution.
Results from an observational study
The most pertinent application for the framework laid out above is an observational study with incomplete or noisy covariates and a continuous treatment variable.
More concretely, the treatment variable should be transformed and scaled into the unit interval such that T = 0 signiﬁes a control with a complete lack of treatment.
Every kind of individual should be about equally likely to fall in the ( T = 0) cohort (Assumption 2.)
As for shaping the ( T > 0) regime, the domain should inform whether a linear scale is employed, versus an empirical or parametric cumulative density function.
[Figure]
We obtained real observational data from the UK Biobank and performed one semi-synthetic and one fully empirical experiment.
In the latter, we relied on discarding covariates to induce greater hidden confounding.
details.
The objective.
We chose to investigate the coverage
[McCandless et al., 2007] of E
[ Y t ] from ignorance intervals on counterfactual models trained with inadequate covariates.
In the semi-synthetic case (Figure 3), the covariates were real but the outcome was simulated, and the treatment effect was known to be linear after a logit transformation.
Coverage in the empirical case (Figure 4) was more difﬁcult in the absence of a ground truth.
There, we trained an uncensored model on an expanded set of covariates to act as an approximate target for the smaller model’s ignorance intervals.
In both cases, the recall was expressed as the portion of the dose-response curve that satisﬁed the relevant objective; ignorance was, the in for the semi-
, and b normalized to the 95%-conﬁdence intervals of the uncensored for the empirical study.
The comparisons.
We compared our δ MSM with r = 32 throughout ( solid in the ﬁgures on this page) to other sensitivity models: namely, ( dotted ) an analogue developed independently and in parallel to the present work, with just one free parameter [Jesson et al., 2022]; ( dashed ) the product of shoehorning a continuous model into the binary MSM by triggering a binary treatment at T > 0 .and
discretizing the propensity at the threshold; and ( dot/dash ) a baseline sensitivity model that emerges from Γ -scaling the Algorithm 1 weights without any propensity.
[Figure]
Discussion
The utility of our framework is evident in the above showcased results.
We demonstrated that, in the presence of a semi-synthetic ground truth, the δ MSM covers the full dose-response curve most efﬁciently.
In the empirical study, we showed that discrepancies in the potential-outcomes continuum between a censored model and a full model are most efﬁciently bridged under the δ MSM assumption.
Moving forward, we plan on generalizing the permissible anchoring points that were restricted by
Assumption 2 at T = 0 .
Even further, the same δ MSM can be applied to treatment variables in ( −∞ , + ∞ ) or [0 , + ∞ ) , inviting new propensity/weight parametrizations.
We shall also investigate the superﬁcial similarity of the δ MSM to the Riesz representers for the average causal derivative under the general framework of Chernozhukov et al.
Ethical implications.
Sensitivity models for hidden confounders can help to guard against erroneous conclusions from observational studies.
We generalized this line of analysis to the regime of continuous treatments, thereby increasing its practical applicability.
The method also bears utility in the context of fairness in machine learning.
It can help decision makers identify subpopulations for whom the sample is too biased to reliably draw conclusions.
Nevertheless, researchers must be careful to maintain a healthy degree of skepticism towards observational results even after properly calibrating the partially identiﬁed effects.
Acknowledgments and Disclosure of Funding
This material is based upon work supported, in part, by the Defense Advanced Research Projects
Agency (DARPA) and Army Research Ofﬁce (ARO) under Contract No.
W911NF-21-C-0002.
A Additional derivations
We will expand the denominator of Equation 4 ﬁrst, and then repurpose the results for the derivatives of Equation 2 that appear in the numerator.
This part shows how the assumed model serves to characterize the unknown quantities, but the impatient reader may skip to §3.2.
Without loss of generality, consider
[Equation]
We may attempt to integrate both sides;
[Equation]
by Assumption 2)
[Equation]
Clearly | λ ( τ | y t , x ) | ≤ τ log Γ because it integrates γ , bounded by ± log Γ , over a support with length τ .
Subsequently Λ( τ | y, t ) is bounded by Γ ± τ .
We are now equipped with the requisite tools to properly bound p ( y t | x ) —or an approximation thereof, erring on ignorance via reliability weights w t ( τ ) .
Consider Equation 3.A:
[Equation]
( A. 2) , where g k ( y t | t, x ) (cid:44) ∂ kτ p ( y t | τ, x ) | τ = t .
Lightening the notation with a shorthand for the weighted expectations, (cid:104)·(cid:105) τ (cid:44) (cid:82) 1 0 w t ( τ )( · ) p ( τ | x ) d τ, it becomes apparent that we must grapple with the pseudo-moments (cid:104) 1 (cid:105) τ , (cid:104)
τ − t (cid:105) τ , and (cid:104) ( τ − t ) 2 (cid:105) τ .
Note that t should not be mistaken for a “mean” value.
Furthermore, we have yet to fully characterize g k ( y t | t, x ) .
Observe that
[Equation]
The p ( y t | x ) will be moved to the other side of the equation as needed; by Equation 8,
[Equation]
Expanding,
[Equation]
Appropriate bounds will be calculated for g 2 ( y t | t, x ) next, utilizing the ﬁnding above as their main ingredient.
Let
[Equation]
The second derivative may be calculated in terms of the ignorance quantities γ, Λ :
[Equation]
And ﬁnally we address ˜ p ( y t | x ) .
Carrying over the components of Equation 9 into Equation 3, ˜ p
[Equation]
y t
[Equation]
=
[Equation]
where these expectations E τ [ · ] are with respect to the implicit distribution q ( τ | t, x ) ∝ w t ( τ ) p ( τ | x ) .
The notation ˙ γ denotes a derivative in the ﬁrst argument of γ ( t | y t , x ) .
To make use of this formula, one ﬁrst procures the set of admissible d ( t | y t , x ) ∈
[ d ( t | y t , x ) , d ( t | y t , x )] that violate ignorability up to a factor Γ according to the δ MSM.
Then, considering their reciprocals as importance weights [Tokdar and Kass, 2010], tight bounds on the partially identiﬁed expectations over ˜ p
( y t | x ) may be optimized.
B Analytical solutions for the Beta parametrization
The remaining degree of freedom disappears by a precision constraint a t + b t − 2 =
r for some r > 0 .
Constraining a more complex dispersion statistic like variance is much more difﬁcult.
The expectations found in Equation 10 are now available in closed form, and can be bounded in terms of just two extra free parameters, Γ and r .
Guidance on setting the violation factor Γ is discussed elsewhere, e.g. §4; as for the class of weights, high r conveys poor trust in the Equationapproximation, as studied in §B.
[Figure]
The ﬁndings.
We pose a third and ﬁnal assumption, which enables us to state Proposition 1.
The main insight to unlocking those expectations is that each one of them involves an integral with the product w t ( τ ) p ( τ | x ) over its normalization constant, yielding the moments of a Beta distribution.
Assumption 3 (Second-order Simpliﬁcation) .
The quantity ˙ γ ( τ | y t , x ) cannot be characterized as-is.
We grant that γ 2 dominates, and consequently (cid:12)(cid:12) (˙ γ + γ 2 )Λ (cid:12)(cid:12) ≤ (cid:12)(cid:12) γ 2 Λ (cid:12)(cid:12) + ε for small ε ≥ 0 .
Proposition 1 (Beta Parametrizations) .
The formulations in Equations 5 & 6 admit analytical solutions to the ignorance denominator in Equation 10.
With α, β implicitly referring to α ( x ) , β ( x ) ,
[Equation]
where again the operator E τ is employed as in Equation 10, and 1 F 1 denotes Kummer’s conﬂuent hypergeometric function [Mathews Jr. et al., 2021].
In addition, E τ [ τ − t ] and E τ [( τ − t ) 2 ] can be readily computed by means of the ﬁrst two moments of the Beta distribution.
C Computing the ignorance intervals
After deriving ˜
p
( y t | x ) in Equation 10 and a speciﬁc solution with Proposition 1, we must ﬁnd a way to bound the partially identiﬁed expectations with respect to this distribution.
Concretely, we seek to characterize the Individual Treatment Effect (ITE) E [ f ( Y t ) | X
= x ] or Average Treatment Effect (ATE) E [ F ( Y t )] for any task-speciﬁc f ( y ) .
This is accomplished with a Monte Carlo importance sampler of n outcome realizations y
i drawn from proposal q ( y ) :
[Equation]
For the ATE, one additionally averages over covariates with sample size m :
[Equation]
Even though ˜ p ( y t | x ) is a normalized probability density, it contains partially identiﬁed quantities.
It is untenable to constrain a search along the candidate values for each d ( t | y t
=
y
i , x ) to even approximately ensure (cid:82) Y ˜ p
( y t = y | x ) d y = 1 .
For this reason the bias of an estimator without the corrective denominator of Equation 11 would be uncontrollable [Tokdar and Kass, 2010].
A greedy algorithm may be deployed to maximize ˜ E
[ f ( Y t ) | X = x ] in the form above by optimizing weights w i attached to each f ( y i ) , within the range
[Equation]
The minimum may be achieved by a trivial adaptation.
Maximizing and minimizing ˜
[Equation]
[ f ( Y t ) | X = x ] with respect to the bounding quantities ( γ, Λ) enables the resolution of ignorance bounds on the basis of Γ from Deﬁnition 2.
Our Algorithm 1 adapts the method of Jesson et al.
[2021] to heterogeneous weight bounds [ w i , w i ] per draw i .
input : { ( w i , w i , f i ) } ni =1 ordered by ascending f i .
output : max w E [ f ( X )] estimated by importance sampling with n draws.
Initialize w i ← w i for all i = 1 , 2 , . . .
n ; for j = 1 , 2 , . . .
n do
Compute ∆ j (cid:44) (cid:80) ni
=
w i ( f j − f i ) ; if ∆ j < 0 then w j ← w j ; else break; end end
Return (cid:80) i w i f i / (cid:80) i w i ;
Algorithm 1: The expectation maximizer intermediate
D Experimental details
Data from the UK Biobank were accessed under application 11559.
From the brain Magnetic
Resonance Imaging (MRI) data we extracted the 74 ﬁelds corresponding to parcelized cortical volumes on the left and the right hemispheres each [Miller et al., 2016].
Semi-synthetic evaluation.
In our ﬁrst experiment, the synthetic binary outcome was generated by linearly combining the covariates and treatment and then applying a logistic curve for a Bernoulli probability.
As the logit-transformed ATE was known to be linear, “recall” was evaluated as the portion of the ignorance intervals that permitted the actual linear effect along each section of the dose response.
Semi-synthetic dataset.
The 148 MRI ﬁelds were normalized such that values ﬂoored at each variable’s 25% quantile and ceiled at the 95% quantile were mapped to the range [0 , 1] .
In each of the four replications, a random quarter of the covariates were assigned a nonzero, normally distributed coefﬁcient, and one of them was deemed the treatment variable with unit coefﬁcient.
After computing the outcomes, we randomly resampled a third of the feature values, with replacement, in order to introduce noise to the covariates while preserving the marginals.
Semi-synthetic estimators.
Both the predictor and the propensity model, censored and uncensored, were trained in ensembles of 32 artiﬁcial neural networks with one inner residual layers of 32 activation units each.
A dropout of 0 .
05 was imposed on these layers.
Additionally, an L 2 -regularization on the inner layers with weight 10 − 3 was applied.
All predictors were trained for 10,000 epochs and propensities for 5,000 epochs.
Empirical evaluation.
Our test relies on approximating an unconfounded model by collecting a large set of covariates, and then learning another model on a heavily censored version.
Our reasoning is that the censored model would suffer from a greater degree of hidden confounding.
The censored model could then be assessed along all potential-outcome predictions, by pretending that the full model represented the real dose-response curves.
A pertinent metric would be how much a sensitivity model with Γ ≥ 1 swallows the “real” dose responses, as a trade-off against the sheer area of the ignorance bounds.
These competing quantities are a form of recall and (the opposite of) precision, respectively.
Denote ( y c , y c ) the partially identiﬁed bounds of the censored model and ( y, y )
the full model’s bounds for ˜ E
[ Y t ] , both at 95% conﬁdence from percentile bootstrapping.
For some Γ := s and a t ∈ [0 , 1] grid, of length 17 in our case, ignorance is (cid:80) t
[ y c ( s, t ) − y c ( s, t )] / (cid:80) t [ y ( t ) − y ( t )] , and recall is the normalized intersection between the bounds:
[Equation]
Empirical dataset.
We summed each left/right pair to arrive at 74 positively valued outcomes.
Six groups of semantically related ﬁelds composed the long covariate vector:
• 6 basic details: age, weight, sex, standing height, seated height, and month of birth.
• 3 reported activity measurements: weekly minutes spent walking, engaged in moderate activity, and vigorous activity.
• 27 environment al variables surveying the pollution and greenery surrounding the person’s life.
• 42 blood measurements from cell counts to calcium concentration.
• 15 cardiac measurements including ECG and PWA modalities.
• 8 welfare indices for English citizens assessing the following: deprivation, income, employment, health
, education, housing, crime, and living environment.
Listwise deletion was employed to handle any missing value.
To censor the covariates, the four largest sectors ( italicized ) encompassing various confounding variables were omitted, one at a time.
The treatment variable was the walking ﬁeld taken from the triad of activity measurements, scaled to the unit interval such that any recording of at least two hours per day was set to T = 0 and any lesser amount had T increase up to 1 according to an empirical CDF.
Empirical estimators.
Both the predictor and the propensity model, censored and uncensored, were trained in ensembles of 32 artiﬁcial neural networks with four inner residual layers of 32 activation units each.
A dropout of 0 .
05 was imposed on these layers.
Additionally, an L 2 -regularization on the inner layers with weight 10 − 3 was applied.
All predictors were trained for 10,000 epochs and propensities for 5,000 epochs.
The outcome predictor parametrized a Gamma distribution, and the propensity model parametrized a Beta distribution.
See Figure 6.
epochs -0.
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Environ.
Predictor 5000epochs -0.
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Blood Predictor 5000epochs -0.
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Cardiac Predictor 5000epochs -0.
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d
Welfare Predictor   epochs 0.
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Environ.
Propensity   epochs 0.
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Blood Propensity   epochs 0.
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Cardiac Propensity   epochs 0.
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Welfare Propensity
[Figure]