Partial Identification of Policy-Relevant Treatment Effects with Instrumental Variables via Optimal Transport

Jiyuan Tan; Jose Blanchet; Vasilis Syrgkanis

arxiv: 2604.12263 · v2 · submitted 2026-04-14 · 📊 stat.ME · econ.EM

Partial Identification of Policy-Relevant Treatment Effects with Instrumental Variables via Optimal Transport

Jiyuan Tan , Jose Blanchet , Vasilis Syrgkanis This is my paper

Pith reviewed 2026-05-10 15:56 UTC · model grok-4.3

classification 📊 stat.ME econ.EM

keywords policy-relevant treatment effectsinstrumental variablesoptimal transportpartial identificationRoy modeldouble machine learningcausal inferencetreatment effects

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The pith

Policy-relevant treatment effects can be partially identified by recasting the problem as constrained conditional optimal transport, which reduces to one-dimensional problems with closed-form sharp bounds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

When an instrument only shifts treatment probabilities over a limited range, standard IV assumptions leave policy-relevant treatment effects only partially identified. The paper reformulates the identification problem in the generalized Roy model as a constrained conditional optimal transport task that matches the joint law of potential outcomes and latent resistance types to the observed conditional distributions. This multidimensional transport problem separates analytically into independent one-dimensional optimal transport calculations under product costs, which deliver explicit sharp bounds. The approach also supplies double machine learning estimators that attain root-n rates with high-dimensional covariates for discrete instruments and explicit nonparametric rates for continuous instruments, and it produces tighter sets than moment-relaxation methods in simulations and a bed-net subsidy application.

Core claim

PRTE partial identification in the generalized Roy model can be formulated as a Constrained Conditional Optimal Transport (CCOT) problem over the joint conditional law of the potential outcome and the latent resistance. The resulting multidimensional CCOT problem reduces analytically to separable one-dimensional OT problems with product costs, yielding sharp closed-form bounds and avoiding direct solution of the original high-dimensional CCOT problem.

What carries the argument

Constrained Conditional Optimal Transport (CCOT) over the joint conditional law of potential outcomes and latent resistance, which reduces to separable one-dimensional OT problems with product costs.

If this is right

The bounds are sharp and available in closed form without solving the full high-dimensional transport problem.
Double machine learning estimators based on Neyman-orthogonal scores achieve the parametric root-n rate and asymptotic normality for discrete instruments even with high-dimensional covariates.
For continuous instruments the framework supplies explicit nonparametric convergence rates.
The resulting bounds are substantially tighter than those from moment-relaxation methods in both simulations and the bed-net subsidy application.
The formulation extends to covariates, continuous instruments, and general treatment settings.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The separability into one-dimensional problems may allow similar optimal-transport reductions in other partial-identification settings that involve latent variables and conditional distributions.
The closed-form nature of the bounds could be used to optimize the choice of instruments or policies that maximize the width of the identified set for a target policy effect.
Integrating flexible machine-learning models for the conditional distributions inside the transport step could further improve finite-sample performance beyond the current DML procedure.

Load-bearing premise

The data are generated by a generalized Roy model in which the instrument produces only limited variation in treatment propensity.

What would settle it

Simulate data from the generalized Roy model with a binary instrument that shifts propensity only between 0.3 and 0.7, compute the true PRTE, and verify whether it always lies inside the closed-form bounds obtained from the observed conditional distributions; systematic violations outside those bounds would falsify sharpness.

Figures

Figures reproduced from arXiv: 2604.12263 by Jiyuan Tan, Jose Blanchet, Vasilis Syrgkanis.

**Figure 1.** Figure 1: Decomposition of the unit interval [0, 1] of the latent variable U for the sharp lower bound in Theorem 3.4 (K = 3 propensity levels). The key distinction is driven by where ω(u) drops. LIV regions (I0, I1): ω is constant on the entire interval because its step falls exactly at a propensity level pk; the integral therefore reduces to a directly identifiable conditional mean ωk Eµ1,k [Y ]. OT region (I2): ω… view at source ↗

**Figure 2.** Figure 2: Decomposition of the unit interval [0, 1] for the sharp lower bound in the general instrument setting (Theorem 3.7), illustrated with K = 2 in the order OT / LIV / OT / trivial: the non-degenerate LIV interval L1 = [p 1 , p1 ] (teal strip) lies between two OT gaps, and the degenerate LIV singleton L2 = {p2} (teal line) sits at the boundary of the trivial tail. OT gaps G0 = (0, p1 ) and G1 = (p1 , p2): only… view at source ↗

**Figure 3.** Figure 3: IVOT versus IVMTE identified sets for θα = E[Y qα − Y ] across policy shifts α ∈ [−0.12, 0.12]. Solid line: ground truth. Shaded ribbons: pointwise 95% confidence intervals for the estimated bound endpoints (orange = IVMTE, blue = IVOT). IVOT bounds are tighter in both settings. 6.2 Effect of Price Subsidies for Bed Nets Background and setup. We apply our method to the bed net subsidy experiment of Dupas [… view at source ↗

**Figure 4.** Figure 4: IVOT versus IVMTE identified sets for PRTEα = E[Y qα − Y ]/α on the bed net data of Dupas [2014a]. IVMTE shown for degree-10 (orange) and degree-20 (red) u-spline sieves; degree 20 widens the bound, reflecting the more nonparametric MTR class. Shaded ribbons: pointwise 95% confidence intervals for the estimated bound endpoints (orange = IVMTE deg. 20, blue = IVOT). IVOT substantially tightens the identifie… view at source ↗

**Figure 5.** Figure 5: Bounds of the moment relaxation approach and the CCOT approach by different [PITH_FULL_IMAGE:figures/full_fig_p064_5.png] view at source ↗

**Figure 6.** Figure 6: Continuous instrument (n = 10,000): IVOT versus IVMTE identified sets and IVMTE 95% backward CI for θα = E[Y qα − Y ] across α ∈ [−0.12, 0.12]. The IVOT bounds cover the truth for all α. {−0.02, −0.01, 0.01}), yielding an overall coverage rate of 88% (22 out of 25 grid points). The near-misses occur where the IVOT interval is extremely tight (width ≈ 0.002) and the true θα falls just outside the estimated … view at source ↗

**Figure 7.** Figure 7: Continuous instrument (n = 5,000): IVOT versus IVMTE identified sets for θα = E[Y qα − Y ] across α ∈ [−0.12, 0.12]. The three values near α = 0 where coverage fails are visible as points where the truth (solid line) falls just outside the IVOT bounds. 103 [PITH_FULL_IMAGE:figures/full_fig_p103_7.png] view at source ↗

read the original abstract

Policy-Relevant Treatment Effects (PRTEs) are generally not point-identified under standard Instrumental Variable (IV) assumptions when the instrument generates limited support in treatment propensity. We show that PRTE partial identification in the generalized Roy model can instead be formulated as a Constrained Conditional Optimal Transport (CCOT) problem over the joint conditional law of the potential outcome and the latent resistance. The resulting multidimensional CCOT problem reduces analytically to separable one-dimensional OT problems with product costs, yielding sharp closed-form bounds and avoiding direct solution of the original high-dimensional CCOT problem. We also develop estimation and inference procedures for these bounds: for discrete instruments, we use a Double Machine Learning (DML) approach based on Neyman-orthogonal scores that accommodates high-dimensional covariates while achieving the parametric $\sqrt{n}$ rate and asymptotic normality; for continuous instruments, we explicitly characterize the corresponding nonparametric convergence rates. The framework accommodates covariates, discrete and continuous instruments, and extensions to general treatment settings. In simulations and a bed-net subsidy application, the resulting bounds are substantially tighter than the moment-relaxation method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper reduces the PRTE partial ID problem to separable 1D optimal transport problems that give closed-form sharp bounds, then adds workable DML estimation.

read the letter

The core move here is to recast partial identification of policy-relevant treatment effects in the generalized Roy model as a constrained conditional optimal transport problem over the joint law of potential outcomes and latent resistance. Because the cost is a product, the high-dimensional CCOT separates into independent one-dimensional OT problems, which admit explicit solutions instead of numerical optimization. That is the actual novelty relative to moment-relaxation methods, and the abstract claims the resulting bounds are sharp under the maintained assumptions. The simulations and bed-net subsidy example show the bounds are materially tighter than the alternatives, which is the practical payoff. Estimation follows the usual Neyman-orthogonal DML route for discrete instruments, delivering sqrt(n) rates with high-dimensional covariates, and they spell out the nonparametric rates for continuous instruments. Those pieces look standard but are cleanly executed for this setting. The framework also handles covariates and general treatments without extra machinery. The main soft spot is that everything sits on the generalized Roy model plus the limited-support IV condition. If either is violated in ways the data cannot detect, the sharpness guarantee does not transfer to the real world, and the paper does not appear to provide extensive sensitivity checks on that front. One application is helpful but thin for establishing robustness. The derivations are presented as analytical, yet without the full proofs it is impossible to confirm there are no hidden steps that restrict the result further. This is for econometricians and applied researchers who already work with IV partial identification and want tighter, computable bounds without solving large transport problems. A reader who needs the closed-form expressions and the rate results will find it directly useful. The paper is coherent on its own terms and formally grounded enough to merit referee time, even if the empirical claims will need more scrutiny in review.

Referee Report

0 major / 3 minor

Summary. The manuscript claims that partial identification of Policy-Relevant Treatment Effects (PRTEs) under the generalized Roy model with instrumental variables (when the instrument has limited support in treatment propensity) can be reformulated as a Constrained Conditional Optimal Transport (CCOT) problem over the joint conditional law of potential outcomes and latent resistance. This multidimensional CCOT reduces analytically to separable one-dimensional OT problems with product costs, delivering sharp closed-form bounds without directly solving the high-dimensional problem. The authors develop Double Machine Learning estimators for discrete instruments that achieve sqrt(n) rates and asymptotic normality, characterize nonparametric rates for continuous instruments, accommodate covariates and general treatments, and show tighter bounds than moment-relaxation methods in simulations and a bed-net subsidy application.

Significance. If the central claims hold, the work provides a meaningful advance in partial identification for policy-relevant parameters by leveraging optimal transport to obtain tractable, sharp bounds that are otherwise difficult to compute. The analytical separability result is a clear strength, as is the integration with Neyman-orthogonal DML scores for feasible estimation under high-dimensional covariates. The framework's flexibility across instrument types and its empirical demonstration of improved bound tightness add practical value to the causal inference and econometrics literature.

minor comments (3)

The abstract and introduction would benefit from a brief explicit statement of the closed-form expression for the bounds (or at least their functional dependence on the marginals) to allow readers to immediately assess sharpness without consulting the full derivation.
Notation for the latent resistance variable and the product-cost structure in the CCOT formulation should be introduced with a short notational table or explicit definition early in the identification section to improve readability for readers less familiar with optimal transport.
In the simulation section, the comparison to the moment-relaxation method would be strengthened by reporting the width of the bounds or a normalized tightness metric in addition to qualitative statements.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The summary accurately captures the core contribution of reformulating PRTE partial identification as a CCOT problem that reduces to separable one-dimensional OT problems, along with the associated estimation procedures.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via external OT theory

full rationale

The paper formulates PRTE partial identification as a CCOT problem over the joint law of potential outcomes and latent resistance, then applies standard properties of optimal transport (specifically, separability under product costs) to reduce the multidimensional problem to independent one-dimensional OT problems. This reduction is a direct mathematical consequence of the product-cost structure and the limited-support IV assumption, not a redefinition or fit of the target quantity itself. No parameters are estimated from the target bounds and then relabeled as predictions; no load-bearing uniqueness theorem or ansatz is imported via self-citation; and the subsequent DML estimation follows conventional semiparametric theory. The central claim therefore rests on independent mathematical structure and external OT results rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on the generalized Roy model and standard IV assumptions with limited propensity support; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Generalized Roy model for potential outcomes and latent resistance
Invoked to set up the joint conditional law for the CCOT formulation.
domain assumption Instrumental variable assumptions with limited treatment propensity support
Standard IV conditions that prevent point identification and motivate partial ID via OT.

pith-pipeline@v0.9.0 · 5494 in / 1202 out tokens · 19267 ms · 2026-05-10T15:56:18.269746+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Also available as arXiv:2202.10665

URL https://proceedings.mlr.press/v177/guo22a/guo22a.pdf. Also available as arXiv:2202.10665. Peter Hall, Rodney C. L. Wolff, and Qiwei Yao. Methods for estimating a conditional distribution function.Journal of the American Statistical Association, 94(445):154–163, 1999. Sukjin Han and Shenshen Yang. A computational approach to identification of treatment...

work page doi:10.1111/1468-0262 1999
[2]

Similarly, P(Y∈ · |W = 0, Z = z) = δ0(·)forces π0(· |u ) = δ0 for almost all u∈ (1/4, 1)

Because the conditional distribution of the observed outcome is a Dirac measure at zero, P(Y∈ · |W = 1, Z = z) = δ0(·)for each z, the observational constraint forcesπ1(· |u ) = δ0 for almost allu∈ (0, 3/4). Similarly, P(Y∈ · |W = 0, Z = z) = δ0(·)forces π0(· |u ) = δ0 for almost all u∈ (1/4, 1). Since ω is supported on(1/4, 1/2) ⊂ (0, 3/4) ∩ (1/4, 1), bot...

work page 2018
[3]

Z 1 0 1 u∈ U id(x, w) Eobs[Y|w, Z=z u, X]ω(w, X, u) du # dw + Z W EX

The resulting bounds[−M/2, M/ 2]are arbitrarily wider than the sharp value of0. By strictly enforcing the full distributional constraint, the CCOT approach eliminates these mathematically feasible but structurally impossible counterfactuals. Figure 5 illustrates this gap numerically. 64 C Proofs in Section 4 Proof of Theorem 4.2.Let {eπw,x}w∈W,x∈X ∈ Γgen(...

work page 1996
[4]

−γ full,k(X)1(j = 0), where theγfull,k(X)term collects the "full" contribution acting on the entire interval andγj,k(X)collects the additional sub-interval mass. Multiplying by QY,k|X and integrating over each sub-interval[κj,k(X), κj+1,k(X)]in the normalized scale u= (v−p k(X))/(p k+1(X)−p k(X))yields (pk+1(X)−p k(X)) Z 1 0 QY,k|X (u)Q ω,k|X (1−u) du=γ f...

work page
[5]

We first compute the empirical marginal averages over the auxiliary sample:¯p(z) = 1 |I1| P i∈I1bp(z, Xi)and ¯q(z) = 1 |I1| P i∈I1bq(z, Xi)

Level Sets:By Theorem 5.2, the true score levels are separated by at least cgap. We first compute the empirical marginal averages over the auxiliary sample:¯p(z) = 1 |I1| P i∈I1bp(z, Xi)and ¯q(z) = 1 |I1| P i∈I1bq(z, Xi). Let V = {¯p(z)}z∈Z ∪ {¯q(z)}z∈Z. We group the elements inV to disjoint subsetsV1,· · ·, V E such that the radius of each subset is less...

work page
[6]

Similarly, estimate the weighting probabilitiesγj,k(x)and γK(x)by regressing their respective set indicators onX

Instrument Assignment:Estimate πk(x) = Pobs(Z∈S k |X = x)by regressing 1(Z∈ bSk)on X. Similarly, estimate the weighting probabilitiesγj,k(x)and γK(x)by regressing their respective set indicators onX

work page
[7]

Conditional Quantiles:By definition, νj,k(x) = QY,k|x(κj,k(x))is the κj,k(x)-quantile of the conditional distributionµ1,k|x, where µ1,k|x is identified in (3.4). Applying the quantile characterization Pµ1,k|x(Y⩽ν j,k(x)) = κj,k(x) = qj,k(x)−pk(x) pk+1(x)−pk(x) and expanding Pµ1,k|x in terms of the observed conditional distributions yields the following mo...

work page
[8]

PropensityScoresandLevelSets:Estimate p(z, x), computebq(z, x) = ϕ(z, x,bp(z, x)), and construct the level setsbSk, bTj,k exactly as in the continuous outcome procedure (Steps 1–3 in Section D.3)

work page
[9]

Similarly, estimateγfull,k(x), γj,k(x), and γK(x)by regressing their respective set indicators onX

Conditional Joint Probability:Estimate P1,k(x) = E[Y W|Z∈S k, X = x]by regressing Y W on X conditional on Z∈ bSk, using any flexible machine learning method. Similarly, estimateγfull,k(x), γj,k(x), and γK(x)by regressing their respective set indicators onX

work page
[10]

Z p(X) p(X) ∂g1(u, X) ∂u ω(X, u) du # =E Z,X

Rearrangement:In the population, P1,k(X)is monotone non-decreasing in k and the gaps are bounded byP1,k+1(X) −P 1,k(X) ⩽p k+1(X) −p k(X), since Jfull,k(X) = (pk+1(X)−p k(X))θ k(X)withθ k(X)∈[0,1]. However, because each bP1,k is estimated on a disjoint subsample (Z∈ bSk) by a separate regression, the finite-sample estimates need not preserve either constra...

work page 2009
[11]

the integration boundarypk+1 −J full,k, corrected byα(j,k) p,k+1Rp,k+1

work page
[12]

the lower level-set boundaryq0,k =p k whenj= 1, corrected byα (1,k) p,k Rp,k

work page
[13]

92 For channel (a), a perturbationpk+1 →p k+1 + t δpk+1(X)shifts the boundary by δpk+1, giving ∂th− j,k = −D− j,k δpk+1

the alternative policy values qj,k and qj−1,k through the mapping ϕ(z, X, p(z, X)), corrected by the policy terms involvingTj,k andT j−1,k. 92 For channel (a), a perturbationpk+1 →p k+1 + t δpk+1(X)shifts the boundary by δpk+1, giving ∂th− j,k = −D− j,k δpk+1. The correction α(j,k) p,k+1Rp,k+1 = −D− j,k · 1(Z∈Sk+1) πk+1 (W−p k+1) contributes∂ tE[α(j,k) p,...

work page 2018

[1] [1]

Also available as arXiv:2202.10665

URL https://proceedings.mlr.press/v177/guo22a/guo22a.pdf. Also available as arXiv:2202.10665. Peter Hall, Rodney C. L. Wolff, and Qiwei Yao. Methods for estimating a conditional distribution function.Journal of the American Statistical Association, 94(445):154–163, 1999. Sukjin Han and Shenshen Yang. A computational approach to identification of treatment...

work page doi:10.1111/1468-0262 1999

[2] [2]

Similarly, P(Y∈ · |W = 0, Z = z) = δ0(·)forces π0(· |u ) = δ0 for almost all u∈ (1/4, 1)

Because the conditional distribution of the observed outcome is a Dirac measure at zero, P(Y∈ · |W = 1, Z = z) = δ0(·)for each z, the observational constraint forcesπ1(· |u ) = δ0 for almost allu∈ (0, 3/4). Similarly, P(Y∈ · |W = 0, Z = z) = δ0(·)forces π0(· |u ) = δ0 for almost all u∈ (1/4, 1). Since ω is supported on(1/4, 1/2) ⊂ (0, 3/4) ∩ (1/4, 1), bot...

work page 2018

[3] [3]

Z 1 0 1 u∈ U id(x, w) Eobs[Y|w, Z=z u, X]ω(w, X, u) du # dw + Z W EX

The resulting bounds[−M/2, M/ 2]are arbitrarily wider than the sharp value of0. By strictly enforcing the full distributional constraint, the CCOT approach eliminates these mathematically feasible but structurally impossible counterfactuals. Figure 5 illustrates this gap numerically. 64 C Proofs in Section 4 Proof of Theorem 4.2.Let {eπw,x}w∈W,x∈X ∈ Γgen(...

work page 1996

[4] [4]

−γ full,k(X)1(j = 0), where theγfull,k(X)term collects the "full" contribution acting on the entire interval andγj,k(X)collects the additional sub-interval mass. Multiplying by QY,k|X and integrating over each sub-interval[κj,k(X), κj+1,k(X)]in the normalized scale u= (v−p k(X))/(p k+1(X)−p k(X))yields (pk+1(X)−p k(X)) Z 1 0 QY,k|X (u)Q ω,k|X (1−u) du=γ f...

work page

[5] [5]

We first compute the empirical marginal averages over the auxiliary sample:¯p(z) = 1 |I1| P i∈I1bp(z, Xi)and ¯q(z) = 1 |I1| P i∈I1bq(z, Xi)

Level Sets:By Theorem 5.2, the true score levels are separated by at least cgap. We first compute the empirical marginal averages over the auxiliary sample:¯p(z) = 1 |I1| P i∈I1bp(z, Xi)and ¯q(z) = 1 |I1| P i∈I1bq(z, Xi). Let V = {¯p(z)}z∈Z ∪ {¯q(z)}z∈Z. We group the elements inV to disjoint subsetsV1,· · ·, V E such that the radius of each subset is less...

work page

[6] [6]

Similarly, estimate the weighting probabilitiesγj,k(x)and γK(x)by regressing their respective set indicators onX

Instrument Assignment:Estimate πk(x) = Pobs(Z∈S k |X = x)by regressing 1(Z∈ bSk)on X. Similarly, estimate the weighting probabilitiesγj,k(x)and γK(x)by regressing their respective set indicators onX

work page

[7] [7]

Conditional Quantiles:By definition, νj,k(x) = QY,k|x(κj,k(x))is the κj,k(x)-quantile of the conditional distributionµ1,k|x, where µ1,k|x is identified in (3.4). Applying the quantile characterization Pµ1,k|x(Y⩽ν j,k(x)) = κj,k(x) = qj,k(x)−pk(x) pk+1(x)−pk(x) and expanding Pµ1,k|x in terms of the observed conditional distributions yields the following mo...

work page

[8] [8]

PropensityScoresandLevelSets:Estimate p(z, x), computebq(z, x) = ϕ(z, x,bp(z, x)), and construct the level setsbSk, bTj,k exactly as in the continuous outcome procedure (Steps 1–3 in Section D.3)

work page

[9] [9]

Similarly, estimateγfull,k(x), γj,k(x), and γK(x)by regressing their respective set indicators onX

Conditional Joint Probability:Estimate P1,k(x) = E[Y W|Z∈S k, X = x]by regressing Y W on X conditional on Z∈ bSk, using any flexible machine learning method. Similarly, estimateγfull,k(x), γj,k(x), and γK(x)by regressing their respective set indicators onX

work page

[10] [10]

Z p(X) p(X) ∂g1(u, X) ∂u ω(X, u) du # =E Z,X

Rearrangement:In the population, P1,k(X)is monotone non-decreasing in k and the gaps are bounded byP1,k+1(X) −P 1,k(X) ⩽p k+1(X) −p k(X), since Jfull,k(X) = (pk+1(X)−p k(X))θ k(X)withθ k(X)∈[0,1]. However, because each bP1,k is estimated on a disjoint subsample (Z∈ bSk) by a separate regression, the finite-sample estimates need not preserve either constra...

work page 2009

[11] [11]

the integration boundarypk+1 −J full,k, corrected byα(j,k) p,k+1Rp,k+1

work page

[12] [12]

the lower level-set boundaryq0,k =p k whenj= 1, corrected byα (1,k) p,k Rp,k

work page

[13] [13]

92 For channel (a), a perturbationpk+1 →p k+1 + t δpk+1(X)shifts the boundary by δpk+1, giving ∂th− j,k = −D− j,k δpk+1

the alternative policy values qj,k and qj−1,k through the mapping ϕ(z, X, p(z, X)), corrected by the policy terms involvingTj,k andT j−1,k. 92 For channel (a), a perturbationpk+1 →p k+1 + t δpk+1(X)shifts the boundary by δpk+1, giving ∂th− j,k = −D− j,k δpk+1. The correction α(j,k) p,k+1Rp,k+1 = −D− j,k · 1(Z∈Sk+1) πk+1 (W−p k+1) contributes∂ tE[α(j,k) p,...

work page 2018