Partial identification with entropy regularized optimal transport

Bruno N. Costa; Florian F. Gunsilius

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Partial identification with entropy regularized optimal transport

Bruno N. Costa, Florian F. Gunsilius

arXiv:2609.40156v1Sep 30, 2026econ.EMmath.STstat.MLstat.TH

TL;DR

With continuous instruments, response paths and distribution-matching constraints are both infinite-dimensional, complicating estimation of sharp bounds on partially identified averages. The paper formulates the problem as transport over admissible paths and develops an entropic approximation; it establishes consistency for the sharp bounds and a central limit theorem for fixed approximation settings.

Problem
With continuous instruments, response paths and distribution-matching constraints are both infinite-dimensional, complicating estimation of sharp bounds on partially identified averages.
Source: §2.1 · p. 5, §3.2 · p. 18
Method
The paper represents the identification problem as constrained transport over admissible response paths and makes it finite through discretization, support relaxation, and entropy regularization.
Source: §2.3 · p. 13, §8 · p. 44
Results
The analysis establishes consistency for the original sharp bounds with suitably chosen approximation parameters and a central limit theorem with estimable variance for fixed regularization and discretization.
Source: §8 · p. 44
Takeaways & Limitations
The path formulation preserves identifying information in observable distributions while making structural assumptions explicit.
Source: §8 · p. 44
Multi-marginal Sinkhorn computation still faces a curse of dimensionality in the number of marginals, while discretization and approximate solution introduce bias and nonlinear systems to solve.
Source: §5 · p. 38

Abstract

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In many statistical settings, the available data and maintained assumptions do not suffice to uniquely identify the model parameters of interest. In such cases, one can only identify sets which are guaranteed to contain the true parameters. These are often characterized through linear programs that optimize over models compatible with the observed data. These programs can be infinite-dimensional in the optimizer and the number of constraints. We provide a unified way to characterize and solve such optimization problems by phrasing them as optimal transport problems on path spaces. This allows us to regularize the problem with an entropy penalty, recasting it as a multi-marginal entropic optimal transport problem, which can be solved efficiently via Sinkhorn iterations. In addition, it allows us to establish convergence of the regularized value to the sharpest bound, derive consistency rates for a plug-in estimator, and obtain asymptotic distribution for approximate bounds. The method is general and accommodates settings ranging from instrumental variable models with continuous variables to welfare estimation in heterogeneous demand models. We verify the statistical and computational properties in simulations and provide an application to demand estimation.

1 Contribution and related literature

The paper frames partially identified averages as constrained optimal transport over structural response paths, then develops an entropic computational approximation with convergence and inference guarantees. It positions this method among work on partial identification, transport, and inference while targeting continuous distributional restrictions and functional heterogeneity.

Motivation

Unobserved heterogeneity makes finite representations grow exponentially with observable supports, while continuous variables can require optimization over distributions of entire response functions.The paper develops a method for approximating and estimating sharp bounds that accommodates functional heterogeneity and restrictions across continuous observed conditions.Source: §1 · p. 1

Transport representation

Under the model assumptions, the identification problem is equivalent to constrained optimal transport over admissible response paths, which encode compliance types in binary IV models.The formulation uses distributions over paths to reproduce the full observable distributions.Source: §1 · p. 2

Computation

Deterministic discretization, structural-support relaxation, and relative-entropy regularization yield a multi-marginal entropic transport problem with a smooth dual solvable by generalized Sinkhorn or IPFP iterations.The construction avoids sampling latent response paths and permits joint analysis of computational approximation and statistical estimation.Source: §1 · p. 2

Statistical guarantees

The regularized value converges to the sharp bound, and the method establishes consistency under joint conditions on approximation parameters and sample size, plus a fixed-approximation central limit theorem.For fixed discretization and regularization, sufficiently accurate numerical optimization gives a consistent variance estimator for the regularized population value.Source: §1 · p. 2

Contribution relative to computational work

Unlike methods based on non-convex optimization, the method provides a convex entropic approximation whose discretization and regularization errors can be analyzed jointly with sampling error.It combines structural response-path spaces, continuously indexed distributional constraints, and deterministic approximation with joint statistical and numerical error analysis.Source: §1 · pp. 2–3

Scope and related work

The framework complements Gao et al. (2025) and Tan, Blanchet, and Syrgkanis (2026) by targeting structural response paths and continuous restrictions without relying on finite marginals or a Roy-model reduction.Applications include continuous-treatment and continuous-instrument IV bounds, simulations against exact linear-program benchmarks, and an application to food and leisure budget shares.Source: §1 · p. 3

2 Setup and approach

The section formulates average effects of latent types as a partial-identification problem and characterizes the sharp bound through optimal transport over potential endogenous paths, with conditional exogeneity allowing bounds to be averaged across covariate values. It then regularizes and discretizes the problem for computation and estimation, establishing consistency and asymptotic inference for the resulting estimator.

2.1 Setup

The observed conditional distributions constrain the latent law’s instrument-wise pushforwards, so the mean effect is generally only partially identified and its sharp lower bound is an infinite-dimensional linear program.The upper bound follows by applying the same program to −Λ and reversing the sign of the value.Source: §2.1 · p. 5
The identification problem is equivalently a constrained optimal transport problem, and simplifying that formulation yields a tractable identification and estimation routine.The transport problem minimizes the expected cost of transporting the instrument distribution to the profile of marginal endogenous distributions.Source: §2.1 · p. 5
With covariates, conditional exogeneity Z ⊥ω | W allows the sharp bound to be computed fiber by fiber and averaged over the covariate distribution.The conditional latent law is the unknown component, and each covariate-specific problem uses its conditional data and instrument distribution.Source: §2.1 · pp. 5–7

2.2 Examples

The examples share a two-stage model in which latent types specify complete treatment-response and outcome profiles, while observed data reveal only the response selected by the realized instrument.Restrictions such as continuity, differentiability, or excluding response patterns shrink the feasible set and can tighten bounds without necessarily point-identifying effects.Source: §2.2 · pp. 7–8
In the binary imperfect-compliance example, the sharp lower bound recovers the Balke–Pearl bound (Balke and Pearl, 1994, 1997), while many-to-one paths prevent recovery of the latent type distribution.Excluding outcome-defiers through a weak-benefit restriction shrinks the feasible set but does not generally collapse it to a point.Source: §2.2 · p. 8
The framework also covers continuous-instrument demand, where average elasticities and welfare effects have sharp lower bounds, and covariate applications where bounds are averaged across cells.It also accommodates returns to schooling, imperfect program take-up, and examiner or judge designs under conditional exogeneity.Source: §2.2 · pp. 9–11

2.3 The method in six steps

Reformulating the bound as transport over achievable endogenous paths replaces the unstructured latent space with couplings of observed marginals, whose joint coupling is the remaining unknown.Feasible latent laws and transport plans correspond with equal objective values.Source: §2.3 · pp. 11–12
Discretizing instrument values and paths yields a finite-resolution transport problem that is itself the sharp bound under a correspondingly coarsened mechanism and marginals.The discretization therefore changes model requirements as well as numerical resolution.Source: §2.3 · p. 12
Replacing the costly achievable-support constraint with a penalty for path shortfalls gives a tractable unconstrained problem whose value converges to the constrained value as δ → 0.The penalty also makes the objective Lipschitz in the path, supporting dual analysis and asymptotic distribution results.Source: §2.3 · p. 12
Adding relative entropy to the unconstrained transport objective produces a strictly convex multi-marginal problem with a unique solution, computable by Sinkhorn iterations.As ε → 0, its value decreases to that of the unregularized problem.Source: §2.3 · p. 13
Discretizing the endogenous space makes the entropic problem finite-dimensional, with n×m equations and unknowns, and its value converges to the continuous-space value as m → ∞.The finite system uses the same Sinkhorn iteration as the preceding step.Source: §2.3 · p. 13
Choosing the five tuning controls jointly with sample size yields L1 consistency, while fixing four controls and selecting Sinkhorn iterations appropriately gives asymptotic normality and estimable variance for inference.The variance is the instrument-weighted variance of the dual potentials under their respective marginals; the method supports confidence intervals and hypothesis tests.Source: §2.3 · pp. 13–14

3 From infinite-dimensional linear programs to the multi-marginal entropic optimal transport

The section recasts the partially identified bound as optimal transport on path spaces and progressively uses discretization and entropy regularization to obtain a tractable finite-dimensional formulation. It establishes the formulation’s well-posedness and convergence to the sharp bound, and shows how the finite problem can be approximated by Sinkhorn iterations.

3.1 Constrained Optimal Transport Bound

For a non-injective mechanism, the path cost is the worst effect among types generating that path, so minimizing expected path cost recovers the lower-bound logic.At an optimum, mass on a path can be assigned to its least favorable compatible type without changing feasibility.Source: §3.1 · pp. 15–16
The lower bound equals a constrained transport problem over achievable paths, and the mechanism and co-mechanism map feasible and optimal solutions between path and latent spaces.This replaces optimization over unobserved latent laws with optimization over couplings of the observed marginal profile.Source: §3.1 · p. 17

3.2 Discretization

Continuous instruments make both the path space and marginal constraints infinite-dimensional; discretization projects paths to piecewise-constant functions and averages conditional marginals within instrument cells.The resolution n controls the number of instrument values represented.Source: §3.2 · pp. 18–20
Under Assumptions 1 and 2, the discretized lower-bound and transport problems share a value, and both values converge to their original bounds as n increases.Assumption 2 requires every feasible latent distribution to be weakly approximable by distributions feasible for the discretized problems.Source: §3.2 · p. 21

3.3 Relaxing the support constraint

After instrument discretization, the transport plan must still be supported on achievable paths, a shape restriction that the next step addresses.For the discretized problem, the required support is the projected achievable set.Source: §3 · p. 14, §3.3 · p. 21
Penalizing paths by their distance from the achievable set recasts the problem as multi-marginal transport; its value approaches the original bound as δ shrinks and n grows.The relaxed value is a tractable proxy, not itself a partial-identification bound; the mass assigned to unachievable paths vanishes as δ ↓ 0.Source: §3.3 · pp. 22–23

3.4 Entropic regularization

Relative-entropy perturbation makes the problem strictly convex with a unique solution; strong duality and a smooth dual yield a fixed-point condition solvable numerically and efficiently by Sinkhorn iteration.With δ and n fixed, uϵ,δ,n(P, λ) ↓ uδ,n(P, λ) as ϵ ↓ 0; jointly sending δ ↓ 0 and ϵ/δ → 0 with n fixed gives uϵ,δ,n(P, λ) → un(P, λ).Source: §3.4 · pp. 24, 26

3.5 Discretizing the endogenous space

Discretizing the endogenous space onto m representative points makes the problem finite-dimensional, reduces noisy plug-in estimation, and yields at most n · m dual equations.The discretized value converges to the full-support value at a rate controlled by mesh size ρm and the cost’s δ−1 Lipschitz constant.Source: §3.5 · pp. 28–29
Under Assumptions 1–2, the discretized entropic transport problem is well-posed, satisfies strong duality, and converges to the continuous-grid value as the partition mesh vanishes.The mesh-convergence result holds for fixed ε, δ, and n.Source: §3.5 · p. 29
The discretized problem reduces to solving at most n·m equations for the dual potentials, which recover the program value and optimal coupling in closed form.The potentials can be represented by an n × m matrix, with the objective determined by the discretized marginals, weights, and grid costs.Source: §3.5 · pp. 29–30
Sinkhorn iterations combine row-wise block updates with normalization, monotonically increase the dual objective, and converge to the program value at a geometric rate.An iterate with sufficiently many iterations approximates the value within any desired tolerance without exactly solving the nonlinear system.Source: §3.5 · p. 31
Replacing population marginals and weights with their sample analogs yields a consistent estimator that is asymptotically normal when its parameters are held fixed.The estimator controls the errors introduced by the successive approximations.Source: §3.5 · p. 31

4 Estimation and inference

Section 4 turns the population lower bound into a sample-based plug-in estimator, establishing consistency under jointly chosen tuning parameters and an asymptotic normal result when regularization parameters are fixed.

4 Estimation and inference

The section extends population-level bound results to a sample-based plug-in estimator without imposing parametric structure on the observed distributions.The estimator is consistent when its regularization parameters are suitably selected; with fixed parameters, it has a central limit theorem and identifiable asymptotic variance.Source: §4 · p. 32, §4.1 · p. 32

4.1 Data and Empirical Marginals

Instrument-cell weights are estimated by sample shares, while conditional endogenous-cell probabilities are estimated by within-cell frequencies.These empirical estimates have consistency, joint asymptotic normality at rate √N, and an L1-error bound.Source: §4.1 · p. 32

4.2 Lower Bound Estimation and Sinkhorn Algorithm

The estimator uses empirical marginals and I Sinkhorn iterations to approximate the lower-bound objective without solving its nonlinear system exactly.The objective-value approximation has an explicit, controllable error, and the iterations converge regardless of the sample realization.Source: §4.2 · p. 33

4.3 Consistency

With compatible sample-size-dependent tuning rates, the estimator is L1-consistent as discretization, entropy smoothing, localization, and finite-iteration errors jointly vanish.The theorem requires the approximation parameters to be selected at compatible relative rates, rather than requiring each to vanish independently.Source: §4.3 · pp. 34–35

4.4 Asymptotic distribution

With fixed (ϵ, δ, n, m) and Sinkhorn iterations growing faster than ln N, the estimator is asymptotically normal with plug-in-estimable variance.The limiting distribution supports confidence intervals and hypothesis tests for the regularized value; the distribution when the regularization parameters vary with N is left for future research.Source: §4.4 · pp. 35–36

5 Applications, Limitations, and Future Directions

The section develops inference for the partially identified bound and examines computational limits of its discretized transport approximation. It outlines practical strategies and structural directions for managing those limits.

Confidence intervals for the partially identified bound

Theorem 3's Wald inference targets the regularized value; a known κ(β) bias bound widens its interval into an asymptotically valid interval for v(P, λ).The bias-rate bound need not be optimal, though a sharper rate yields a tighter interval.Source: §5 · p. 37
Smaller ϵ and δ tend to increase asymptotic variance, so regularization choice balances deterministic bias against sampling variance.These parameters control smoothness of the entropic dual objective and stability of its maximizer.Source: §5 · p. 37

Hypothesis testing

The test rejects H0 when the estimated bound is at or below v_N and is asymptotically valid at level τ.Source: §5 · p. 37

The curse of dimensionality

Discretized paths can grow exponentially in instrument cells, and Sinkhorn iteration complexity also suffers a curse of dimensionality in marginal count.Source: §5 · p. 38
Mechanism-achievable paths and pruning high-cost or unlikely paths can shrink the active problem dimension without changing the underlying discretization.Source: §5 · p. 38
Finer endogenous partitions shrink mesh bias but enlarge the grid, while a precise bias rate can help select a coarser, manageable discretization.Source: §5 · p. 38

Future directions

Future work proposes exploiting shape restrictions, separability, or low-rank structure, and deriving sharper bias rates and joint-limit results.Source: §5 · pp. 38–39

The curse of dimensionality

Fixing regularization and discretization while increasing iterations sidesteps exponential iteration growth but leaves discretization bias uncontrolled; slowly growing n instead requires many observations per instrument cell.Source: §5 · p. 39

6 Simulations

The simulations examine approximation accuracy, inference, and computational cost across binary and continuous settings and a five-value instrument. Results show close agreement with sharp LP benchmarks in selected settings, with computational burden rising for the larger instrument.

Simulation designs

The simulation suite proceeds from a binary benchmark to continuous treatment and outcomes with a binary instrument, then to continuous treatment and outcomes with a multi-valued instrument.The binary-instrument continuous setting is used to assess statistical properties.Source: §6 · pp. 39–40

Binary benchmark

At (δ, ϵ) = (0.25, 0.0002), the binary benchmark’s maximum absolute difference from the sharp LP is 1.48 × 10−4.The benchmark admits all sixteen binary response types.Source: §6 · p. 39, Table 1 · p. 39
At δ = 0.25, removing entropy gives sharp bounds to full numerical precision, while reducing entropy lowers error approximately proportionally under sufficiently strong support penalties.A substantial error persists at δ = 0.75.Source: §6 · p. 40, Figure 10 · p. 40

Continuous treatment and outcome with binary instrument

Across N = 500, 2,000, and 10,000, individual 95% Wald interval coverage ranges from 94.4% to 95.8%, and estimated SE/SD ranges from 0.949 to 1.031.The intervals use the fixed-regularization central limit theorem and dual-potential standard errors.Source: §6 · p. 40, Table 2 · p. 41

Five-value instrument

With outcome monotonicity, the twelve-bin outer interval width falls by 44.7%, and its upper endpoint moves from 0.5045 to −0.1377.The strict separation from zero uses both the data and the shape restriction; comparison with analytical sharp continuous-population bounds shows remaining approximation error.Source: §6 · p. 41, Table 3 · p. 42, Figure 11 · p. 42
The complete five-value comparison across costs, two resolutions, two outcome specifications, and three entropy levels takes approximately three minutes on the reported MacBook Pro.It uses full cost tensors of up to 227,136 entries per sign.Source: §6 · pp. 41–42

7 Application to Demand Estimation

The application uses UK household expenditure data to bound how food and leisure budget-share distributions change across expenditure levels. Bounds become informative about food as a necessity and leisure as a luxury under second-stage monotonicity restrictions.

Data

The estimation sample has 1,320 households after 330 households are used to calibrate income cutoffs; normalized log-income is divided into three terciles as the instrument.The outcome is food or leisure expenditure share, and the treatment is normalized log total expenditure.Source: §7 · pp. 42–43, Figure 12 · p. 43, Table 4 · p. 43

Target and interpretation

For each good, the target is the population-averaged distributional contrast in whether household expenditure shares cross a fixed threshold q as expenditure moves from 0.25 to 0.75.Source: §7 · p. 43
A positive contrast means the higher-expenditure intervention raises the probability of a share at or below the threshold, corresponding to necessity; a negative contrast corresponds to luxury.The baseline assumes both latent stages are 1-Lipschitz without monotonicity; the secondary specification imposes nonincreasing food shares and nondecreasing leisure shares.Source: §7 · pp. 43–44

Inference

The confidence intervals hold the income partition, observation grid, structural restrictions, distance penalty δ = 0.025, and positive entropy parameter fixed.Each interval concerns its own fixed-entropy endpoint and uses the normal critical value 1.96.Source: §7 · p. 44

Results

With second-stage monotonicity, the bounds suggest that food is a necessity and leisure a luxury, even when each bound’s 95% confidence interval is considered separately.Without that restriction, the bounds contain zero and are largely uninformative about necessity versus luxury.Source: §7 · p. 44, Table 5 · p. 44

8 Conclusion

The paper formulates sharp bounds on partially identified averages as constrained optimal transport over admissible response paths. It makes the formulation computational and establishes consistency and fixed-regularization inference, while identifying growing discretization demands and varying-regularization inference as open challenges.

Contributions

Under the paper’s regularity conditions, sharp bounds on partially identified averages are equivalent to constrained optimal transport over admissible response paths.The path formulation preserves identifying information in observable distributions and makes structural assumptions explicit.Source: §8 · p. 44

Computation and inference

Deterministic discretization, support relaxation, and relative entropy regularization yield problems solvable by generalized Sinkhorn iterations.The analysis establishes consistency for sharp bounds under suitable approximation parameters and a central limit theorem with estimable asymptotic variance for fixed regularization and discretization.Source: §8 · p. 44

Open directions

Computation grows more demanding as the number of discretizations in Z increases, and inference allowing discretization and regularization to vary with sample size remains open.The paper points to exploiting transport-cost and path structure, low-rank tensor approximations, or Markov structures as possible computational directions.Source: §8 · pp. 44–45

A Auxiliary Mathematical Results

The appendix assembles mathematical objects used throughout the paper, beginning with discretization projections and cost constructions and then defining marginal operators and constraint sets.

Appendix overview

The appendix organizes supporting objects into discretization projections, cost functions and associated correspondences, and marginal operators with their constraint sets.It follows the notation and terminology of the main text.Source: A Auxiliary Mathematical Results · p. 53

Discretization projections

The product space X^n is metrized so it is a compact Polish space, with the metric chosen so π_n acts as an isometry.The construction fixes the cell probabilities of the partition A_n.Source: A Auxiliary Mathematical Results · p. 53

Path space

The path space is L1(Z, λ; X), the Polish space of functions taking values in X almost surely, with the restricted Bochner-space norm.The appendix introduces remaining objects in its three subsections.Source: A Auxiliary Mathematical Results · p. 53

A.1 Projections and Discretizations

A refining projection maps paths to finite-resolution, piecewise-constant approximations while preserving the structure needed for finite-dimensional transport. Its compact range, isometry, refinement compatibility, and uniform convergence support well-posed discretization and recovery of the original paths.

A.1 Projections and Discretizations

The projection πn simultaneously reduces paths, exogenous variables, and marginal constraints to a common finite resolution n.Its compact range makes the discretized transport problem well posed, and its isometry with X n yields a finite-dimensional program.Source: §A.1 · p. 53
Nested partitions with vanishing mesh ensure piecewise-constant approximations track local path behavior rather than only the number of cells.Each cell has positive measure, and every cell in one partition is a union of cells in the finer partition.Source: §A.1 · p. 53
On H, πn is idempotent with a convex compact range, and ˜πn maps that range isometrically onto X n.The projections are compatible across refinements, and πn converges uniformly to the identity over achievable paths.Source: §A.1 · p. 54

A.2 Cost Functions and Correspondences

The appendices establish regularity for the costs, constraints, and entropic transport value used in the paper. These properties include convergence under discretization, compact feasible sets, stability to marginal perturbations, and differentiability under finite-support conditions.

A.2 Cost Functions and Correspondences

The latent-type fibers define the original and discretized costs, which are bounded and lower semicontinuous, with Ψn increasing to Ψ on achievable paths.The moment-penalized cost is bounded and δ−1-Lipschitz, and increases to Ψn as δ decreases for paths whose projection is achievable.Source: §A.2 · pp. 54–55
A log-sum-exp smoothing replaces minimization over the latent space in the practical regularized cost construction.Under the stated small-ball and Lipschitz conditions, this regularized cost converges to the original as ζ tends to zero.Source: §A.2 · pp. 55–56

A.3 Marginal Operators and Constraint Sets

The marginal operators preserve positivity and constants, and their resulting feasible-measure sets are convex and compact, ensuring transport infima are attained.Under Assumptions 1 and 2, the discretized feasible sets converge to the original feasible set in the sense that the original sets are exactly weak limit points.Source: §A.3 · pp. 57–58

A.4 Properties of the Entropic Optimal Transport Problem

For Lipschitz costs, entropic transport values are stable to marginal perturbations; with finite common support, the value is differentiable and the normalized dual optimum is unique.The stability result is used to establish convergence of the entropic transport value with estimated marginals to the value with true marginals.Source: §A.4 · pp. 59–60

A.5 Sample Properties of the Discretized Marginals

The discretized marginal and cell-weight estimates are constructed from empirical joint-cell counts. They are consistent and asymptotically normal, with asymptotic independence and small L1 error under the stated sample-size condition.

A.5 Sample Properties of the Discretized Marginals

The empirical average of indicator matrices estimates the joint probabilities of the discretized cells.Each observation marks exactly one joint cell, and row totals count observations in each exogenous cell.Source: §A.5 · p. 61
The marginal-probability and cell-weight estimates are consistent and jointly asymptotically normal, and are pairwise asymptotically independent.The estimates are well-defined when every cell receives observations; the probability of any empty cell is bounded by ne^(−Nλn).Source: §A.5 · p. 61
When Nλn > 1, the marginal and cell-weight estimates have small L1 error.Source: §A.5 · p. 61

B.1 Proofs of Section 3.1

The proofs connect the latent-law and path-distribution formulations and establish equality of their optimal values. For the discretized problem, the proof also reduces the optimization to projected path distributions and matches the latent and path program values.

B.1 Proofs of Section 3.1

The latent-law and path-distribution programs have equal values, and pushforward maps carry optimal solutions between their feasible formulations.The proof uses a measurable selection of minimizing latent types and the corresponding right-inverse mapping.Source: §B.1 · pp. 62–63

B.2 Proofs of Section 3.2

The discretized transport value depends only on projected feasible path distributions because its cost is invariant to πn and projection preserves feasibility.Source: §B.2 · p. 63
A minimizing latent selection for each projected path is a right-inverse of the discretized mechanism, yielding equality between the discretized latent and path program values.Source: §B.2 · pp. 63–64

B.3 Proofs of Section 3.3

The proof establishes attainment of the regularized optimization and shows that its value converges to the unregularized bound as the tolerance vanishes, including in the joint limit.

B.3 Proofs of Section 3.3

The projected path-space optimization reduces to the corresponding projected feasible-set problem through the stated projection and cost correspondence.Taking the infimum after applying the projection establishes the first claim.Source: §B.3 · p. 64
The infimum defining uδ,n(P, λ) is attained because the projected feasible set is compact and the cost is bounded and Lipschitz.The cost and constraints depend on a path only through its projection.Source: §B.3 · pp. 64–65
An optimal feasible measure yields a latent law whose projected model distribution lies within Wasserstein distance δ of the projected measure.The construction pushes the optimizer through a measurable selection of minimizing latent types.Source: §B.3 · p. 65
As δ ↓0, uδ,n(P, λ) increases to un(P, λ); the joint-limit argument also establishes convergence to the limiting bound.The proof extracts weakly convergent subsequences and uses feasibility and continuity to force equality at the limit.Source: §B.3 · pp. 65–66

B.4 Proofs of Section 3.4

The proof derives the entropic transport dual and its optimizer properties, then shows that removing entropy and approximation penalties recovers the unregularized and limiting values.

B.4 Proofs of Section 3.4

The cost is bounded, and the dual potentials are δ^-1-Lipschitz with sup-norm bounds controlled by the cost bound and marginal weights.The cost bound is 1 + 2δ^-1 diam(X).Source: §B.4 · pp. 66–67
As ϵ ↓0, uϵ,δ,n(P, λ) decreases to uδ,n(P, λ), and further limits recover un(P, λ) and u(P, λ).The stated upper envelope is uδ,n(P, λ) + d ϵ log ϵ + ϵ/δ.Source: §B.4 · p. 66
The entropic dual has an attained maximizer, unique up to additive normalization, and its associated primal optimizer is unique.Uniqueness of the primal optimizer follows from strict convexity of the minimization.Source: §B.4 · p. 66

B.5 Proofs of Section 3.5

These proofs establish convergence and statistical properties of finite-resolution entropic transport estimators, while also supplying the projection and penalty results used to justify the approximations.

B.5 Proofs of Section 3.5

As the cell radius ρm shrinks, the finite-support transport values converge; jointly, the entropic discretized values converge to v(P, λ).The joint limit requires ρm/δ →0, ϵ/δ →0, and n →∞.Source: §B.5 · p. 67
The Sinkhorn iterates correspond term by term to Carlier’s block-coordinate-descent iterates under the stated rescaling.The correspondence uses Carlier (2022, Eqs. (2.7)–(2.14)) and starts from φ0 = 0.Source: §B.5 · pp. 67–68

B.6 Proofs of Section 4.3

The plug-in estimator is L1-consistent for the partially identified bound when the Sinkhorn and empirical-marginal errors vanish and the population approximation converges.The proof decomposes the error into Sinkhorn error, empirical-marginal error, and population approximation error.Source: §B.6 · p. 68

B.7 Proofs of Section 4.4

The asymptotic variance is expressed as a weighted population variance involving the optimal dual potential.The proof obtains this form by combining the block structure of the covariance matrix with the value-function gradient.Source: §B.7 · p. 69

B.8.1 Proofs of Section A.1

The projection operators are contractions, and the finite-partition projections converge uniformly to the identity on the compact set K.The proof establishes the operator-norm bound and then uses compactness to obtain uniform convergence.Source: §B.8.1 · pp. 69–71
The projected path space is isometrically represented by X^n through piecewise-constant paths on the partition cells.The map from cell values in X^n to paths is a two-sided inverse of the projection on its range.Source: §B.8.1 · p. 70

B.8.2 Proofs of Section A.2

The penalized path cost is bounded and attainable, and its minimizing correspondence is nonempty and compact-valued.The bound is 1 + 2δ^-1 diam(X).Source: §B.8.2 · p. 72
For fixed n, Ψδ,n increases to Ψn as δ ↓0, with limiting optimizers feasible for the unregularized cost.The proof uses compactness to extract a limiting optimizer and show its constraint discrepancy vanishes.Source: §B.8.2 · pp. 72–73

B.8.3 Proofs of Section A.3

The proof establishes that the operators preserve probability measures and that the feasible sets are compact and closed, with limiting feasible laws characterized by finite-resolution sets.

B.8.3 Proofs of Section A.3

The adjoint of a positive, continuous, linear operator that preserves constants maps probability measures to probability measures.The proof establishes that the adjoint measure is positive and has total mass one.Source: §B.8.3 · p. 74
The feasible sets are convex and closed as preimages under continuous affine maps, with projected sets handled by expressing constraints directly in terms of projected measures.The projected constraints depend on μ only through πn#μ.Source: §B.8.3 · p. 74
The relevant feasible-measure sets are compact because they are closed and supported on compact spaces.The proof invokes equitightness and Prokhorov’s theorem.Source: §B.8.3 · p. 74
The limiting latent-law and path-measure feasible sets equal intersections of their finite-resolution counterparts.The proof establishes both the limiting characterization and the converse approximation using Assumption 2.Source: §B.8.3 · p. 75

B.8.4 Proofs of Section A.4

The proofs establish coupling-based stability for entropic values, well-posed finite-support dual optimization, and consistency and asymptotic normality for plug-in quantities.

B.8.4 Proofs of Section A.4

A shadow coupling transfers an optimizer to perturbed marginals; the shared kernel and entropy data processing control relative entropy, while the reverse inequality completes the claimed identity.The coordinatewise couplings are chosen to be optimal for the W1 distances.Source: §B.8.4 · pp. 75–76
With finitely supported marginals, the dual reduces to a finite-dimensional objective whose maximum exists and is unique on the specified feasible set.The proof establishes existence by compactness and continuity, then strict concavity for uniqueness.Source: §B.8.4 · pp. 76–77
A bounded dual optimizer can be chosen by setting inactive entries to zero and adjusting shifts; Danskin’s theorem gives directional differentiability, and uniqueness yields convergence of near-optimizers.The proof bounds the optimizer and shows that every convergent subsequence of near-optimizers has the unique optimizer as its limit.Source: §B.8.4 · p. 77

B.8.5 Proofs of Section A.5

The plug-in empirical cell probabilities and weights converge almost surely to their population values as sample size grows.The proof uses the strong law and shows the probability of an empty cell vanishes.Source: §B.8.5 · p. 78
The delta method yields a multivariate normal limit for the scaled plug-in quantities, with covariance matrix Σ as stated.The proof applies the central limit theorem to empirical cell counts and differentiates the map Γ.Source: §B.8.5 · p. 78