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Domain-Adapted Diffusion Models for Conditional Independence Testing

Yanfeng Yang, Junda Zhao, Yijie Gao, Jiaqi Yang, Xinyu Shi, Ziqi Chen, Shunyu Zhao, Shuai Li, Wei Huang, Eshant English, Kenji Fukumizu

arXiv:2609.33490v1stat.MLcs.LG

TL;DR

Conditional independence tests based on estimated conditional distributions face Type I error bounds that depend on estimation accuracy, and target-domain data may be limited. The paper proposes DA-Diff, which uses weighted empirical risk across target and source domains, and builds DA-CIT on it. The authors report improved conditional-generation quality over transfer-learning diffusion baselines, with reliable Type I error control and competitive power for DA-CIT.

  • Problem

    CRT validity depends on accurate conditional-distribution estimation, while target-only estimation may be too slow for existing sufficient conditions for asymptotic Type I error control.

    Source: §1 · p. 2

  • Method

    DA-Diff fits a target-oriented conditional score model by weighted empirical score matching over target and source domains, and DA-CIT uses it to generate CRT samples.

    Source: §1 · p. 2

  • Results

    The authors report that DA-Diff improves conditional-generation quality using heterogeneous source domains, while DA-CIT provides reliable Type I error control and competitive power.

    Source: §6 · pp. 11–12

  • Takeaways & Limitations

    Under suitable transferability conditions, informative sources can improve target-domain estimation while the framework controls transfer bias and supports asymptotic Type I error control.

    Source: §1 · p. 2

    DA-Diff and DA-CIT have relatively high computational cost, mainly from source-model pretraining and repeated weight optimization during weighted ERM.

    Source: §6 · p. 12

Abstract

from arXiv · show

Conditional independence (CI) is a fundamental concept in statistics and machine learning. Recent advances in conditional generative modeling provide flexible tools for generative-model-based CI tests, which rely on an estimated conditional distribution to generate randomized samples. However, errors in estimating this distribution accumulate in existing Type I error bounds, and consistency of the generative estimator alone does not guarantee asymptotic Type I error control. To address this limitation, we formulate conditional generative modeling as a domain adaptation problem and leverage auxiliary data from multiple source domains to improve estimation in the target CI testing domain. We propose Domain-Adapted Diffusion (DA-Diff), a multi-source domain adaptation framework for conditional diffusion models based on weighted empirical risk minimization over both target and source domains. We establish the convergence rate of DA-Diff and show how transferable source data can improve target-domain estimation through an increased effective sample size while controlling transfer bias. Building on DA-Diff, we further propose Domain-Adapted Conditional Independence Testing (DA-CIT) and show that its Type I error satisfies $P(p \leq \alpha) \leq \alpha + o(1)$. Experiments demonstrate that DA-Diff improved conditional generation quality compared with transfer-learning diffusion baselines, while DA-CIT provides strong Type I error control and competitive power.

1 INTRODUCTION

Conditional randomization tests depend on accurate estimates of the target conditional distribution, but target-only estimation may be too slow for existing Type I error guarantees. The paper proposes DA-Diff and DA-CIT to use heterogeneous source data while controlling transfer bias and to establish asymptotic Type I error control.

1 INTRODUCTION

  • Conditional independence testing concerns whether variables remain independent conditional on another variable, a concept with applications including variable selection and causal structure learning.The introduction situates CI in statistics and machine learning, including sufficient dimension reduction.Source: §1 · p. 1
  • When the conditional distribution is estimated from data, CRT validity depends critically on estimation accuracy, and existing error bounds include an excess Type I error term.The target-only estimation rates described here may be insufficient to make that excess term vanish.Source: §1 · p. 2
  • Target-only conditional generative estimation may converge too slowly for existing sufficient conditions for asymptotic CRT validity, motivating auxiliary data.Pooling heterogeneous sources naively can introduce transfer bias, so the paper asks how to select sources while controlling distribution-shift bias.Source: §1 · p. 2
  • DA-Diff uses weighted empirical score matching across target and source domains to balance the effective-sample-size benefit of informative sources against transfer bias.The paper establishes a target-domain TV error bound for the learned conditional distribution.Source: §1 · p. 2
  • DA-CIT uses DA-Diff for CRT randomization and establishes asymptotic excess Type I error control under suitable source–target transferability conditions.The paper evaluates conditional-generation accuracy, Type I error, and power in synthetic experiments, and DA-CIT on the multi-source Sachs dataset.Source: §1 · pp. 2–3

2 PRELIMINARY

The preliminary section introduces CRT testing, conditional diffusion modeling, and weighted ERM for adapting a conditional score model across domains. These components underpin DA-CIT's randomized samples and DA-Diff's use of source data while balancing estimation gains against transfer bias.

2.1 CONDITIONAL RANDOMIZATION TESTS (CRT)

  • The CRT compares an observed test statistic with statistics from randomized samples generated using an estimated conditional distribution, yielding a p-value.With the true conditional distribution under the null, the statistics are exchangeable and finite-sample Type I error is controlled; estimation error can add excess Type I error.Source: §2.1 · p. 3

2.2 DIFFUSION MODEL

  • A conditional diffusion model estimates the conditional score and uses it in a reverse-time SDE to generate samples approximating the target conditional distribution.The forward Ornstein–Uhlenbeck process approaches a standard normal marginal for sufficiently large terminal time; reverse sampling starts from that normal distribution.Source: §2.2 · pp. 3–4

2.3 DOMAIN ADAPTATION THROUGH WEIGHTED EMPIRICAL RISK MINIMIZATION (ERM)

  • Weighted ERM combines empirical diffusion losses across target and source domains, with source weights controlling each domain's contribution to training.The domain sample sizes are accounted for separately from the weights.Source: §2.3 · p. 4
  • The source weights are chosen to minimize expected target-domain population loss, balancing additional-sample benefits against bias from source–target discrepancies.Under suitable conditions, informative sources can improve the target-domain convergence rate relative to training only on target data.Source: §2.3 · p. 5

3 THEORETICAL RESULTS

The theoretical results connect weighted multi-domain score matching to target-domain distributional accuracy and asymptotic Type I error control. DA-Diff’s bounds separate estimation error from transfer bias, while DA-CIT requires a stronger growth regime to make excess Type I error vanish.

3 THEORETICAL RESULTS

  • The analysis extends diffusion-model convergence to weighted empirical risk minimization across multiple domains, separately controlling transfer bias and weighted estimation error.It connects DA-Diff’s score-matching risk bound to conditional-distribution approximation and then to CI-test validity.Source: §3 · p. 5

3.1 DOMAIN-ADAPTED DIFFUSION MODEL (DA-DIFF)

  • DA-Diff’s target-domain risk bound has an estimation-error term that decreases with effective sample size and a separate term for source-induced transfer bias.Under the stated conditions, its TV error tends to zero when effective sample size grows and the transfer-bias growth condition holds; a sufficient regime allows source-to-target score discrepancy to decay at an arbitrarily slow polynomial rate.Source: §3.1 · pp. 6–7

3.2 DOMAIN-ADAPTED CONDITIONAL INDEPENDENCE TEST (DA-CIT)

  • Under condition (32), DA-CIT’s excess Type I error vanishes, yielding P(pw ≤ α) ≤ α + o(1).The sufficient regime requires γ > 1 and a stronger upper-growth condition on effective sample size than DA-Diff’s TV consistency regime, because TV error is accumulated n0 times in the Type I error bound.Source: §3.2 · p. 7

4 SYNTHETIC EXPERIMENTS

The synthetic experiments assess DA-Diff’s conditional-generation accuracy and DA-CIT’s Type I error and power. DA-Diff performs strongly against diffusion baselines, while DA-CIT maintains Type I error control with comparable power across conditioning dimensions.

4.1 GAUSSIAN MIXTURE EXAMPLE

  • The Gaussian-mixture evaluation varies the number of source domains K with 500 samples per domain and assesses conditional-distribution quality using average W2 distance.For each of 200 test values, the evaluation compares 1,000 samples from the true and estimated conditional distributions.Source: §4.1 · p. 8
  • DA-Diff has the lowest mean W2 error at K = 2 and 5, and the second-lowest at K = 10 and 20; it also beats Target-only across all four settings.At K = 10 and 20, Finetune performs best; at larger sample sizes, DA-Diff outperforms Target-only, Cat-all, and Finetune for all considered K values.Source: §4.1 · p. 8, Table 1 · p. 8
  • Cat-all performs worse than Target-only for K = 2, 5, and 10, illustrating that pooling heterogeneous sources can produce negative transfer.TGDP is also worse than Target-only for K = 2, 5, and 10; DA-Diff has lower W2 error than DoG and TGDP for every K.Source: §4.1 · p. 8

4.2 POST-NONLINEAR EXAMPLE

  • The post-nonlinear evaluation varies conditioning dimension dz from 5 to 100 and reports Type I error and power over 100 repetitions at significance level α = 0.05.It uses one target and 10 source domains, each with 500 observations.Source: §4.2 · p. 9
  • DA-CIT is compared with CI tests spanning kernel, conditional-distribution, score-matching, robustness, and p-value-calibration approaches.CRT∗ uses external and unlabeled data but assumes a sparse linear conditional-distribution model; the diffusion framework accommodates more general nonlinear and non-Gaussian conditional distributions.Source: §4.2 · p. 9
  • Across conditioning dimensions, DA-CIT’s Type I error is no larger than CDCIT’s and generally stays near nominal while maintaining comparable power.KCIT’s Type I error is inflated in most settings; ECCIT-GCM and CRT∗ also show substantial inflation, while SGMCIT and RBPT have lower power than DA-CIT and CDCIT.Source: §4.2 · p. 9, Figure 1 · p. 9

5 REAL DATA ANALYSIS

The Sachs analysis treats experimental conditions as heterogeneous domains, using cd3cd28 as the target and the other 13 conditions as sources. DA-CIT achieves the highest F1 score among compared methods while maintaining high precision and recall.

5 REAL DATA ANALYSIS

  • The 14 Sachs sub-datasets are treated as separate domains, with cd3cd28 as target and the remaining 13 experimental conditions as sources.Each sub-dataset contains the same 11 variables and 707–927 single-cell observations.Source: §5 · pp. 9–10, Table 2 · p. 10
  • The t-SNE visualization shows target samples overlapping several source domains while other sources have visibly shifted structures.This pattern motivates selectively exploiting source information rather than naively pooling all source observations.Source: §5 · p. 10, Figure 2 · p. 11
  • The Sachs evaluation selects 50 CI and 40 non-CI triples, with one-dimensional X and Y and conditioning dimension Z ranging from 1 to 9.Methods are evaluated using precision, recall, and F1; predictions use majority voting across five random-seed runs.Source: §5 · p. 10
  • DA-CIT achieves the highest F1 score among compared methods while maintaining high precision and high recall.CRT∗ has high precision but substantially lower recall, while CDCIT and RBPT have high recall at the cost of lower precision.Source: §5 · p. 10, Table 3 · p. 11
  • For most competing methods, Cat-all performs worse than Target-only, particularly in recall and F1, whereas DA-CIT selectively uses source data.The comparison evaluates most baselines under both Target-only and Cat-all settings; DA-CIT and CRT∗ directly leverage external sources.Source: §5 · p. 10

6 CONCLUSION

The conclusion presents DA-Diff for weighted multi-source conditional diffusion estimation and DA-CIT for conditional independence testing, with asymptotic control and empirical support, while identifying computational cost as a limitation and future-work priority. It also describes tuning that selects adaptation parameters by predicted first-order effects on target validation loss, favoring domains whose training gradients support descent and flagging potentially conflicting directions as negative transfer.

6 CONCLUSION

  • DA-Diff’s target-domain TV error converges to zero under suitable assumptions, while DA-CIT’s excess Type I error vanishes asymptotically and it maintains competitive power.The conclusion reports that DA-Diff improves conditional distribution estimation by leveraging heterogeneous sources, and that DA-CIT was assessed on simulated and real-world data.Source: §6 · pp. 11–12
  • DA-Diff and DA-CIT have relatively high computational cost from source-model pretraining and repeated weight optimization during weighted ERM training.Improving their computational efficiency is identified as an important direction for future work.Source: §6 · p. 12

A.1 DIFFUSION MODELS FOR TRANSFER LEARNING

  • DA-Diff uses weighted ERM to incorporate multiple source datasets, unlike DoG (Zhong et al., 2025) and TGDP (Ouyang et al., 2024), which mainly transfer from a single source model.It avoids explicit density-ratio estimation; its effective-sample-size term captures gains from transferable source samples, while source weights and the discrepancy matrix characterize transfer bias without requiring a shared representation.Source: §A.1 · p. 16

A.2 DOMAIN ADAPTATION AND WEIGHTED ERM

  • DA-Diff measures source discrepancies through score-function outputs, making its discrepancy measure invariant to parameter reparameterization and independent of aligned parameters or identical network architectures.The related-work discussion situates the approach within diffusion-model transfer learning, domain adaptation, and weighted ERM.Source: A RELATED WORK · p. 16, §A.2 · p. 18

B INTERPRETATION OF THE ASSUMPTION 1 AND 2

  • The transferability assumptions require overlap between target and source Z marginals and sufficient similarity between source and target conditional scores.The score discrepancy parameter γ quantifies transferability: larger γ corresponds to faster discrepancy decay and smaller transfer bias.Source: B INTERPRETATION OF THE ASSUMPTION 1… · pp. 18–19

C OPTIMIZATION OF THE WEIGHTED-ERM WEIGHTS

  • Because the optimal weighted-ERM weights lack a closed-form solution, the method uses a surrogate criterion based on the target score-matching risk bound.It calibrates estimation and transfer-bias terms and selects tuning parameters, including ρ and Creg, from finite candidate sets using gradient alignment.Source: C OPTIMIZATION OF THE WEIGHTED-ERM… · p. 19, §C.1 · p. 20, §C.3 · p. 22

C.2 MAJORIZATION–MINIMIZATION QUADRATIC PROGRAM (MM-QP) UPDATE FOR FIXED HYPERPARAMETERS

  • For fixed hyperparameters, the MM-QP procedure handles the nonconvex weight objective and constraints, with updates that guarantee monotonic descent of the surrogate objective.The method majorizes the objective terms and uses an inner approximation of the regularity constraint that retains the current feasible iterate.Source: §C.2 · pp. 20–22

C.3 SELECTION OF TUNING PARAMETERS BY GRADIENT ALIGNMENT

  • The method jointly selects ρ, Creg, and λ by predicted first-order improvement in target validation loss, rather than by comparing surrogate objective values across candidates.The criterion accounts for changes in scale across candidate ρ and λ values.Source: §C.3 · pp. 22–23
  • Domains with gradients aligned to target-validation descent are favored, while negative alignment signals a potentially conflicting direction and negative transfer.The criterion favors configurations whose MM-QP solutions assign greater weights to domains with beneficial training gradients.Source: §C.3 · p. 24

D.1 GAUSSIAN MIXTURE DATA

The appendices specify synthetic settings for evaluating DA-Diff and DA-CIT, then describe ablation and sensitivity analyses that omit UOWQ after its negative-transfer performance in Table 1.

D.1 GAUSSIAN MIXTURE DATA

  • The Gaussian-mixture benchmark uses ten 5D components, 500 observations per domain, and five near plus five far sources with specified shift strengths.The near-source shifts span 0.1–0.2 and the far-source shifts span 0.2–0.5.Source: §D.1 · pp. 24–25

D.2 POST-NONLINEAR MODEL

  • Post-nonlinear experiments create multiple sources by perturbing covariate distributions and the conditional mechanisms linking Z to X and Y.The setup uses seven near sources with shifts from 0.01 to 0.1 and three far sources with shifts from 0.2 to 0.5.Source: §D.2 · pp. 25–26
  • The alternative-strength parameter h1 controls conditional dependence: h1 = 0 gives conditional independence, and larger values strengthen dependence.Source: §D.2 · p. 25

E ABLATION AND SENSITIVITY STUDIES

E.1 SENSITIVITY ANALYSIS ON GAUSSIAN MIXTURE DATA

DA-Diff is evaluated across sample sizes, source counts, and hyperparameter settings on Gaussian-mixture data. Its reported performance is robust across these variations.

E.1 SENSITIVITY ANALYSIS ON GAUSSIAN MIXTURE DATA

  • The analyses vary target and source sample sizes, the number of source domains, and key DA-Diff hyperparameters.Source: §E.1 · p. 26
  • DA-Diff performs best across considered K values at sample sizes 1,000 and 1,500, and remains among the top-performing methods at 2,000.Source: Table 4 · p. 26, §E.1 · p. 26
  • Across the tested time-bin counts, update intervals, grid resolutions, and warmup epochs, W2 distance varies moderately with largely overlapping error bars.The study characterizes DA-Diff as having limited sensitivity to these hyperparameter choices.Source: §E.1 · pp. 26–27, Figure 4 · p. 27

E.2 SENSITIVITY ANALYSIS ON THE POST-NONLINEAR MODEL

On post-nonlinear data, DA-CIT remains stable across source and sample-size variations, usually controls Type I error at or below nominal, and outperforms comparison tests on calibration or power in most settings.

E.2 SENSITIVITY ANALYSIS ON THE POST-NONLINEAR MODEL

  • DA-CIT remains stable as source count, near–far composition, and target or source sample sizes change.Source: §E.2 · pp. 27–28
  • As h1 increases, DA-CIT’s power increases, consistent with stronger conditional dependence being more readily tested.Source: §E.2 · pp. 27–28
  • Across most settings, DA-CIT has lower Type I error than CDCIT and higher power than CDCIT and SGMCIT.The sensitivity study varies source count and composition, target and source sample sizes, and alternative strength with dz = 20.Source: §E.2 · pp. 27–28, Figure 5 · p. 28
  • Naively concatenating target and source samples generally inflates competitors’ Type I error, while DA-CIT remains substantially better calibrated.Source: §E.2 · p. 28

E.3 EMPIRICAL CUMULATIVE DISTRIBUTION FUNCTIONS OF CI-TEST p-VALUES

The section assesses DA-CIT’s null calibration by comparing its p-value distributions with the uniform distribution. It also notes that algorithms for DA-Diff and DA-CIT are provided.

E.3 EMPIRICAL CUMULATIVE DISTRIBUTION FUNCTIONS OF CI-TEST p-VALUES

  • Across conditioning dimensions, DA-CIT p-value ECDFs closely follow the uniform CDF, and no KS test rejects uniformity at the 0.05 level.The KS tests use 100 null p-values.Source: §E.3 · p. 29, Figure 7 · p. 30
  • Across sensitivity settings, DA-CIT p-value ECDFs remain close to uniform, with no KS test rejecting uniformity at the 0.05 level.Settings vary source-domain counts, far-source counts, and target and source sample sizes.Source: §E.3 · p. 29

F ALGORITHMS OF DA-DIFF AND DA-CIT

G TRAINING BEHAVIOR OF DA-DIFF AND MM-QP

DA-Diff is examined under a Gaussian-mixture setup with periodic updates to its weighted empirical risk and hyperparameters. Its training remains stable as domain weights and selected hyperparameters vary over time.

G TRAINING BEHAVIOR OF DA-DIFF AND MM-QP

H COMPUTATIONAL EFFICIENCY

The section breaks down DA-CIT’s runtime and reports that its computation is substantially higher than CDCIT’s. A Gaussian-mixture figure also reports W2 distances for generated samples.

H COMPUTATIONAL EFFICIENCY

  • DA-CIT’s adaptation and MM-QP optimization account for most runtime; more frequent weight optimization, additional time bins, or a finer λ grid increase MM-QP cost.Source-model pretraining, CRT sampling, and statistic computation are the other identified components.Source: H COMPUTATIONAL EFFICIENCY · p. 34
  • W2 distances in the Gaussian-mixture experiment were 0.040 for Target only, 0.047 for Cat-all, and 0.023 for DA-Diff.Source: Figure 13 · p. 35
  • DA-CIT has substantially higher computational time than CDCIT, mainly from source-specific score-model pretraining and repeated MM-QP optimization.The passage identifies computational efficiency of these steps as future work.Source: H COMPUTATIONAL EFFICIENCY · p. 35

I IMPLEMENTATION DETAILS

This section specifies DA-Diff’s experimental implementation and develops theoretical analyses of its weighted score-matching and excess risk. It establishes bounds and analytical tools for transfer bias, score approximation, effective sample size, and a condition used to bound B.

I IMPLEMENTATION DETAILS

  • DA-Diff uses a fully connected score network with hidden dimension 256 and three hidden layers, and trains source pretraining and adaptation for 500 epochs each at learning rate 10^-3.Adaptation uses a 100-epoch warm-up and five batches per epoch; domain weights are learned across five diffusion-time bins and updated every 10 epochs.Source: I IMPLEMENTATION DETAILS · pp. 35–36

J PROOF OF THEOREMS

  • The theoretical analysis defines Hölder function classes and ReLU neural-network classes to parameterize the score functions.The network input dimension is dx + dz + 1 and its output dimension is dx.Source: J PROOF OF THEOREMS · pp. 37–38

J.1 GLOBAL POPULATION MINIMIZER OF THE WEIGHTED SCORE MATCHING RISK

  • The weighted score-matching risk has a unique global population minimizer almost everywhere on the support of the weighted mixture density.The theorem characterizes the minimizer under the weighted risk; strict convexity establishes uniqueness.Source: §J.1 · pp. 38–40

J.2 EXCESS-RISK DECOMPOSITION

  • The excess score-matching risk separates target-domain estimation error from transfer bias between the weighted-risk minimizer and the target-domain true score.This decomposition provides the two components used to bound the score-matching risk.Source: §J.2 · p. 40

J.3 PROPERTIES OF pw,t

  • Under the stated assumptions, the mixture has a conditional-density structure and score representation, with regularity bounds independent of the number of domains and mixture weights.The proof also establishes a valid conditional-density version and a sub-Gaussian tail for the mixture’s Z marginal.Source: §J.3 · pp. 41, 44

J.5.1 TURN H0 TO Hw

  • Any convergence rate for estimation error under H_w is preserved under H_0.Lemma 2 establishes this change-of-measure result under Assumptions 1 and 2.Source: §J.5.1 · pp. 48–49

J.5.2 DECOMPOSITION OF THE EXPECTED EXCESS RISK ED0,...,DK

  • The expected excess risk is decomposed into A2, B, A1, and C using ghost samples and truncated losses.The proof introduces an independent ghost sample for each domain and truncates the loss to control the tail region.Source: §J.5.2 · pp. 49–50

J.5.3 BOUNDING A1 AND A2

  • Theorem 7 bounds the combined truncation terms |A1| + |A2|, building on uniform score-matching loss and truncation bounds.The supplied passages state these bounds but omit their expressions; the truncation control holds uniformly across component distributions.Source: §J.5.3 · pp. 51, 54, 55

J.5.4 BOUNDING C

  • Under the stated conditional-density and network conditions, a ReLU network approximates the diffused conditional score at rate N^-2β/(dx+dz)(log N)^(β+1).The rate is stated for every t in [t0, T].Source: §J.5.4 · p. 56
  • The weighted-score approximation error is bounded by a smoothness-dependent network approximation term with logarithmic factors, and the Ornstein–Uhlenbeck time integral is O(log N).Source: §J.5.4 · pp. 57–58

J.5.5 BOUNDING B

  • The excess-risk analysis reduces the truncated loss class to a finite cover, controls its empirical process with self-normalized and Bernstein-type bounds, and defines effective sample size to characterize weighted-risk variance.Source: §J.5.5 · pp. 61, 62, 64, 67
  • The derivation defines W_N as the weighted sum of source sample sizes and uses the stated definitions and assumption (26) to establish an equivalent form of condition (381).The excerpt refers to equations (376)–(378), but their expressions are incomplete in the supplied text.Source: §J.5.5 · pp. 67–68
  • Under condition (381), the derivation combines the preceding result with (379) to obtain a further consequence.The consequence’s expression is not present in the supplied excerpt.Source: §J.5.5 · p. 68

J.5.6 PROOF OF THEOREM 1

Theorem 1’s proof combines component bounds to obtain a target-domain risk bound, then selects tuning parameters so remainder terms are absorbed into the leading terms.

J.5.6 PROOF OF THEOREM 1

  • The target-domain risk bound includes an approximation term of order N^-2β/(dx+dz)(log N)^(β+1), plus transfer and remainder terms.The displayed bound also includes MℓρwL_N, δ, and an exponential term.Source: §J.5.6 · pp. 68–69
  • Choosing δ=N^-2β/(dx+dz) and R²=CR log N lets the proof absorb the truncation remainder and δ into the leading terms.CR is chosen sufficiently large.Source: §J.5.6 · p. 69
  • The proof balances the polynomial parts of the approximation and estimation terms before substituting the choice into the bound.Source: §J.5.6 · p. 69

J.6.1 PROOF OF THEOREM 2

The appendix derives a target-domain total-variation convergence rate, uses it to bound DA-CIT’s excess Type I error, and establishes consistency of the estimated source discrepancy matrix under stated conditions.

J.6.1 PROOF OF THEOREM 2

  • Theorem 2’s proof yields a target-domain total-variation convergence rate under the conditions of Lemma 7 and Theorem 1.The corollary states the rate, but the supplied excerpt does not preserve its full expression.Source: §J.6.1 · pp. 70–71

J.7 TYPE I ERROR CONTROL OF DA-CIT

  • DA-CIT’s excess Type I error is bounded by the total-variation error of the estimated target conditional distribution, with the DA-Diff rate substituted into the bound.The CRT samples are independent of the data used to train the conditional diffusion model; source datasets may also be used for that training.Source: §J.7 · p. 71

J.8 ESTIMATION OF SOURCE DISCREPANCY MATRIX G

  • The source-specific score-estimation analysis assumes individual source overlap: the target covariate density is bounded by a constant multiple of each source density.Under this overlap condition and Assumption 2, the appendix derives score-estimation error bounds used to estimate G.Source: §J.8 · p. 71
  • Theorem 9 combines population plug-in and empirical approximation errors to bound estimation error for the source discrepancy matrix G.Under the stated feasible regime and growth condition, the estimation error converges to zero.Source: §J.8 · p. 75

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