{"id":"170e6c16-e5cd-4ccf-90d7-196a66d499e7","arxiv_id":"2412.11744","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Conditional diffusion models can replace GANs in the conditional randomization test, giving asymptotic type I error control and strong power on synthetic and real benchmarks.","lead":"This paper combines conditional diffusion models with the conditional randomization test to decide whether two variables stay dependent after accounting for a third set. The result is a conditional independence test that controls false positives in high-dimensional and mixed continuous-discrete settings, with code available.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's proof bounds the continuous-time reverse SDE, not the Euler-Maruyama sampler in Algorithm 2, so Corollary 1's α+o(1) guarantee is not established for the implemented test.","rationale":"The reader identified exactly the same load-bearing concern: the imported convergence guarantee from Fu et al. (2024) covers the continuous-time reverse process, while Algorithm 2 uses a finite-step Euler-Maruyama discretization, and the paper provides no bound on the discretization error. This concern lands because the proof of Theorem 1 in the supplementary material explicitly uses Proposition 4.5 of Fu et al. to bound dTV(p, bp) with terms involving tmin, T, and score estimation error R(bs), but no term involving the step size Δt = (T - tmin)/K. Thus the theorem statement, which defines bp as the density of the output of Algorithm 2, is not actually proven. Since Corollary 1 applies Theorem 1 to obtain the α + o(1) type I error guarantee, the central claim is not established for the implemented algorithm. The paper does have real strengths: it provides a coherent CRT framework, a classifier-based CMI statistic, extensive synthetic experiments, a public code repository, and an analysis of the score-matching error inherited from Fu et al. The empirical evidence suggests the discretization may be benign in the tested settings, but the theoretical guarantee is the paper's main advertised contribution, and the missing discretization bound is a correctness risk rather than a matter of consensus. I considered other possible concerns, such as the consistency of the CMI estimator or the real-data evaluation design, but those are either addressed by cited results or less directly tied to the central claim. Therefore the appropriate verdict remains CONDITIONAL: the method is promising and empirically supported, but the asymptotic validity theorem must be reconciled with the actual sampling algorithm before the claim is accepted at face value.","tokens_in":21794,"tokens_out":7133,"duration_ms":66801,"concrete_test":"Derive a total-variation bound for the Euler-Maruyama discretization of the reverse SDE in Eq. (3) under Assumptions 1-2, and add it to the RHS of Theorem 1. Use standard SDE weak-error estimates with the score Lipschitz/smoothness conditions implied by Assumptions 1-2. Then check whether the combined bound is o(1) under the Corollary 1 condition n·N^{-Γ1(k,α)}(log N)^{Γ2(k,α)} = o(1), with K either fixed at 1000 or chosen as K(N). If the discretization term is O(Δt^{1/2}) with Δt ≈ 0.01 and is not o(1), Theorem 1 fails for Algorithm 2 and the type I error guarantee is invalid; if the term is o(1) for a defensible choice of K(N), the concern is resolved.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Algorithm 2 (steps 5-6) is a K-step Euler-Maruyama discretization of the reverse SDE in Eq. (3). Theorem 1's proof in Supplementary S2 invokes Proposition 4.5 of Fu et al. (2024) to bound dTV(p(·|Z), bp(·|Z)) by terms involving sqrt(tmin), exp(-T), and the score-estimation error R(bs). None of these terms accounts for the discretization error introduced by replacing the continuous-time reverse process with finite-step Euler updates. As stated, Theorem 1 claims the bound holds for the density of bX sampled from Algorithm 2, but the proof only covers the continuous-time process. Since Corollary 1 rests on Theorem 1, the central claim P(p ≤ α | H0) ≤ α + o(1) is not proven for the algorithm as actually implemented. With the paper's settings (T=10, tmin=0.01, K=1000), Δt ≈ 0.01; without a discretization bound, the TV gap could be non-negligible. This is the most load-bearing gap because the paper's headline contribution is the asymptotic type I error control of Algorithm 3, and the implemented sampler is exactly the Euler-Maruyama scheme in Algorithm 2.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes CDCIT, a conditional randomization test in which the conditional distribution X|Z is learned with a score-based conditional diffusion model trained on unlabeled data. Pseudo-replicates of X are generated from the learned conditional distribution by a K-step Euler-Maruyama discretization of the reverse SDE, the test statistic is a classifier-based conditional mutual information estimator, and the p-value is computed by the CRT formula. The main theoretical claims are Theorem 1, a total-variation bound between the true and generated conditional distributions; Theorem 2, an excess type I error bound in terms of that total-variation distance; and Corollary 1, stating asymptotic type I error control for Algorithm 3. The paper also reports quantile-based comparisons of generated samples, synthetic experiments against seven baselines, and two real-data studies.","tokens_in":22101,"tokens_out":9739,"duration_ms":98543,"significance":"If the stated guarantees hold, this is a practically valuable combination: conditional diffusion models provide a more stable and higher-quality sampler for the CRT than the GAN-based and nearest-neighbor alternatives, and the classifier-based CMI statistic is well suited to high-dimensional and mixed-type conditioning sets. Theorem 2's exchangeability argument is clean, the empirical comparison is broad and includes public code, and the computational cost is reported. The main caveat is that the headline asymptotic guarantee is currently a theorem about a continuous-time process rather than the implemented Euler-Maruyama sampler, and the mixed-type setting is outside the stated assumptions; these gaps affect the central claim as stated.","major_comments":[{"comment":"Theorem 1 is stated for the density bp(·|Z) of bX sampled from Algorithm 2, but the proof in Supplementary S2 bounds the continuous-time reverse SDE in Eq. (3), not the K-step Euler-Maruyama recursion in Algorithm 2 (lines 3-7). The three terms in the bound, C2 sqrt(tmin)(log(1/tmin))^{(dx+1)/2}, C3 exp(-T), and C4 sqrt(T R(bs)), come from Proposition 4.5 of Fu et al. (2024) and contain no discretization error term. With the implementation settings T=10, tmin=0.01, K=1000, the step size is Delta t = 0.00999, and the gap between the continuous-time process and the discretized chain is not controlled. Moreover, Theorem 1 specifies tmin = N^{-4Γ1(k,α)-1} and T = 2Γ1(k,α) log N, whereas Algorithm 2 uses fixed values of tmin and T, so the theorem's scaling is not matched by the implemented algorithm. Since Corollary 1 rests on Theorem 1, the claim P(p ≤ α | H0) ≤ α + o(1) is not currently established for Algorithm 3 as implemented. The authors should either add a discretization error bound to Theorem 1 or state Theorem 1 for the continuous-time sampler and clearly separate the guarantee for the implemented algorithm.","section":"Theorem 1 and Algorithm 2"},{"comment":"Theorem 2 states P(p ≤ α | Y, Z) ≤ α + dTV(p(n)(·|Z), bp(n)(·|Z)). In Algorithm 3, bp is random because it is determined by the trained score network bs, which depends on the unlabeled data DU. As written, the left-hand side conditions only on Y and Z while the right-hand side is a random variable, so the inequality is not on a common conditioning. The proof works if the statement is read conditional on DU, or equivalently on bs: then \\dot X and cX(b) are i.i.d. draws from bp(n) and the exchangeability argument is valid; taking expectations afterward gives the unconditional bound. Please restate Theorem 2 with this conditioning made explicit.","section":"Theorem 2 and Algorithm 3"},{"comment":"The paper repeatedly claims that the method handles conditioning sets that mix continuous and discrete variables (Abstract, Model M3, Scenario II, and the real-data clinical covariates). However, Assumption 2 in Supplementary S1 assumes that pZ has a density with respect to Lebesgue measure and is sub-Gaussian, which fails when any coordinate of Z is discrete. Consequently Theorem 1 and Corollary 1 do not cover the mixed-type simulations and real-data analyses. The empirical results may stand on their own, but the theoretical type I error guarantee should be either restricted to continuous Z or extended to distributions with atomic components.","section":"Supplementary S1, Assumption 2 and mixed-type claims"}],"minor_comments":[{"comment":"The quantities k and α appearing in Γ1(k,α) and Γ2(k,α) are defined only in the Supplementary Material; they should be defined in the main text where Theorem 1 is stated.","section":"Theoretical Guarantee for Sampling Quality"},{"comment":"There are typos, for example 'Moerever' in the Introduction; a careful proofreading pass is needed.","section":"Introduction"},{"comment":"In Eq. (2), the coefficient of ϵ should read sqrt(1 - exp(-t)); the current typesetting with 'p 1 - exp(-t)' is easy to misread.","section":"Methodology for Sampling"},{"comment":"The caption says the smallest MSEs and SDs are bolded, but this is not consistently implemented across all rows (for example, the M2 0.05 row); the caption should also state whether boldface indicates best MSE, best SD, or both.","section":"Table 1"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the paper is a reasonable application of conditional diffusion models to the conditional randomization test, but the theoretical novelty is limited: Theorem 1 is imported from Fu et al. (2024) and the CMI estimator is from Mukherjee et al. (2020), with 1-NN sampling from Li et al. (2024). The new contribution is the combination, the explicit TV-based type I error transfer, and the empirical study. The discretization gap in the central guarantee is the main obstacle; if the authors can close it, the paper may meet the bar for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something genuinely useful: it replaces the GAN or kNN sampler inside the conditional randomization test with a conditional diffusion model, and pairs that with the classifier-based CMI estimator. The combination is new, the synthetic experiments are thorough and fairly compared against seven baselines, and the code is public. The theoretical core, Theorem 2, is a clean standard argument: excess type I error is bounded by the TV distance between the true and estimated conditional distributions. That part holds up. The paper also honestly attributes the score approximation bound to Fu et al. (2024). So the contribution is incremental in architecture but solid in execution.\n\nThe soft spot is exactly the one the stress-test note flags. Theorem 1 bounds the TV error of the continuous-time reverse SDE, with terms involving tmin, T, and the score estimation error. Algorithm 2, however, is a K-step Euler-Maruyama discretization, and nowhere in the proof is the discretization error accounted for. As stated, Corollary 1's alpha + o(1) claim therefore does not formally cover the implemented test. The paper's settings (T=10, tmin=0.01, K=1000) give a step size around 0.01; without a discretization bound, the TV gap is simply uncontrolled. This is a real gap, not a manufactured one, and it is load-bearing because the paper's headline is asymptotic type I error control. It is also fixable: either prove a discretization bound, or restate the guarantee for the continuous-time idealization and present the algorithm as an approximate implementation.\n\nA second, minor issue: in the breast cancer study, genes are selected using the full dataset before the prediction-based evaluation is run on a train/test split. That is a form of selection leakage, and it weakens the interpretability of the MSE comparison. The authors do describe the split, so the reader's worry about a missing split is overstated, but the leakage is real. The flow-cytometry evaluation is cleaner.\n\nOverall: this is a competent paper, honestly written, with empirical evidence that the diffusion-based sampler controls type I error better than GAN and NN baselines in the tested settings. The theoretical gap should not be ignored, and a serious referee should demand either a discretization-aware bound or a careful restatement of the claim. I would send it out for review rather than desk-reject, and I would expect the authors to tighten the theory in revision. It is a useful paper for anyone working on CRT or CI testing, and I will likely cite it as a baseline.","headline":"A useful empirical CRT variant with a real, fixable gap between the stated type I error guarantee and the discretized sampler actually implemented.","tokens_in":22598,"tokens_out":1615,"would_cite":true,"duration_ms":18425,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62G10","62H15","62G20"],"pacs":[],"model":"deepseek-v4-flash","headline":"Conditional diffusion models can supply the unknown X given Z distribution that conditional randomization tests need, giving asymptotic type I error control.","keywords":["conditional independence testing","conditional randomization test","conditional diffusion models","score-based generative models","conditional mutual information","type I error control","high-dimensional mixed data","generative model approximation"],"falsifier":"Under a null model with known Gaussian $X|Z$ satisfying the paper's smoothness assumptions, run Algorithm 3 with large $N$ and $n$ chosen so the corollary's rate applies, and record the rejection frequency at $\\alpha=0.05$; if the empirical type I error stays above $\\alpha$ by a nonvanishing amount when the number of diffusion steps $K$ is fixed, the discretization gap is a genuine break in the argument.","tokens_in":21636,"feed_emoji":"📊","tokens_out":11674,"duration_ms":95618,"temperature":0.7,"pith_summary":"The paper proposes CDCIT, a conditional independence test for $X \\perp\\!\\!\\!\\perp Y \\mid Z$ that does not require the conditional distribution of $X$ given $Z$ to be known. A conditional diffusion model trained on unlabeled data produces pseudo-copies of $X$, and a classifier-based conditional mutual information estimate provides the test statistic. The central claim is that the generated distribution converges to the true conditional distribution in total variation at a polynomial rate, which makes the conditional randomization test p-value satisfy $P(p \\le \\alpha \\mid H_0) \\le \\alpha + o(1)$ under mild smoothness assumptions. If correct, the test controls type I error asymptotically in high-dimensional settings with mixed continuous and discrete conditioning variables, and it does so without assuming a specific distributional form or dependency structure.","feed_headline":"Diffusion model supplies the missing X|Z for CI tests","feed_subtitle":"Rejection rates stay near alpha even with high-dimensional, mixed continuous-discrete confounders.","key_machinery":"The load-bearing machinery is the score-based conditional diffusion model: an Ornstein-Uhlenbeck forward process corrupts $X$ into noise, a ReLU neural network is trained by conditional score matching to approximate $\\nabla \\log p_t(x|z)$, and a reverse stochastic differential equation started at $N(0, I)$ generates pseudo-samples from $\\hat{P}(\\cdot|Z)$. What carries the statistical argument is the total-variation guarantee of Theorem 1, the product-space inequality $d_{\\mathrm{TV}}(p^{(n)}(\\cdot|Z), \\hat{p}^{(n)}(\\cdot|Z)) \\le n\\, d_{\\mathrm{TV}}(p(\\cdot|Z), \\hat{p}(\\cdot|Z))$, and the exchangeability of the null triples in the CRT p-value. The test statistic is the classifier-based conditional mutual information estimator, which estimates the likelihood ratio $p(x,y,z)/(p(x,z)p(y|z))$ from a binary classifier's predicted probabilities, with 1-nearest-neighbor resampling used to build the negative samples.","core_discovery":"On the paper's own terms, the discovery is a transfer theorem for conditional randomization tests: when the true conditional law $P(\\cdot|Z)$ is replaced by an estimated law $\\hat{P}(\\cdot|Z)$, the excess type I error is no larger than the total-variation distance between the product laws. Theorem 2 states $P(p \\le \\alpha \\mid Y,Z) \\le \\alpha + d_{\\mathrm{TV}}(p^{(n)}(\\cdot|Z), \\hat{p}^{(n)}(\\cdot|Z))$, and Theorem 1 gives the sampler a rate $O(N^{-\\Gamma_1(k,\\alpha)}(\\log N)^{\\Gamma_2(k,\\alpha)})$ for that distance. Corollary 1 then concludes that the p-value from Algorithm 3 controls type I error at level $\\alpha$ asymptotically whenever $n \\cdot N^{-\\Gamma_1(k,\\alpha)}(\\log N)^{\\Gamma_2(k,\\alpha)} \\to 0$.","pith_inferences":["The proof chain is sampler-agnostic: any generative model with a total-variation guarantee could be substituted for the diffusion model in the CRT validity argument, so the contribution is a template as well as a specific test.","A concrete open gap is the Euler-Maruyama discretization in Algorithm 2: the stated TV rate is proven for the continuous-time reverse process, and the implemented finite-step sampler would need a matching discretization bound to close the argument.","In finite samples, experiments show that a small unlabeled sample $N$ inflates type I error, so the rate in Corollary 1 is not merely a formality; it predicts when the test becomes usable.","In constraint-based causal discovery, a high-dimensional mixed-data CI test with valid error rates could serve as the oracle in algorithms that build graphs from conditional independence decisions."],"forward_implications":["CDCIT removes the main practical obstacle to conditional randomization tests: the conditional distribution $X|Z$ no longer has to be known, because the diffusion sampler supplies the pseudo-copies.","The test controls type I error asymptotically even when $Z$ is high-dimensional and mixes continuous and discrete variables, provided the smoothness and sub-Gaussian assumptions hold.","Because the diffusion model is trained only on unlabeled pairs $(X,Z)$, the method can exploit large unlabeled databases and keep the labeled sample size $n$ small relative to $N$.","The excess type I error is controlled by the total-variation rate of the sampler, so improving the diffusion approximation directly tightens the validity guarantee.","The classifier-based CMI statistic makes power available under complex, nonparametric dependence structures between $X$ and $Y$ given $Z$."],"supporting_citations":[{"why":"Introduces the conditional randomization test and proves p-value validity when $P(X|Z)$ is known.","marker":"Candès et al. 2018"},{"why":"Supplies the score-approximation bound that Theorem 1 imports for conditional diffusion models.","marker":"Fu et al. 2024"},{"why":"Provides the score-based SDE formulation and reverse-process sampler used in Algorithm 2.","marker":"Song et al. 2021"},{"why":"Gives the denoising score-matching objective used to train the conditional score network.","marker":"Ho, Jain, and Abbeel 2020"},{"why":"Defines the classifier-based CMI estimator and proves its consistency as the test statistic.","marker":"Mukherjee, Asnani, and Kannan 2020"},{"why":"Provides the local-sampling construction for negative samples in the CMI estimator.","marker":"Li et al. 2024"},{"why":"Confirms the exchangeability and validity properties of the CRT procedure that Theorem 2 relies on.","marker":"Berrett et al. 2020"},{"why":"Supplies the rank-test exchangeability bound used to prove Theorem 2.","marker":"Kuchibhotla 2020"}],"fun_headline_variants":["Diffusion models fill in unknown conditional for CI testing","Diffusion-based test for conditional independence without known X|Z","How diffusion models make conditional randomization tests practical","CDM-based CRT: accurate X|Z approximation for valid CI tests"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole validity argument rests on the assumption that the finite-step discretized sampler used in Algorithm 2 inherits the total-variation convergence proven for the continuous-time reverse diffusion process, so the generated pseudo-samples really are close to $P(X|Z)$.","fun_headline_variants_meta":{"raw":{"variants":["Diffusion models fill in unknown conditional for CI testing","Diffusion-based test for conditional independence without known X|Z","How diffusion models make conditional randomization tests practical","CDM-based CRT: accurate X|Z approximation for valid CI tests"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00063,"raw_usage":{"total_tokens":2933,"prompt_tokens":989,"completion_tokens":1944,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":605,"completion_tokens_details":{"reasoning_tokens":1877}},"tokens_in":605,"tokens_out":1944,"duration_ms":11685,"temperature":1.0,"reasoning_tokens":1877,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T14:37:57.510420+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Under a null model with known Gaussian $X|Z$ satisfying the paper's smoothness assumptions, run Algorithm 3 with large $N$ and $n$ chosen so the corollary's rate applies, and record the rejection frequency at $\\alpha=0.05$; if the empirical type I error stays above $\\alpha$ by a nonvanishing amount when the number of diffusion steps $K$ is fixed, the discretization gap is a genuine break in the argument.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the classifier-based CMI estimator and proves its consistency as the test statistic."},{"cited_title":"B.; Wang, Y.; Barber, R","cited_arxiv_id":null,"evidence_quote":"Confirms the exchangeability and validity properties of the CRT procedure that Theorem 2 relies on."}],"review_version":1}