{"id":"a70b9555-75b1-4b6b-ab04-b73bbaedb539","arxiv_id":"2507.16529","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For linear Gaussian causal models, the posterior probability of the true DAG converges to 1 exponentially if the DAG is maximal, and no faster than 1/sqrt(n) otherwise.","lead":"This paper proves that a Bayesian method's confidence in the true causal graph grows exponentially fast when the graph is maximal, but at best as the square root of the sample size otherwise. It also connects the Bayesian posterior to an optimal edge-detection test and checks the theory on simulations.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3's constant-1 lower bound is false for large prior weight variance; the determinant sign error in Lemma 5 hides a prior-dependent factor that can make the posterior gap c/√n with c<1.","rationale":"The reader correctly identified a sign inconsistency in the determinant expansion of Lemma 5, but their weakest-assumption diagnosis (causal minimality) is not where the argument breaks. The load-bearing flaw is that Theorem 3's exact lower bound fails: the determinant term in Lemma 5 has the wrong sign, and once corrected it contributes a prior-dependent constant. In the d=2 empty-DAG example the posterior deficiency is asymptotically (2/σ_w)/√n, so for σ_w>2 the claimed (1+o(1))/√n bound is false. The paper's main qualitative dichotomy — exponential concentration for maximal DAGs versus only polynomial concentration for non-maximal DAGs — is likely repairable by weakening Theorem 3 to c/√n with a positive, prior-dependent c, and the abstract's rate-only wording is consistent with that. The reader's conditional verdict therefore remains appropriate, but the required revision is more substantive than just fixing a sign: the theorem and Lemma 5 must be restated, and the dependency on σ_w should be made explicit. No ad hominem or theatrical language is intended; this is a precise mathematical objection supported by an explicit counterexample.","tokens_in":19779,"tokens_out":25238,"duration_ms":244610,"concrete_test":"Evaluate the exact closed-form Bayes factor for d=2, true S*=empty DAG, σ^2=1, σ_w^2=10: compare √n(1-π(S*|Xn)) along a simulated sample path to the predicted liminf 2/σ_w ≈ 0.632; if it is not ≥1+o(1), Theorem 3 is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing weakness is in Lemma 5 and Theorem 3, not in the causal-minimality assumption. Equation (44) in the proof of Lemma 5 assigns the wrong sign to the log-determinant difference: from (32), log|Σ_w^(j)(S,Xn)| = -s_j log n + O(1), so for S+ with one extra parent the difference is -(1/2)log n + c, not +(1/2)log n. With the sign corrected, the remainder in Lemma 5 is not eventually nonnegative: it contains c = (1/2)log(σ^2/(σ_w^2 E[Z^2])) in the simplest case, plus half the squared score statistic. For d=2, true S* the empty DAG, σ^2=1, and prior variance v=σ_w^2, the exact Bayes factor for adding either edge is v^{-1/2} n^{-1/2} exp(z_n^2/2), where z_n is the usual t-statistic. By the LIL, liminf exp(z_n^2/2)=1, so the a.s. liminf of √n(1-π(S*|Xn)) is 2/√v, which is <1 whenever v>4. This directly contradicts Theorem 3's (1+o(1))/√n bound. The qualitative exponential-versus-polynomial rate dichotomy may survive if the theorem is weakened to c/√n with a prior-dependent c>0, but the sharp statement as written is false.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies posterior concentration on DAG model space for linear structural equation models with Gaussian noise. With a uniform prior on DAGs and independent Gaussian priors on edge weights, it claims a sharp dichotomy: if the true DAG is maximal, the posterior probability of the true model tends to 1 exponentially fast with exponent n D(A*), while if the true DAG is non-maximal, the posterior probability converges no faster than 1/sqrt(n), specifically 1 - pi(S*|X^n) >= (1+o(1))/sqrt(n) almost surely. A binary-weight version gives exponential convergence with exponent 1/2 for every true DAG. The paper also reformulates an optimal Neyman-Pearson edge detector in terms of posterior probabilities and reports simulation experiments supporting the rates.","tokens_in":20101,"tokens_out":9972,"duration_ms":100210,"significance":"If correct, the maximal/non-maximal dichotomy would be a valuable qualitative contribution, linking Bayesian causal discovery to the phenomenon that avoiding overfitting is harder than identifying existing structure. The proof strategy via a BIC-type expansion and the law of the iterated logarithm is natural, and the exact small-d simulations are a useful check. The reformulation of the optimal detector in Proposition 3 is clean and potentially useful. However, the sharp constant in Theorem 3 is false as stated, and the error originates in the sign of the determinant contribution in Lemma 5; the qualitative dichotomy may survive with a prior-dependent constant, but the main theorem in its present form needs substantial correction.","major_comments":[{"comment":"","section":"Section IV, Lemma 5 and Eqs. (35), (44)"},{"comment":"","section":"Theorem 3"},{"comment":"","section":"Lemma 5 proof, remainder term"}],"minor_comments":[{"comment":"","section":"Eq. (35)"},{"comment":"","section":"Section VI-A"},{"comment":"","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The central qualitative dichotomy is plausible and the paper contains several useful ideas, but the sharp statement of Theorem 3 is false for large prior variance, and the proof of Lemma 5 has a sign error that hides a prior-dependent constant. This is fixable by weakening the constant to a prior-dependent c/sqrt(n) and revising Lemma 5 accordingly, so I do not recommend rejection. The self-citation to [29] is appropriate and Proposition 3 is an independent algebraic reformulation."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the paper's central dichotomy is real, but Theorem 3 as stated is false. The sign error in the determinant expansion (equations (35)/(44)) turns the correct −(1/2)log n contribution into +1/2 log n, which flips the constant in the 1/√n rate. In the d=2 empty-graph example with prior variance v, the Bayes factor for each spurious edge is v^{-1/2} n^{-1/2} exp(z_n^2/2), so the a.s. liminf of √n(1−π(S∗|Xn)) is 2/√v. For v>4 that is below 1, contradicting Theorem 3's (1+o(1))/√n lower bound. The qualitative rate is still polynomial, but the sharp constant is prior-dependent.\n\nWhat is genuinely new and good: Theorem 1 for the binary fixed-weight model is clean and correct, with the exponent 1/2 from a simple KL computation. Theorem 2 for maximal DAGs looks solid—the log-odds expansion is standard BIC-type, and the sign of the determinant term is irrelevant there because nD dominates. Proposition 3 is a neat reformulation of the Neyman-Pearson detector in [29] in terms of posterior edge probabilities. The paper is honest about prior work, crediting the dichotomy in BIC and Bayesian context trees.\n\nSoft spots: the sign error is load-bearing, not a typo. Lemma 5's claim that δ_n is eventually nonnegative is exactly where the wrong sign bites; with the correct sign the log-odds is −log n + T_diff + O(1), where T_diff is O(1) and can be negative. The empirical section is illustrative only, with no error bars or calibration, and the claimed optimality of the MCMC detector is inherited from [29]. The causal-minimality assumption is standard and clearly stated; the counterexample satisfies it, so that's not the issue.\n\nWho is this for: statisticians and causal discovery researchers who want rates for Bayesian structure learning. The dichotomy is worth knowing, and the exponential rate for maximal DAGs is likely to survive. But Theorem 3 must be restated with a correct constant (e.g., c/√n with prior-dependent c>0) or proved with a universal constant under restricted priors—which I doubt.\n\nRecommendation: send to peer review, but with major revision expected. The ideas are worth referee time; the printed Theorem 3 is wrong and should not be published in that form.","headline":"A genuine and interesting dichotomy, but Theorem 3's constant-1 lower bound is wrong due to a determinant sign error; the correct rate is c/√n with a prior-dependent c.","tokens_in":20609,"tokens_out":9072,"would_cite":false,"duration_ms":85224,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["62F15","62C10","62F12"],"pacs":[],"model":"deepseek-v4-flash","headline":"Posterior probability of the true causal DAG converges to 1 exponentially fast when the DAG is maximal, and at best polynomially fast otherwise.","keywords":["Bayesian causal discovery","posterior concentration","linear structural equation model","Gaussian DAG model","maximal DAG","edge detection","Neyman-Pearson hypothesis testing","law of the iterated logarithm"],"falsifier":"Take a known non-maximal three-node DAG, such as $X_3 = aX_1 + \\epsilon_3$ with $a\\neq0$ and no other edges, generate $n$ samples, and compute the exact posterior over all 25 DAGs for $n = 10^4, 10^5, 10^6$. Plot $\\log(1-\\pi(S^*|X_n))$ against $\\log n$; Theorem 3 predicts a slope approaching $-1/2$, while an exponential decay or a slope much steeper than $-1/2$ would contradict it. Repeating the same comparison with a maximal three-node DAG should show the linear-in-$n$ decay of Theorem 2 after subtracting the $O(\\sqrt{n\\log\\log n})$ fluctuation term.","tokens_in":19601,"feed_emoji":"📈","tokens_out":9764,"duration_ms":97493,"temperature":0.7,"pith_summary":"Bayesian causal discovery with linear Gaussian structural equations and standard spike-and-slab priors has two very different sample-size regimes. The paper proves that if the true directed acyclic graph is maximal, meaning that adding any single edge would create a cycle, then the posterior probability of the true graph approaches 1 exponentially fast in the number of observations, almost surely. If the true graph is not maximal, the same posterior probability cannot approach 1 faster than $1/\\sqrt{n}$, almost surely, no matter how many observations are collected. The bottleneck is the possibility of one spurious extra edge carrying an arbitrarily small but nonzero coefficient, so the hard part of causal discovery is avoiding overfitting rather than finding the existing structure. The paper also proves that the Bayesian posterior yields an optimal edge-detection rule in the Neyman-Pearson sense.","feed_headline":"Posterior finds maximal causal graphs exponentially fast","feed_subtitle":"When the true graph is not maximal, ruling out one extra edge takes roughly the square root of the sample size.","key_machinery":"The engine of the proof is the posterior log-odds ratio $\\log[\\pi(S|X_n)/\\pi(S^*|X_n)]$ between a candidate DAG $S$ and the true DAG $S^*$. Lemma 4 shows that whenever $S^*$ is not a subgraph of $S$, this log-odds equals $-nD(P_{A^*}\\|P_{m(S,\\mu_\\infty(S))}) + O(\\sqrt{n\\log\\log n})$ almost surely, so every such competitor is killed at an exponential rate. Lemma 5 shows that when $S^*$ is non-maximal, there is a one-edge extension $S^+$ of $S^*$ whose log-odds is only $-\\frac12\\log n + \\delta_n$ with $\\delta_n$ eventually nonnegative, which produces the $1/\\sqrt{n}$ floor. The asymptotic limits $\\mu_\\infty(S)$ come from least-squares projections, the fluctuation bounds come from the law of the iterated logarithm, and the one-edge analysis uses the Sherman-Morrison inversion formula.","core_discovery":"The central result is a pair of almost-sure asymptotic statements for the posterior on DAG space under the model $X = AX + \\epsilon$, with Gaussian noise, a uniform prior over DAGs, and independent Gaussian priors on nonzero edge weights. Under causal minimality of the true pair $(S^*,w^*)$, Theorem 2 gives $\\pi(S^*|X_n) = 1 - \\exp\\{-nD(A^*) + O(\\sqrt{n\\log\\log n})\\}$ when $S^*$ is maximal, where $D(A^*) > 0$ is the minimum Kullback-Leibler divergence between the true distribution and the distribution obtained from any other DAG with its limiting least-squares weights. Theorem 3 gives $1 - \\pi(S^*|X_n) \\geq (1+o(1))/\\sqrt{n}$ when $S^*$ is non-maximal, so consistency holds but at a strictly slower rate. In the binary version of the model with all nonzero weights fixed to 1, the error is $1 - \\exp\\{-n/2 + O(\\sqrt{n\\log\\log n})\\}$ for every true DAG. The paper further shows that the optimal Neyman-Pearson edge detector is exactly the posterior rule that declares an edge absent when $\\pi(\\chi_{ij}(S)=0|X_n)$ exceeds a threshold.","pith_inferences":["The $1/\\sqrt{n}$ floor is what one expects when testing a zero edge against a local alternative with coefficient of order $n^{-1/2}$; the authors do not state this connection explicitly, but it suggests the same rate should appear whenever an edge weight is allowed to shrink with $n$.","The exponent $D(A^*)$ is a population quantity, so a practical extension would be to estimate it from data and use it as a sequential stopping rule for MCMC-based causal discovery; the paper does not develop this.","Because the optimal detector is a posterior threshold, any approximate posterior sampler that converges to $\\pi(S|X_n)$ should inherit near-optimal edge-detection performance; this is testable on larger graphs even though the theoretical rates are proved for fixed dimension.","In high-dimensional settings with $d$ growing in $n$, the factorial-size factor $|G_d|$ in the lower-bound argument may change the dichotomy, so the paper's fixed-$d$ rates should not be extrapolated without further analysis."],"forward_implications":["For a maximal true DAG, sample size only needs to grow logarithmically in the desired posterior error, because the error is $\\exp\\{-nD(A^*)+O(\\sqrt{n\\log\\log n})\\}$.","For a non-maximal true DAG, the posterior assigns probability at least roughly $1/\\sqrt{n}$ to a one-edge-larger DAG for every $n$, so exact structure recovery is qualitatively harder.","The slow regime is caused by the prior's ability to place arbitrarily small nonzero weights on a spurious edge; fixing all nonzero weights to 1 removes the problem and restores exponential convergence with exponent $1/2$.","The optimal edge-detection rule from hypothesis testing can be evaluated by thresholding the posterior probability of each edge's absence, so MCMC approximations to the posterior are directly useful for constrained error detection.","Posterior consistency holds for both maximal and non-maximal true DAGs, so the issue is only the rate at which uncertainty disappears, not whether it disappears."],"supporting_citations":[{"why":"Establishes identifiability of linear Gaussian structural equation models, grounding the causal-minimality assumption that makes the posterior concentration results meaningful.","marker":"[25]"},{"why":"Defines the Neyman-Pearson edge-detection problem and the optimal detector that Proposition 3 rewrites as a posterior threshold.","marker":"[29]"},{"why":"Schwarz's BIC derivation is the asymptotic template for the order-n log-likelihood plus $(\\log n)/2$ penalty that drives Lemmas 4 and 5.","marker":"[28]"},{"why":"Schwartz's Bayes-procedure theory supplies the posterior consistency invoked in Theorem 3 for non-maximal models.","marker":"[27]"},{"why":"Earlier tree-model example of the same polynomial-versus-exponential posterior-rate dichotomy that motivates the paper's framing.","marker":"[23]"},{"why":"Law of the iterated logarithm in Banach spaces supplies the $O(\\sqrt{n\\log\\log n})$ fluctuation bounds used throughout the proofs.","marker":"[18]"},{"why":"Sherman-Morrison formula is used in Lemma 5 to compute the posterior log-odds of adding one edge.","marker":"[30]"},{"why":"Least-squares projection formula used in Propositions 1 and 2 to identify limiting posterior means and signal sizes.","marker":"[13]"},{"why":"Provides the definition of causal minimality that is assumed for identifiability throughout the paper.","marker":"[38]"}],"fun_headline_variants":["Posterior nails maximal DAGs exponentially","Exponential posterior concentration only on maximal DAGs","Bayesian posterior: optimal edge detection from DAG rates","Maximal causal graphs: posterior goes exponential, else sqrt","Posterior's sharp rate split on DAGs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the true DAG and its edge weights are causally minimal, meaning no subgraph of the true DAG generates the same distribution; if this fails, the posterior need not concentrate on the true graph, and the exponential-versus-$1/\\sqrt{n}$ dichotomy can break down.","fun_headline_variants_meta":{"raw":{"variants":["Posterior nails maximal DAGs exponentially","Exponential posterior concentration only on maximal DAGs","Bayesian posterior: optimal edge detection from DAG rates","Maximal causal graphs: posterior goes exponential, else sqrt","Posterior's sharp rate split on DAGs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000162,"raw_usage":{"total_tokens":1271,"prompt_tokens":1011,"completion_tokens":260,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":627,"completion_tokens_details":{"reasoning_tokens":184}},"tokens_in":627,"tokens_out":260,"duration_ms":3660,"temperature":1.0,"reasoning_tokens":184,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T15:09:56.856301+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a known non-maximal three-node DAG, such as $X_3 = aX_1 + \\epsilon_3$ with $a\\neq0$ and no other edges, generate $n$ samples, and compute the exact posterior over all 25 DAGs for $n = 10^4, 10^5, 10^6$. Plot $\\log(1-\\pi(S^*|X_n))$ against $\\log n$; Theorem 3 predicts a slope approaching $-1/2$, while an exponential decay or a slope much steeper than $-1/2$ would contradict it. Repeating the same comparison with a maximal three-node DAG should show the linear-in-$n$ decay of Theorem 2 after subtracting the $O(\\sqrt{n\\log\\log n})$ fluctuation term.","supporting_citations":[{"cited_title":"Peters and P","cited_arxiv_id":null,"evidence_quote":"Establishes identifiability of linear Gaussian structural equation models, grounding the causal-minimality assumption that makes the posterior concentration results meaningful."},{"cited_title":"Shaska and U","cited_arxiv_id":null,"evidence_quote":"Defines the Neyman-Pearson edge-detection problem and the optimal detector that Proposition 3 rewrites as a posterior threshold."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Schwarz's BIC derivation is the asymptotic template for the order-n log-likelihood plus $(\\log n)/2$ penalty that drives Lemmas 4 and 5."},{"cited_title":"Schwartz","cited_arxiv_id":null,"evidence_quote":"Schwartz's Bayes-procedure theory supplies the posterior consistency invoked in Theorem 3 for non-maximal models."},{"cited_title":"Papageorgiou and I","cited_arxiv_id":null,"evidence_quote":"Earlier tree-model example of the same polynomial-versus-exponential posterior-rate dichotomy that motivates the paper's framing."},{"cited_title":"Ledoux and M","cited_arxiv_id":null,"evidence_quote":"Law of the iterated logarithm in Banach spaces supplies the $O(\\sqrt{n\\log\\log n})$ fluctuation bounds used throughout the proofs."},{"cited_title":"Sherman and W.J","cited_arxiv_id":null,"evidence_quote":"Sherman-Morrison formula is used in Lemma 5 to compute the posterior log-odds of adding one edge."},{"cited_title":"Hastie, R","cited_arxiv_id":null,"evidence_quote":"Least-squares projection formula used in Propositions 1 and 2 to identify limiting posterior means and signal sizes."},{"cited_title":"Zhang and P","cited_arxiv_id":null,"evidence_quote":"Provides the definition of causal minimality that is assumed for identifiability throughout the paper."}],"review_version":1}