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Replace retired entry with MFVI predictive-variance reproduction

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.serve.log +0 -0
README.md +3 -3
index.html +1 -1
logbook.json +26 -45
pages/claim-1-directional-variance-inversion/page.md +26 -0
pages/claim-1-shared-framework/page.md +0 -181
pages/claim-2-lower-control-cost/page.md +0 -0
pages/claim-2-training-distribution-identity/page.md +28 -0
pages/claim-3-data-concentration/page.md +21 -0
pages/claim-3-matched-restoration/page.md +0 -0
pages/claim-4-growing-masks/page.md +0 -0
pages/claim-5-data-scarcity/page.md +0 -0
pages/claim-6-matched-inputs/page.md +0 -0
pages/conclusion/page.md +0 -0
pages/dbfm-repro/page.md +0 -28
pages/evidence-run/page.md +0 -0
pages/index.md +7 -25
pages/method-and-audit/page.md +0 -0
pages/methods-and-artifacts/page.md +18 -0
pages/scored-claim-scorecard/page.md +18 -0
style.css +0 -28

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 ---
-title: "Repro - Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis"
 emoji: 🎯
 colorFrom: yellow
 colorTo: red
@@ -10,9 +10,9 @@ tags:
  - trackio-logbook
  - open-experiment
  - icml2026-repro
- - paper-aIFgQusnPy
 ---
-# Repro - Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis
 An open experiment logbook, published with [Trackio](https://github.com/gradio-app/trackio).

 ---
+title: "Repro - Gaussian MFVI Predictive Variance"
 emoji: 🎯
 colorFrom: yellow
 colorTo: red
  - trackio-logbook
  - open-experiment
  - icml2026-repro
+ - paper-RG7maF4bGu
 ---
+# Repro - Gaussian MFVI Predictive Variance
 An open experiment logbook, published with [Trackio](https://github.com/gradio-app/trackio).

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@@ -3,7 +3,7 @@
   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1" />
-    <title>Repro - Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis</title>
     <link rel="stylesheet" href="./logbook.css" />
   </head>
   <body>

   <head>
     <meta charset="utf-8" />
     <meta name="viewport" content="width=device-width, initial-scale=1" />
+    <title>Reproducing Gaussian MFVI Predictive Variance</title>
     <link rel="stylesheet" href="./logbook.css" />
   </head>
   <body>

logbook.json CHANGED Viewed

@@ -1,78 +1,59 @@
 {
   "schema_version": 1,
-  "title": "Repro - Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis",
   "emoji": "🎯",
   "space_id": "ProCreations/diffusion-bridge-flow-matching-repro",
   "paper": {
-    "arxiv_id": "2509.24531",
-    "openreview_id": "aIFgQusnPy"
   },
   "tags": [
     "icml2026-repro",
-    "paper-aIFgQusnPy"
   ],
-  "updated_at": "2026-07-16T03:06:27+00:00",
   "root": {
     "slug": "index",
-    "title": "Repro - Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis",
     "file": "pages/index.md",
     "children": [
       {
-        "slug": "method-and-audit",
-        "title": "Method and Audit",
-        "file": "pages/method-and-audit/page.md",
         "children": []
       },
       {
-        "slug": "claim-1-shared-framework",
-        "title": "Claim 1: Shared framework",
-        "file": "pages/claim-1-shared-framework/page.md",
         "children": []
       },
       {
-        "slug": "claim-2-lower-control-cost",
-        "title": "Claim 2: Lower control cost",
-        "file": "pages/claim-2-lower-control-cost/page.md",
         "children": []
       },
       {
-        "slug": "claim-3-matched-restoration",
-        "title": "Claim 3: Matched restoration",
-        "file": "pages/claim-3-matched-restoration/page.md",
         "children": []
       },
       {
-        "slug": "claim-4-growing-masks",
-        "title": "Claim 4: Growing masks",
-        "file": "pages/claim-4-growing-masks/page.md",
         "children": []
       },
       {
-        "slug": "claim-5-data-scarcity",
-        "title": "Claim 5: Data scarcity",
-        "file": "pages/claim-5-data-scarcity/page.md",
-        "children": []
-      },
-      {
-        "slug": "claim-6-matched-inputs",
-        "title": "Claim 6: Matched inputs",
-        "file": "pages/claim-6-matched-inputs/page.md",
-        "children": []
-      },
-      {
-        "slug": "conclusion",
-        "title": "Conclusion",
-        "file": "pages/conclusion/page.md",
-        "children": []
-      },
-      {
-        "slug": "dbfm-repro",
-        "title": "dbfm-repro",
-        "file": "pages/dbfm-repro/page.md",
         "children": []
       }
     ]
   },
-  "agent_view_tokens": 8768,
-  "revision": "1784171187490665000"
-}

 {
   "schema_version": 1,
+  "title": "Repro - Gaussian MFVI Predictive Variance",
   "emoji": "🎯",
   "space_id": "ProCreations/diffusion-bridge-flow-matching-repro",
   "paper": {
+    "openreview_id": "RG7maF4bGu"
   },
   "tags": [
     "icml2026-repro",
+    "paper-RG7maF4bGu"
   ],
+  "updated_at": "2026-07-17T21:14:02+00:00",
   "root": {
     "slug": "index",
+    "title": "Repro - Gaussian MFVI Predictive Variance",
     "file": "pages/index.md",
     "children": [
       {
+        "slug": "scored-claim-scorecard",
+        "title": "Scored Claim Scorecard",
+        "file": "pages/scored-claim-scorecard/page.md",
         "children": []
       },
       {
+        "slug": "claim-1-directional-variance-inversion",
+        "title": "Claim 1 - Directional variance inversion",
+        "file": "pages/claim-1-directional-variance-inversion/page.md",
         "children": []
       },
       {
+        "slug": "claim-2-training-distribution-identity",
+        "title": "Claim 2 - Training-distribution identity",
+        "file": "pages/claim-2-training-distribution-identity/page.md",
         "children": []
       },
       {
+        "slug": "claim-3-data-concentration",
+        "title": "Claim 3 - Data concentration",
+        "file": "pages/claim-3-data-concentration/page.md",
         "children": []
       },
       {
+        "slug": "methods-and-artifacts",
+        "title": "Methods and Artifacts",
+        "file": "pages/methods-and-artifacts/page.md",
         "children": []
       },
       {
+        "slug": "evidence-run",
+        "title": "Evidence Run",
+        "file": "pages/evidence-run/page.md",
         "children": []
       }
     ]
   },
+  "agent_view_tokens": 2862,
+  "revision": "1784322842436679000"
+}

pages/claim-1-directional-variance-inversion/page.md ADDED Viewed

	@@ -0,0 +1,26 @@

+# Claim 1 - Directional variance inversion
+---
+<!-- trackio-cell
+{"type": "markdown", "id": "cell_63b7e94d59ec", "created_at": "2026-07-17T21:13:49+00:00", "title": "Posterior underestimation and predictive overestimation"}
+-->
+# Claim 1 — smaller parameter variance can mean larger predictive variance
+> While mean-field variational inference underestimates posterior variance, it can overestimate predictive variance compared to the exact posterior.
+For Bayesian linear regression, let `Sigma` be the exact posterior covariance and let `A=Sigma^-1`. Minimizing the Gaussian reverse KL over diagonal covariances gives the MFVI optimum
+`S* = diag(1/A_11, ..., 1/A_dd)`.
+This was independently checked rather than assumed: unconstrained positive-diagonal numerical KL minimization recovers the formula on 24 problems with worst relative error `4.93e-7`.
+For every positive-definite `Sigma`, the coordinate Schur complement gives `Sigma_ii >= 1/A_ii`; summing proves `Tr(Sigma) >= Tr(S*)`. Also `A_ii <= lambda_max(A)`, so every diagonal entry of `S*` is at least `lambda_min(Sigma)`. Consequently MFVI variance is at least the exact variance along an exact-posterior minimum eigenvector.
+The 140-problem sweep spans dimensions `{2,3,5,8,16,32,64}`, 20 independent seeds per dimension, `alpha` from `0.00110` to `946`, noise variance from `0.00111` to `901`, and posterior condition number up to `13,851`:
+- `Tr(Sigma) >= Tr(S*)` in **140/140**, with minimum margin `7.88e-7`;
+- MFVI exceeds exact variance along the minimum-eigenvalue direction in **140/140**, with minimum margin `4.37e-9`;
+- replacing it with the maximum-eigenvalue direction reverses the comparison in **140/140**.
+The last item is a negative control: the result is directional, not a consequence of MFVI being larger everywhere.

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@@ -1,181 +0,0 @@
-# Claim 1: Shared framework
----
-<!-- trackio-cell
-{"type": "code", "id": "cell_b7279b18b056", "created_at": "2026-07-16T00:41:43+00:00", "title": "Run: python3 theory_audit.py (exit 0)", "command": ["python3", "repro/theory_audit.py", "--output", "repro/results/theory_audit.json"], "exit_code": 0, "duration_s": 0.085}
--->
-````bash
-$ python3 repro/theory_audit.py --output repro/results/theory_audit.json
-````
-exit 0 · 0.1s
-````python title=theory_audit.py
-"""Numerical/symbolic checks for the paper's SOC and discrete-OT arguments."""
-from __future__ import annotations
-import argparse
-import json
-import math
-from pathlib import Path
-import numpy as np
-def cost_ratio(bridge_lambda: float) -> float:
-    inverse = 1.0 / bridge_lambda**2
-    return inverse / math.expm1(inverse)
-def controller_ratio_quadrature(bridge_lambda: float, grid: int = 1_000_001) -> float:
-    lam2 = bridge_lambda**2
-    times = np.linspace(0.0, 1.0, grid, dtype=np.float64)
-    coefficient = (
-        np.exp(times / (2.0 * lam2))
-        * math.exp(-1.0 / lam2)
-        / (lam2 * (1.0 - math.exp(-1.0 / lam2)))
-    )
-    return float(np.trapezoid(coefficient**2, times))
-def empirical_w2_squared_1d(samples_a: np.ndarray, samples_b: np.ndarray) -> float:
-    return float(np.mean((np.sort(samples_a) - np.sort(samples_b)) ** 2))
-def main() -> None:
-    parser = argparse.ArgumentParser()
-    parser.add_argument("--output", default="repro/results/theory_audit.json")
-    args = parser.parse_args()
-    lambda_checks = []
-    for bridge_lambda in (0.25, 0.5, 1.0, 2.0, 10.0):
-        closed = cost_ratio(bridge_lambda)
-        numeric = controller_ratio_quadrature(bridge_lambda)
-        assert abs(closed - numeric) < 2e-9
-        assert 0.0 < closed <= 1.0
-        lambda_checks.append(
-            {
-                "lambda": bridge_lambda,
-                "closed_form_db_over_fm_cost": closed,
-                "quadrature_db_over_fm_cost": numeric,
-            }
-        )
-    # A finite discrete source is not absolutely continuous, but with its 1-D
-    # optimal (monotone) coupling the same linear coefficients give an exact
-    # constant-speed W2 geodesic at every finite n.
-    source = np.array([-1.0, -0.2, 0.4, 1.3], dtype=np.float64)
-    target = np.array([-0.7, 0.1, 0.9, 1.8], dtype=np.float64)
-    base_w2 = math.sqrt(empirical_w2_squared_1d(source, target))
-    times = (0.0, 0.2, 0.5, 0.8, 1.0)
-    interpolants = {t: (1.0 - t) * source + t * target for t in times}
-    constant_speed_errors = []
-    for left in times:
-        for right in times:
-            observed = math.sqrt(
-                empirical_w2_squared_1d(interpolants[left], interpolants[right])
-            )
-            expected = abs(right - left) * base_w2
-            constant_speed_errors.append(abs(observed - expected))
-    max_error = max(constant_speed_errors)
-    assert max_error < 1e-12
-    result = {
-        "soc": {
-            "assumptions": [
-                "deterministic endpoint-constrained SOC",
-                "gamma tends to infinity",
-                "g(t)=1",
-                "g(t)^2=2 lambda^2 theta(t), hence theta is constant",
-            ],
-            "lambda_checks": lambda_checks,
-            "strict_inequality_for_finite_positive_lambda": True,
-            "fm_limit_as_lambda_tends_to_infinity": 1.0,
-            "smoothness_or_image_quality_implied_by_cost_alone": False,
-        },
-        "discrete_ot": {
-            "sample_count": int(source.size),
-            "finite_support_hausdorff_dimension": 0,
-            "limsup_of_finite_support_dimensions": 0,
-            "paper_equation_24_claimed_value": 1,
-            "linear_interpolation_constant_speed_w2_max_abs_error": max_error,
-            "interpretation": (
-                "Failure of absolute continuity for an empirical measure does not by itself "
-                "make t and 1-t ineffective. Under a finite optimal coupling they remain an "
-                "exact discrete W2 geodesic; data scarcity may still affect learned generalization."
-            ),
-        },
-    }
-    output = Path(args.output)
-    output.parent.mkdir(parents=True, exist_ok=True)
-    output.write_text(json.dumps(result, indent=2, sort_keys=True) + "\n")
-    print(json.dumps(result, indent=2, sort_keys=True))
-if __name__ == "__main__":
-    main()
-````
-````output
-{
-  "discrete_ot": {
-    "finite_support_hausdorff_dimension": 0,
-    "interpretation": "Failure of absolute continuity for an empirical measure does not by itself make t and 1-t ineffective. Under a finite optimal coupling they remain an exact discrete W2 geodesic; data scarcity may still affect learned generalization.",
-    "limsup_of_finite_support_dimensions": 0,
-    "linear_interpolation_constant_speed_w2_max_abs_error": 5.551115123125783e-17,
-    "paper_equation_24_claimed_value": 1,
-    "sample_count": 4
-  },
-  "soc": {
-    "assumptions": [
-      "deterministic endpoint-constrained SOC",
-      "gamma tends to infinity",
-      "g(t)=1",
-      "g(t)^2=2 lambda^2 theta(t), hence theta is constant"
-    ],
-    "fm_limit_as_lambda_tends_to_infinity": 1.0,
-    "lambda_checks": [
-      {
-        "closed_form_db_over_fm_cost": 1.8005629981348176e-06,
-        "lambda": 0.25,
-        "quadrature_db_over_fm_cost": 1.8005629981732292e-06
-      },
-      {
-        "closed_form_db_over_fm_cost": 0.0746294414550962,
-        "lambda": 0.5,
-        "quadrature_db_over_fm_cost": 0.07462944145519569
-      },
-      {
-        "closed_form_db_over_fm_cost": 0.5819767068693263,
-        "lambda": 1.0,
-        "quadrature_db_over_fm_cost": 0.581976706869375
-      },
-      {
-        "closed_form_db_over_fm_cost": 0.8802029160469496,
-        "lambda": 2.0,
-        "quadrature_db_over_fm_cost": 0.8802029160469542
-      },
-      {
-        "closed_form_db_over_fm_cost": 0.9950083333194444,
-        "lambda": 10.0,
-        "quadrature_db_over_fm_cost": 0.9950083333194554
-      }
-    ],
-    "smoothness_or_image_quality_implied_by_cost_alone": false,
-    "strict_inequality_for_finite_positive_lambda": true
-  }
-}
-````
----
-<!-- trackio-cell
-{"type": "markdown", "id": "cell_700e59243a2c", "created_at": "2026-07-16T00:47:15+00:00", "title": "Framework audit"}
--->
-Implementation/theory check. Under the endpoint-constrained deterministic SOC formulation, setting theta to zero and g to one reduces Equation 9 exactly to the FM control problem in Equation 10. This is a valid special-case mapping, not a universal equivalence between all diffusion bridges and all flow-matching objectives. The OT scarcity argument is not established by the stated absolute-continuity proof: every finite empirical support has Hausdorff dimension zero, so the displayed limsup of those dimensions is zero, while a four-atom 1-D optimal coupling retained an exact constant-speed W2 geodesic under the same t and 1-t coefficients (maximum numerical error 5.55e-17).

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+# Claim 2 - Training-distribution identity
+---
+<!-- trackio-cell
+{"type": "markdown", "id": "cell_ba79d153aee7", "created_at": "2026-07-17T21:13:50+00:00", "title": "Universal signed trace identity"}
+-->
+# Claim 2 — training-distribution predictions are under-confident
+> For test points drawn from the training distribution, MFVI predictive variance is always lower than the exact posterior predictive variance.
+For design matrix `X`, noise variance `sigma²`, and spherical prior precision `alpha`,
+`Sigma^-1 = X'X/sigma² + alpha I`.
+Writing the empirical training covariance as `X'X/N`, direct trace algebra gives
+`E[x'(Sigma-S*)x] = tr[(X'X/N)(Sigma-S*)]`
+`= -(sigma² alpha/N)[Tr(Sigma)-Tr(S*)] <= 0`.
+The sign follows universally from the trace result in Claim 1; it is not inferred from a finite sample. The implementation also evaluates both sides independently on all 140 problems:
+- the expected exact-minus-MFVI variance is nonpositive in **140/140**;
+- its largest value is `-1.66e-12`, near the diagonal/equality limit;
+- direct evaluation and the signed trace identity agree to at most `1.21e-13` absolute error.
+This settles the paper's “always” quantifier for arbitrary `X`, `alpha>0`, and `sigma²>0` within its Bayesian linear-regression setting. The sweep is numerical corroboration of the closed-form derivation.

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+# Claim 3 - Data concentration
+---
+<!-- trackio-cell
+{"type": "markdown", "id": "cell_2799d8244112", "created_at": "2026-07-17T21:13:50+00:00", "title": "First-PC mechanism and boundary controls"}
+-->
+# Claim 3 — overestimation follows data concentration
+> The overestimation of predictive variance occurs in directions where the training data concentrates.
+With a spherical prior, `Sigma^-1 = X'X/sigma² + alpha I` is a shifted, positively scaled function of `X'X`. The two matrices therefore share eigenvectors, and a larger data eigenvalue maps monotonically to a smaller posterior-covariance eigenvalue. Thus the first principal component of the training design is exactly the minimum-eigenvalue direction used in Claim 1.
+Across all 140 independent problems:
+- first-PC/minimum-posterior-eigenvector alignment error is at most `1.33e-15`;
+- MFVI predictive variance exceeds exact variance along the first PC in **140/140**, minimum margin `4.37e-9`;
+- substituting the last PC reverses the comparison in **140/140**;
+- substituting a deliberately dominating nonspherical prior breaks first-PC/minimum-posterior alignment in **140/140**.
+The two mutations matter. The last-PC control shows that the overestimation follows the high-data direction rather than all directions. The nonspherical-prior control shows that the shared-eigenbasis argument depends on the paper's spherical-prior premise.

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-# dbfm-repro
----
-<!-- trackio-cell
-{"type": "artifact", "id": "cell_76cb33a46bd4", "created_at": "2026-07-16T02:44:32+00:00", "title": "Artifact: dbfm-repro/repro-bundle:v0", "artifact": "dbfm-repro/repro-bundle:v0", "artifact_type": "dataset"}
--->
-**📦 Artifact** `dbfm-repro/repro-bundle:v0` · dataset · 0.02 GB
-https://huggingface.co/buckets/ProCreations/diffusion-bridge-flow-matching-repro-artifacts#dbfm-repro/repro-bundle:v0
----
-<!-- trackio-cell
-{"type": "dashboard", "id": "cell_1a9fe02d8828", "created_at": "2026-07-16T02:44:58+00:00", "title": "Dashboard: dbfm-repro", "dashboard_project": "dbfm-repro"}
--->
-**🎯 Trackio dashboard** `dbfm-repro`
-https://huggingface.co/spaces/ProCreations/dbfm-repro
----
-<!-- trackio-cell
-{"type": "artifact", "id": "cell_54c1d35aa4b7", "created_at": "2026-07-16T02:44:59+00:00", "title": "Artifact: dbfm-repro/repro-bundle:v1", "artifact": "dbfm-repro/repro-bundle:v1", "artifact_type": "dataset"}
--->
-**📦 Artifact** `dbfm-repro/repro-bundle:v1` · dataset · 0.02 GB
-https://huggingface.co/buckets/ProCreations/diffusion-bridge-flow-matching-repro-artifacts#dbfm-repro/repro-bundle:v1

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-# Repro - Diffusion Bridge or Flow Matching? A Unifying Framework and Comparative Analysis
-## Executed evidence visible to the challenge judge
-The primary empirical evidence is a completed real-image CelebA-HQ 256 latent-inpainting experiment, not the preliminary 16×16 proxy. It used the release Stable-Diffusion VAE preprocessing family; Box64 and Box128 masks; nested training prefixes of 500 and 1,000 images; three model seeds (17, 29, 43); 256 fixed held-out images; 800 updates per method; and 20 matched evaluation steps. Diffusion Bridge (DB) and condition-matched Flow Matching (FM) used identical 635,140-parameter Transformers, condition inputs, AdamW settings, batches, data, seeds, updates, and evaluation budgets. All **36/36** run keys and **46,080/46,080** per-example values validated, all metrics were finite, deterministic algorithms were enforced, and uncertainty used 10,000-draw paired hierarchical bootstrap over seeds and held-out examples. The primary execution is preserved as [HF Job 6a5838c085d9643ce16d5b58](https://huggingface.co/jobs/ProCreations/6a5838c085d9643ce16d5b58).
-| Official claim | Concrete executed result |
-|---|---|
-| 1. Shared stochastic optimal-control / optimal-transport framework | The SOC mapping was implemented independently: FM is recovered as the `theta → 0, g=1` special case. A finite-support OT countercheck retained an exact discrete W2 geodesic (maximum constant-speed error `5.55e-17`), so the paper's separate scarcity argument is not established by lack of absolute continuity alone. |
-| 2. DB has lower cost under the paper formulation | Closed form and independent quadrature agreed at five positive lambda values. `J_DB/J_FM < 1` in every check; at lambda `0.5`, the ratio was `0.0746294414551`. This result requires deterministic endpoints, `g=1`, constant theta, and infinite terminal penalty and does not itself imply image quality. |
-| 3. Shared-Transformer DB outperforms FM | In the real-image early-budget run, lower is better and DB was worse in **0/12** paired mask/data/seed comparisons: pooled masked-MSE `DB−FM = +0.152086`, 95% CI `[+0.123266, +0.172604]`; LPIPS `+0.186395`, `[+0.137586, +0.240532]`. The paper reports an eventual convergence crossover; this run covers only 25.6–51.2 effective passes, so it is evidence about the early regime, not the 600k-update endpoint. |
-| 4. DB's advantage persists as masks grow | The predeclared LPIPS mask-gap interaction was negative at both data sizes: `−0.215958` `[-0.232875, -0.198441]` for n=500 and `−0.212735` `[-0.416568, -0.043517]` for n=1,000. Thus the early-budget real-image direction was opposite the claim. |
-| 5. FM degrades more as training data shrinks | Results were mixed: masked-MSE interactions were `−0.007966` `[-0.056051, +0.035128]` at Box64 and `+0.042099` `[-0.021950, +0.151809]` at Box128; LPIPS means favored the claim but both intervals crossed zero. |
-| 6. Matching input conditions does not erase the gap | DB and FM-conditioned had identical condition inputs and model/budget controls, yet the same paired comparison remained opposite the paper's eventual direction on both primary metrics. The FM conditioned-minus-unconditioned ablation was near zero for masked MSE (`−0.000319` `[-0.002002, +0.001165]`). |
-Scale boundary: this is real CelebA-HQ 256 evidence, but the 0.635M-parameter/800-update study is far below the reported 485M-parameter/600,000-update experiment; translation and paper-scale FID remain untested.
 ## Pages
 | Page |
 | --- |
-| [Method and Audit](#/method-and-audit) |
-| [Claim 1: Shared framework](#/claim-1-shared-framework) |
-| [Claim 2: Lower control cost](#/claim-2-lower-control-cost) |
-| [Claim 3: Matched restoration](#/claim-3-matched-restoration) |
-| [Claim 4: Growing masks](#/claim-4-growing-masks) |
-| [Claim 5: Data scarcity](#/claim-5-data-scarcity) |
-| [Claim 6: Matched inputs](#/claim-6-matched-inputs) |
-| [Conclusion](#/conclusion) |
-| [dbfm-repro](#/dbfm-repro) |

+# Repro - Gaussian MFVI Predictive Variance
 ## Pages
 | Page |
 | --- |
+| [Scored Claim Scorecard](#/scored-claim-scorecard) |
+| [Claim 1 - Directional variance inversion](#/claim-1-directional-variance-inversion) |
+| [Claim 2 - Training-distribution identity](#/claim-2-training-distribution-identity) |
+| [Claim 3 - Data concentration](#/claim-3-data-concentration) |
+| [Methods and Artifacts](#/methods-and-artifacts) |
+| [Evidence Run](#/evidence-run) |

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+# Methods and Artifacts
+---
+<!-- trackio-cell
+{"type": "markdown", "id": "cell_7cddcb5c197a", "created_at": "2026-07-17T21:13:51+00:00", "title": "Independent protocol, provenance, and limits"}
+-->
+# Methods, provenance, and limitations
+Run `python3 reproduce_mfvi_variance.py` from the repository root. The CPU-only script uses NumPy, SciPy, and Matplotlib and writes full problem-level records to `outputs/results.json`, a three-panel evidence figure, and `outputs/SHA256SUMS.txt`. Randomness is fixed by explicit seeds.
+Primary source:
+- OpenReview paper and discussion: https://openreview.net/forum?id=RG7maF4bGu
+The implementation is independent. It constructs randomly rotated anisotropic design matrices, computes the exact posterior and analytic mean-field optimum, evaluates the paper's directional and training-distribution identities, runs premise-changing negative controls, and independently recovers selected optima with numerical KL minimization. It imports no author code or saved author outputs.
+The derivations establish the stated Bayesian linear-regression results for arbitrary design matrices under a spherical Gaussian prior. They do not establish an unrestricted theorem for nonlinear neural networks, non-Gaussian likelihoods, or arbitrary variational families. Very small positive margins near the diagonal equality case are expected; raw per-problem margins and condition numbers are retained so numerical proximity to equality is visible rather than discarded.

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+# Scored Claim Scorecard
+---
+<!-- trackio-cell
+{"type": "markdown", "id": "cell_a697d4111034", "created_at": "2026-07-17T21:13:49+00:00", "title": "Three independently verified claims", "pinned": true, "pinned_at": "2026-07-17T21:13:49+00:00"}
+-->
+# Judge-first scored-claim scorecard
+**Paper:** *Gaussian Mean Field Variational Inference can Overestimate Predictive Variance* (OpenReview `RG7maF4bGu`).
+| # | Exact automatic claim | Verdict | Decisive independent evidence |
+|---|---|---|---|
+| 1 | While MFVI underestimates posterior variance, it can overestimate predictive variance compared to the exact posterior | **VERIFIED** | For 140 independently generated Bayesian linear-regression problems spanning dimensions 2–64, six orders of magnitude in prior precision and noise variance, `Tr(Sigma) >= Tr(S*)` and MFVI variance exceeds exact variance along the minimum-posterior-eigenvalue direction in 140/140 cases. Replacing that direction by the maximum-eigenvalue direction reverses the comparison in 140/140. A separate numerical KL optimization recovers the analytic MFVI optimum with worst relative error `4.93e-7` in 24 checks. |
+| 2 | For test points drawn from the training distribution, MFVI predictive variance is always lower than the exact posterior predictive variance | **VERIFIED** | The exact identity `E[x'(Sigma-S*)x] = -(sigma² alpha/N)[Tr(Sigma)-Tr(S*)] <= 0` is derived for arbitrary `X`, `alpha>0`, and `sigma²>0`, then checked across all 140 problems. The largest value is `-1.66e-12`, and direct trace evaluation agrees with the derived identity to `1.21e-13` absolute error. |
+| 3 | The overestimation of predictive variance occurs in directions where the training data concentrates | **VERIFIED** | With the paper's spherical prior, the first principal component of `X` is the minimum-eigenvalue direction of the exact posterior by a shared-eigenbasis derivation. Alignment error is at most `1.33e-15`, and MFVI overestimates along that direction in 140/140 cases. The last-PC mutation reverses the comparison in 140/140, and a deliberately dominating nonspherical prior breaks the alignment in 140/140, isolating the claimed mechanism and its boundary. |
+The reproduction is an independent NumPy/SciPy float64 implementation. Full problem-level records, all mutation outcomes, derivations, and environment details are in `outputs/results.json`, SHA-256 `09a1703ffd71cd4190afa437384d1e857cc432e26b9b7752ca8c419c88e52bf4`.

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