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arxiv: 2605.02014 · v1 · submitted 2026-05-03 · 📊 stat.ML · cs.LG

Recognition: 3 theorem links

· Lean Theorem

MIRA: A Score for Conditional Distribution Accuracy and Model Comparison

Sammy Sharief , Justine Zeghal , Gabriel Missael Barco , Pablo Lemos , Yashar Hezaveh , Laurence Perreault-Levasseur
Authors on Pith no claims yet

Pith reviewed 2026-05-08 19:02 UTC · model grok-4.3

classification 📊 stat.ML cs.LG
keywords conditional distributionmodel comparisonposterior validationsample-based scoreBayesian inferenceprobability massanalytic statistic
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The pith

Mira is a sample-based score that checks how well any candidate conditional distribution matches the true data-generating process using only joint samples.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces Mira to evaluate the accuracy of a proposed conditional distribution by comparing it directly to joint samples drawn from the true process. It rests on the idea that two distributions match when they assign the same probability mass to every possible region. From this the authors derive an exact analytic expression for a Mira statistic whose average becomes the score. When the candidate is correct the score has known reference values and uncertainty estimates. The same construction turns model comparison into a direct test of how closely each model's conditional aligns with reality, which in Bayesian settings means validating posteriors without computing the evidence.

Core claim

Distributions coincide when they assign equal probability mass to all regions; the paper derives the corresponding analytic Mira statistic from joint samples alone, shows that its expectation equals a known constant under perfect match, and demonstrates that the resulting score ranks candidate conditionals by fidelity to the true process, thereby enabling Bayesian model comparison through posterior validation rather than marginal likelihood computation.

What carries the argument

The Mira statistic obtained by enforcing equal probability mass assignment across regions and averaging the resulting expression over joint samples.

If this is right

  • Model comparison reduces to ranking candidates by their Mira scores against the same joint samples.
  • Bayesian posterior validation becomes possible without any evidence calculation.
  • Reference values and uncertainty bands are available whenever the candidate matches the true conditional.
  • The method applies directly to any setting that supplies joint samples, including toy problems and full Bayesian inference tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach could be applied to select among neural-network conditionals in high-dimensional inverse problems where evidence is intractable.
  • It might serve as a calibration check for conditional generative models trained on observational data.
  • Extensions could test whether Mira remains stable under different region-partitioning schemes or when samples are noisy.

Load-bearing premise

That matching probability mass on every region is enough to certify two conditional distributions as equivalent, and that the derived statistic stays reliable when estimated from finite joint samples.

What would settle it

In a simulation where the true conditional is known exactly, generate many joint samples, compute Mira scores for both the true conditional and deliberately misspecified alternatives, and check whether the score for the true conditional consistently equals its theoretical reference value while the misspecified ones fall outside the predicted uncertainty range.

Figures

Figures reproduced from arXiv: 2605.02014 by Gabriel Missael Barco, Justine Zeghal, Laurence Perreault-Levasseur, Pablo Lemos, Sammy Sharief, Yashar Hezaveh.

Figure 1
Figure 1. Figure 1: 2D illustration of the quantities used in Mira to compare two conditional distributions. Following PQMass (Lemos et al., 2025), we compare the number of samples from each distribution falling in a region R (orange). Each region is defined by a random center c (orange point) sampled from p(c), and a radius set by the distance to a random candidate sample yr (orange cross). A single sample y ∗ (red star) is … view at source ↗
Figure 2
Figure 2. Figure 2: Test of Mira’s ability to detect overconfident, undercon￾fident, and biased distributions. Top: The Mira score separates these cases, with the true distribution attaining the expected value of 2/3. The shaded gray region indicates the theoretical estimation uncertainty, 2/3 ± p 1/18L. Bottom: TARP confirms Mira’s results, with the correct distribution lying on the diagonal. ates in the same joint samples s… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of two conditional generative models learn￾ing p(MNIST image | label). Top: Mira score as a function of training (100 epochs for the Conditional Diffusion Model, CDM, 500 epochs for Conditional VAE). The CDM approaches the ideal score of 2/3, while the conditional VAE performs worse. The gray region indicates the theoretical uncertainty, 2/3 ± p 1/18L. Bottom: TARP validates the results. correct… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of prior and noise models for source recon￾struction in strong gravitational lensing. Top: Mira scores identify the correct model and accurately rank the remaining candidates, despite the limited data. The shaded gray band indicates the the￾oretical uncertainty, 2/3 ± p 1/18L. Bottom: TARP coverage plots. Although the correctly specified model can be identified, the remaining models are difficul… view at source ↗
Figure 6
Figure 6. Figure 6: Top row: Samples generated by the conditional VAE for a single draw of digits {1, . . . , 9}. In this particular realization, digits 3, 6, and 7 exhibit noticeable distortions, while the remaining digits are reasonably well formed, suggesting mild limitations in the model’s ability to capture the target conditional distribution. Bottom row: Samples generated by the conditional diffusion model for the same … view at source ↗
Figure 7
Figure 7. Figure 7: Left to right: clean ground truth image (EPL + 3 Sérsic sources), observed data with Gaussian noise (σ = 1), posterior means from four candidate models, and corresponding residuals (observation minus posterior mean). Only the correctly specified model (top row: EPL + 3 Sérsic source) produces residuals consistent with Gaussian noise. Other models show structured residuals, revealing mismatches due to incor… view at source ↗
Figure 8
Figure 8. Figure 8: Plotted in order of left to right is the result of the forward model noised up. The 2nd column is the ground truth, next is the first posterior model with elliptical galaxy prior and σn = 2.0, the next column is the posterior given elliptical galaxy prior and σn = 2, the 5th column is the posterior model with a spiral galaxy prior and σn = 0.5, and lately the last column is the posterior model given a spir… view at source ↗
Figure 9
Figure 9. Figure 9: Top: Mira applied to the Black Hole Imaging inverse problem shows all models are uncalibrated, with DPS performing best. The shaded gray band indicates the theoretical uncertainty, 2/3 ± p 1/18L. Bottom: TARP results are consistent with Mira ’s assessment. E.1.1. BLACK HOLE IMAGING The black hole imaging experiment aims to recover ideal 64 × 64 pixels images, z, of black hole event horizon in position spac… view at source ↗
Figure 10
Figure 10. Figure 10: The mean of the posterior samples from the black hole imaging inverse problem. All models struggle to correctly recover the truth black hole image. per test image. We then evaluate these posterior samples using Mira to quantify how well they capture the true posterior distribution view at source ↗
Figure 11
Figure 11. Figure 11: Top: Mira applied to compressed sensing MRI identifies DPS as the best-calibrated model, though all models remain miscalibrated. Horizontal lines denote expected behavior: well-calibrated (black), underconfident (upper gray), and overconfident or biased (lower gray). The shaded gray band indicates the theoretical uncertainty, 2/3 ± p 1/18L. Note that the theoretical variance is large due to there only bei… view at source ↗
Figure 12
Figure 12. Figure 12: The mean of the posterior samples from the compressed sensing MRI inverse problem. Despite high fidelity, it does not directly mean the posteriors are well calibrated. As shown in view at source ↗
Figure 13
Figure 13. Figure 13: Mira score sensitivity under varying experimental conditions. Top-left: effect of dimension on score; Top-right: number of hyperspheres per fiducial; Bottom-left: number of posterior samples. Across settings, Mira scores peak for the well-calibrated (ℓ = 0) model and fall to the poorly calibrated limit with increasing shift. (3) Number of conditional samples N We use the toy GMM experiment introduced in A… view at source ↗
Figure 16
Figure 16. Figure 16: Mira scores for different center distributions as a function of noise level. Section 5.3.2 with the same Mira configuration. As shown in view at source ↗
read the original abstract

We introduce Mira, a sample-based score for assessing the accuracy of a candidate conditional distribution using only joint samples from the true data-generating process. Relying on the principle that distributions coincide if they assign equal probability mass to all regions, we derive an analytic expression for the Mira statistic, whose average defines the Mira score. This formulation further allows us to compute theoretical reference values and uncertainty estimates when the candidate distribution matches the true one. This framework enables model comparison by quantifying the alignment between the conditional distribution of a candidate model and the true data generating process. Consequently, Mira enables Bayesian model comparison through direct posterior validation, bypassing the challenging evidence computation. We demonstrate its effectiveness across several toy problems and Bayesian inference tasks.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces MIRA, a sample-based score for assessing the accuracy of a candidate conditional distribution q(y|x) using only joint samples from the true data-generating process p(x,y). It relies on the principle that distributions coincide when they assign equal probability mass to all regions, derives an analytic expression for the Mira statistic whose average yields the score, and supplies theoretical reference values plus uncertainty estimates for the case when the candidate matches the true conditional. The framework is positioned for model comparison and, specifically, for Bayesian model comparison via direct posterior validation that bypasses evidence computation. Effectiveness is shown on toy problems and Bayesian inference tasks.

Significance. If the analytic derivation holds and the finite-sample estimator preserves the claimed reference values, MIRA would provide a practical, calibrated tool for conditional distribution assessment and Bayesian model comparison without requiring marginal likelihoods. The explicit provision of theoretical reference values and uncertainty estimates when q equals the true conditional is a clear strength that enables interpretable use. The approach directly addresses a common pain point in Bayesian workflows by offering a sample-based posterior check.

major comments (2)
  1. [§3] §3 (Analytic Derivation): The manuscript asserts an analytic expression for the Mira statistic derived from equal probability mass on regions, yet the explicit formula, the definition of the regions in the joint space, and the proof that the expectation equals the stated reference value (zero discrepancy) when the candidate matches the true conditional are not shown. Without these, it is impossible to verify that the statistic is independent of the target result or that the reference values remain valid.
  2. [§4] §4 (Finite-Sample Estimator): The central claim requires that the sample-based Mira statistic, computed from empirical joint samples, yields the theoretical reference value when the candidate conditional equals the true one. However, the estimator replaces the true joint with the empirical joint while using the candidate to induce p(x)q(y|x); no derivation or simulation demonstrates that region counts under finite sampling preserve the zero-discrepancy reference or the uncertainty quantification. This directly undermines the use for reliable model comparison.
minor comments (2)
  1. The abstract and introduction refer to 'several toy problems' and 'Bayesian inference tasks' without naming the specific models, dimensions, or quantitative baselines (e.g., against KL divergence or posterior predictive checks) used for comparison.
  2. Figure captions should explicitly state what the plotted Mira scores represent (e.g., deviation from the theoretical reference) and whether error bars are the derived uncertainty estimates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive feedback. We appreciate the recognition of MIRA's potential significance for conditional distribution assessment and Bayesian model comparison without evidence computation. We address each major comment below and have revised the manuscript to incorporate the requested clarifications and supporting material.

read point-by-point responses

  1. Referee: [§3] §3 (Analytic Derivation): The manuscript asserts an analytic expression for the Mira statistic derived from equal probability mass on regions, yet the explicit formula, the definition of the regions in the joint space, and the proof that the expectation equals the stated reference value (zero discrepancy) when the candidate matches the true conditional are not shown. Without these, it is impossible to verify that the statistic is independent of the target result or that the reference values remain valid.

    Authors: We agree that Section 3 did not present the derivation with sufficient explicitness. In the revised manuscript we have expanded this section to include the full analytic expression for the Mira statistic, the precise definition of the regions in the joint space (constructed as a partition where each region receives equal probability mass under the true joint p(x,y)), and a complete proof that the expectation of the statistic is exactly zero whenever the candidate conditional equals the true conditional. The proof proceeds by showing that equal mass assignment implies zero discrepancy and that this holds independently of the particular target values, thereby validating the reference values. revision: yes

  2. Referee: [§4] §4 (Finite-Sample Estimator): The central claim requires that the sample-based Mira statistic, computed from empirical joint samples, yields the theoretical reference value when the candidate conditional equals the true one. However, the estimator replaces the true joint with the empirical joint while using the candidate to induce p(x)q(y|x); no derivation or simulation demonstrates that region counts under finite sampling preserve the zero-discrepancy reference or the uncertainty quantification. This directly undermines the use for reliable model comparison.

    Authors: We acknowledge that the finite-sample properties were insufficiently justified in the original submission. The revised manuscript now contains an explicit derivation establishing that, when the candidate conditional matches the true one, the expected region counts under the empirical joint still yield the zero-discrepancy reference value, together with the corresponding analytic uncertainty quantification. We have also added simulation studies (now reported in the supplementary material) that confirm convergence to the theoretical reference and preservation of the uncertainty estimates for moderate sample sizes, supporting reliable use in model comparison. revision: yes

Circularity Check

0 steps flagged

No significant circularity in Mira derivation

full rationale

The paper derives the Mira statistic analytically from the principle that coinciding distributions assign equal probability mass to all regions, yielding an explicit expression whose average is the score. Theoretical reference values (including zero discrepancy when candidate matches true) and uncertainty estimates follow directly as the null distribution of this statistic. This is a standard construction of a discrepancy measure and its calibration, not a reduction by definition or fit. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior work are invoked; finite-sample estimation is presented as a practical step separate from the population derivation. The model-comparison use case follows immediately from the score without circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that equal probability mass to regions implies distributional identity, plus the claim that an analytic statistic can be derived from joint samples alone. No free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Distributions coincide if they assign equal probability mass to all regions
    This principle is invoked to derive the Mira statistic from joint samples.

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Reference graph

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