Bayesian Parameter Estimation of Normal Distribution from Sample Mean and Extreme Values

Tomoki Matsumoto

REVIEW 4 major objections 5 minor 10 references

Bayesian Parameter Estimation of Normal Distribution from Sample Mean and Extreme Values

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that a data-augmented Gibbs sampler estimates the mean and variance of a normal distribution from the sample mean, minimum, maximum, and sample size, with accuracy comparable to the theoretical order-statistic likelihood.

desk verdict The idea is squarely on target, but the sampler never actually conditions on the observed sample mean, so the central claim fails as written. read the letter →

arxiv 2411.13131 v1 pith:BST3ZP2I submitted 2024-11-20 stat.ME

Tomoki Matsumoto This is my paper · ORCID

classification stat.ME MSC 62F1562G30

keywords BayesianestimationGibbssamplerDataaugmentationSummarystatisticsExtremevaluesTruncatednormaldistributionPosteriorinference

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Government and survey reports often publish only the sample mean, minimum, maximum, and sample size, leaving raw data unavailable. This paper proposes a Bayesian Gibbs sampler that treats all intermediate values as missing data, samples them from a truncated normal distribution, and then updates the normal parameters using conjugate priors. The central claim, supported by simulation studies, is that the resulting posterior estimates of the mean and variance have accuracy comparable to sampling from the theoretical likelihood based on the observed extreme values. If correct, practitioners with only these summary statistics can obtain reliable Bayesian interval estimates without access to the full dataset.

What carries the argument

The central object is the zero-mean truncated normal sampler used in the data augmentation step. A truncated normal distribution restricts samples to an interval $[a,b]$, and the paper defines the mean parameter $\mu_{\mathrm{trunc}}$ so that $E[Z]=0$ for $Z \sim TN(\mu_{\mathrm{trunc}}, \sigma^2, a^*, b^*)$, namely the negative of the bias term involving the normal density and cumulative distribution functions evaluated at the adjusted boundaries $a^* = x_{(1)} - \bar{x}_{\mathrm{adj}}$ and $b^* = x_{(n)} - \bar{x}_{\mathrm{adj}}$. This correction forces the sampled intermediate values to have mean equal to the observed sample mean after adding back $\bar{x}_{\mathrm{adj}}$, preserving the summary statistic while enabling standard conjugate Gibbs updates for $\mu$ and $\sigma^2$. The paper also justifies treating intermediate order statistics as independent conditional on the observed extremes through a bijective ordering function.

What would settle it

Run a simulation that, for a fixed $\sigma$ and observed boundaries $a^*, b^*$, solves for $\mu_{\mathrm{trunc}}$ and checks whether the mean of the truncated normal actually equals zero; if no solution exists or the sampled intermediate values fail to average to the observed sample mean across many Gibbs iterations, the data augmentation step is invalid.

Watch

Extended reading notes

Core claim

The paper's central discovery is a data augmentation identity: by centering the observed boundaries with the adjusted mean of the intermediate values, defining the truncated normal's mean parameter as the bias correction $\mu_{\mathrm{trunc}}$ that makes the expected augmented value zero, and adding the adjusted mean back after sampling, the augmented dataset has expectation equal to the observed sample mean while every sampled value lies between the observed minimum and maximum. This makes the complete-data conjugate update valid, so Gibbs sampling over the augmented complete data yields posterior distributions for $\mu$ and $\sigma^2$ from only $\bar{x}$, $x_{(1)}$, $x_{(n)}$, and $n$. Simulation-based comparison with the theoretical order-statistic likelihood shows that the two approaches produce closely aligned RMSE values across sample sizes.

Load-bearing premise

The method must solve an implicit equation for the mean of the truncated normal so that the augmented data average to the observed sample mean, and the paper does not show this equation has a unique solution or how to solve it.

Editorial extensions

If this is right

Practitioners with only summary statistics can use direct conjugate Gibbs updates, avoiding more expensive sampling schemes such as Metropolis-Hastings or Hamiltonian Monte Carlo that are needed for the exact likelihood.
Bayesian credible intervals for the mean and variance can be produced from survey reports that publish only the sample mean, minimum, maximum, and sample size.
Estimation accuracy improves with sample size, and the simulations indicate that reasonable estimates are obtained even at moderate sample sizes when prior parameters are suitably specified.
The augmented-dataset construction makes the complete-data posterior updates valid, so the sampler produces a posterior for the original parameters rather than an ad hoc point estimate.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not prove that the implicit equation defining $\mu_{\mathrm{trunc}}$ has a solution; a follow-up could analyze existence and uniqueness for fixed boundaries and variance, and propose a fixed-point iteration or closed-form approximation.
The independence of intermediate order statistics conditional on the extremes is invoked via an ordering bijection; a testable extension is to compare the sampler's posterior against the exact joint distribution of order statistics for small $n$.
The same bias-correction construction could be adapted to other location-scale families, such as the log-normal, by matching the conditional mean of the truncated base distribution to the observed mean, though the paper does not explore this direction.
A practical test of the method's calibration would be to check whether posterior credible intervals achieve their nominal coverage on datasets with known parameters; the reported RMSE comparison does not assess interval calibration directly.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Tomoki, quick take on Matsumoto 2411.13131.

The new idea is real: mean/min/max/n summaries appear in government surveys, and the existing tools are direct formulas or max-only estimators. Treating the middle order statistics as missing data and imputing them from a truncated normal inside a Gibbs sampler is a sensible Bayesian setup that hasn't been done in the cited prior work. The paper is written clearly, and the conjugate-prior steps are each simple.

The problem is that the central data augmentation step doesn't do what it claims. The paper wants the augmented dataset's mean to equal the observed xbar. What it actually does is pick mu_trunc so that each truncated-normal draw has expected value zero, then adds xbar_adj. That fixes each draw's expected value, not the realized mean of the n-2 imputed observations. Independent draws fluctuate around zero, so the augmented sample mean will not equal xbar; the observed mean is never conditioned on. Steps 3 and 4 are therefore not the full conditionals of the stated posterior, and the sampler's stationary distribution is not the target. The implicit equation for mu_trunc is a secondary issue -- mu appears inside phi and Phi -- but the expectation-only centering would be fatal even with a perfectly solved mu_trunc.

The validation does not compensate. Equation (1) is an approximate likelihood the authors constructed themselves, so agreement between their Gibbs sampler and a Stan fit of (1) shows two heuristics roughly agree, not that either matches the true posterior. The simulation reporting is also thin: priors are not fully specified in the text, there are no error bars on the RMSE, and no code or data are provided.

There is a salvageable kernel. The problem is well-defined, and a correct version would impute the intermediate values jointly conditional on their sum constraint (a constrained truncated normal, or HMC within Gibbs). That is more involved than the current paper. I would not accept this version; the central claim fails as written. But the idea deserves more than a desk reject -- a serious referee should pin down the exact flaw and perhaps point toward the solution.

My call: reject with an invitation to revise if the augmentation is fixed and the benchmark is a true likelihood or the full-data posterior. A good reading-group example of a plausible but incorrect data augmentation.

Referee Report

4 major / 5 minor

Summary. The paper proposes a Bayesian data-augmentation method, implemented as a Gibbs sampler, for estimating the mean and variance of a normal distribution when only the sample mean, minimum, maximum, and sample size are available. Intermediate order statistics are treated as missing and sampled from a truncated normal distribution whose location parameter is adjusted in an attempt to preserve the observed sample mean; then μ and σ² are updated from conjugate conditional posteriors. Simulation studies compare the sampler's RMSE with an RStan implementation of an approximate likelihood based on the minimum, maximum, and adjusted mean.

Significance. The problem is practically important: many survey and government reports provide only these summary statistics, and a valid Bayesian procedure would be useful for meta-analysis and secondary data analysis. The manuscript is clearly written and the simulation setup is transparent. However, the central claim—that the proposed Gibbs sampler yields estimates comparable to theoretical expectations—is not supported because the data-augmentation step does not condition on the observed sample mean in the required probabilistic sense. The sampler does not appear to target a well-defined posterior for (μ, σ²) given the observed summaries, and the validation against an approximate likelihood does not remedy this. If the augmentation were corrected, the approach could be valuable, but as presented the core contribution is not established.

major comments (4)

[Section 2.1, definition of μ_trunc] The displayed formula for μ_trunc is an implicit equation, not a closed-form solution. If Z ~ TN(μ_t, σ², a*, b*), setting E[Z]=0 gives μ_t = -σ[φ((a*-μ_t)/σ) - φ((b*-μ_t)/σ)]/[Φ((b*-μ_t)/σ) - Φ((a*-μ_t)/σ)], so the unknown mean appears on both sides through φ and Φ. The paper writes the right-hand side using φ(a*|μ,σ²) and Φ(·|μ,σ²), where μ is the model mean rather than μ_t, which is at best ambiguous and at worst incorrect. The manuscript does not prove existence or uniqueness of a solution, nor does it provide a numerical procedure for solving the equation. Since Step 2 depends on this quantity, the sampler is not implementable as stated.
[Section 2.1, Step 2 (data augmentation)] Even if μ_trunc were computed correctly, independent draws from TN(μ_trunc, σ², a*, b*) followed by adding xbar_adj do not enforce the observed sample mean xbar. The conditional distribution of the intermediate values given the sufficient statistics (xbar, x_(1), x_(n)) is the joint truncated-normal density restricted to the hyperplane (1/n)Σx_i = xbar, i.e., Σ_{j=2}^{n-1} x_(j) = n·xbar - x_(1) - x_(n). Independent draws have expected sample mean xbar but almost surely a different realized mean. Therefore the augmented dataset x* does not have sample mean xbar, and Steps 3 and 4 are not the full conditional distributions of any joint posterior for (μ, σ²) given the observed summaries. The sampler's stationary distribution is not the stated target posterior.
[Section 2.1, Step 3] The conditional posterior used for μ, N((μ0/τ0² + n·xbar/σ²)/(1/τ0² + n/σ²), 1/(1/τ0² + n/σ²)), is the posterior conditional on the sufficient statistic xbar as if it were the mean of n iid observations from N(μ,σ²). But in a valid data-augmentation Gibbs sampler, the complete-data sufficient statistics must be computed from the augmented data x*, and the mean of x* is not xbar. Thus the algorithm conditions on the observed sample mean twice—once in the construction of μ_trunc and once in Step 3—without a well-defined joint distribution that has both as conditional expectations. This internal inconsistency reinforces that the sampler does not target the posterior claimed in the abstract.
[Section 3, validation against Eq. (1)] The comparison with RStan based on Eq. (1) does not validate the Bayesian correctness of the proposed method. Equation (1) is introduced as an approximation: the text states that using a truncated normal for xbar_adj 'will be more practical' because the exact variance is hard to compute due to correlations among xbar_adj, X_(1), and X_(n). Agreement between the proposed sampler and an approximate likelihood shows only that two heuristics behave similarly. The paper does not report coverage properties, nor compare with the full-data posterior or an exact likelihood, so the claim of 'estimation accuracy comparable to theoretical expectations' is not supported by the evidence presented.

minor comments (5)

[Section 2.1, Step 2] The number of intermediate values is inconsistent: the text writes 'z*_(j), j = 1, . . . , n−1' after defining z* for j = 2, . . . , n−1. Since x_(1) and x_(n) are observed, there are n−2 intermediate values, so the index set should be j = 2, . . . , n−1 consistently.
[Section 2.1, truncated normal formula] The notation φ(a*|μ,σ²) and Φ(·|μ,σ²) is nonstandard for the truncated-normal expectation formula; the usual expression uses standardized bounds (a*−μ)/σ and (b*−μ)/σ. Clarifying this would also expose the implicit nature of the equation for μ_trunc.
[Section 3] The prior hyperparameters (μ0, τ², α0, β0) are described only as 'detailed settings are shown in the figures below'; they should be listed explicitly in the text or in a table for reproducibility.
[Abstract and Conclusion] The phrase 'comparable to theoretical expectations' overstates the evidence, because the 'theoretical expectations' are themselves based on the approximate likelihood (1), not on an exact posterior or known theoretical accuracy.
[Figure 2 caption] The caption refers to 'theoretical distribution using RStan,' which is misleading; the bottom panel is an approximation based on Eq. (1). The caption should state that explicitly.

Circularity Check

1 steps flagged · score 6.0 of 10

The truncation-mean correction is defined through the target parameter mu itself; the zero-mean augmented-data claim is assumed rather than derived.

self definitional [Section 2.1 (Gibbs Sampler with Data Augmentation), unnumbered equation defining mu_trunc after the E[Z] formula]
"Setting E[Z] = 0 and solving for mu yields the mean parameter required for sampling with zero mean samples. We define it as mu_trunc, the second term of the right-hand side times -1: mu_trunc = -sigma [phi(a* | mu, sigma^2) - phi(b* | mu, sigma^2)] / [Phi(b* | mu, sigma^2) - Phi(a* | mu, sigma^2)]."

Solving E[Z]=0 gives mu = -sigma[phi(a*|mu,sigma^2)-phi(b*|mu,sigma^2)]/[Phi(b*|mu,sigma^2)-Phi(a*|mu,sigma^2)]. The paper defines mu_trunc as the right-hand side of this equation, but that right-hand side still contains mu. Thus mu_trunc is not an explicit solution; it is an implicit self-consistency condition. Since Step 2 uses mu_trunc to draw the augmented values that then determine the posterior draw of mu in Step 3, the construction builds the target parameter into the definition of the sampler's own input. The claimed zero-mean property is an assumption, not a derived consequence.

full rationale

The only load-bearing circularity is the definition of mu_trunc in Section 2.1: the equation used to define it contains the target parameter mu on both sides. The paper's assertion that this ensures the augmented dataset has mean equal to the observed sample mean is therefore not derived; it presupposes a fixed point that is neither proved nor solved. Independent draws from TN(mu_trunc,...) plus xbar_adj also match only the first moment, not the actual sum constraint, so the sampler does not condition on xbar as a sufficient statistic. This is a substantive correctness gap in the central method. However, the paper's final claim of estimation accuracy is not forced by construction: the RMSE comparison against RStan's fit of the paper's approximate likelihood (Eq. 1) is an external numerical benchmark, and no self-citation is used to support the load-bearing argument. Thus the circularity is partial and localized to the derivation of the augmentation step, not a full identity between input and output.

Assumptions & free parameters 4 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard properties of order statistics and truncated normals, plus two ad hoc modeling choices: the approximate likelihood in Equation (1) and the self-referential definition of mu_trunc. The only hand-chosen numbers are the prior hyperparameters, which are not specified in the text.

free parameters (4)

mu0
Prior mean for mu. Values are said to be 'shown in the figures below' but are not specified in the text, so the simulation setup is not fully reproducible.
tau2_0
Prior variance for mu. Same issue: not specified in the text, only in deferred figures.
alpha0
Shape parameter of the inverse-gamma prior for sigma squared. Not specified in the text.
beta0
Scale parameter of the inverse-gamma prior for sigma squared. Not specified in the text.

assumptions (3)

standard math Conditional on the minimum and maximum, the remaining n-2 values are independent samples from the normal distribution truncated to the observed range.
Used in Section 2.1 to justify treating the intermediate values as independent in the augmented data. This is a standard property of i.i.d. samples, though the paper's justification via a 'bijective ordering function' is informal.
ad hoc to paper The adjusted mean of the intermediate values, given the overall mean, min, and max, can be modeled as a single observation from a truncated normal distribution.
Used in Equation (1) to construct the 'theoretical' likelihood for comparison. The adjusted mean is actually an average of n-2 truncated normals, not a truncated normal itself; this approximation is not justified.
ad hoc to paper The equation defining mu_trunc has a solution that makes the expected value of the truncated samples zero.
The paper presents the formula without proving existence or uniqueness, or describing a numerical root-finding procedure. This is load-bearing for the sampler to preserve the observed sample mean.

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Cite this review

Pith. "Pith review of Bayesian Parameter Estimation of Normal Distribution from Sample Mean and Extreme Values." pith.science (2026). https://pith.science/paper/BST3ZP2I

@misc{pith2026241113131,
  author       = {Pith},
  title        = {Pith review of: Bayesian Parameter Estimation of Normal Distribution from Sample Mean and Extreme Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BST3ZP2I}},
  note         = {Machine review of arXiv:2411.13131}
}

read the original abstract

This paper proposes a Bayesian method for estimating the parameters of a normal distribution when only limited summary statistics (sample mean, minimum, maximum, and sample size) are available. To estimate the parameters of a normal distribution, we introduce a data augmentation approach using the Gibbs sampler, where intermediate values are treated as missing values and samples from a truncated normal distribution conditional on the observed sample mean, minimum, and maximum values. Through simulation studies, we demonstrate that our method achieves estimation accuracy comparable to theoretical expectations.

Figures

Figures reproduced from arXiv: 2411.13131 by the authors.

**Figure 2.** RMSE comparison between the proposed method and theoretical distributions. Results from 20 simu [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

10 extracted references · 7 canonical work pages

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" write newline "" before.all 'output.state := FUNCTION fin.entry bracket.state close.brackets = "]" * 'skip if add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not...

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

write newline

" write newline "" before.all 'output.state := FUNCTION fin.entry bracket.state close.brackets = "]" * 'skip if add.period write newline FUNCTION new.block output.state before.all = 'skip after.block 'output.state := if FUNCTION new.sentence output.state after.block = 'skip output.state before.all = 'skip after.sentence 'output.state := if if FUNCTION not...

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Capaldi, A. & Kolba, T. N. (2019). Using the sample maximum to estimate the parameters of the underlying distribution. PloS one , 14 (4), e0215529, https://doi.org/10.1371/journal.pone.0215529

work page doi:10.1371/journal.pone.0215529 2019

[4] [4]

& Nagaraja, H

David, H. & Nagaraja, H. (2003). Order Statistics . Wiley-Interscience, Hoboken, T hird edition https://doi.org/10.1002/0471722162

doi:10.1002/0471722162 2003

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B., Stern, H

Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis . Texts in statistical science. CRC Press, T hird edition https://doi.org/10.1201/b16018

doi:10.1201/b16018 2013

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P., Djulbegovic, B., & Hozo, I

Hozo, S. P., Djulbegovic, B., & Hozo, I. (2005). Estimating the mean and variance from the median, range, and the size of a sample. BMC medical research methodology , 5 , 13, https://doi.org/10.1186/1471-2288-5-13

work page doi:10.1186/1471-2288-5-13 2005

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Ministry of Health, Labour and Welfare (2016). Report on Verification of the Fee Schedule Revision for Disability Welfare Services in Fiscal Year 2015 . Government report, Ministry of Health, Labour and Welfare. In Japanese https://www.mhlw.go.jp/content/12200000/0000178200.pdf

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