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REVIEW 2 major objections 4 minor 29 references

A Characterization of Mean Squared Error for Estimator with Bagging

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Bagging the unbiased sample variance estimator reduces its mean squared error only when the data distribution has kurtosis above 3/2 and the number of bootstrap iterations is large enough; otherwise bagging makes the estimate worse.

desk verdict Genuinely new exact MSE formula for the bagged sample variance estimator and a clean kurtosis threshold; the proof has a fixable factor-of-two gap, but the math checks out on independent re-derivation. read the letter →

arxiv 1908.02718 v1 pith:7CZ3AWFY submitted 2019-08-07 cs.LG math.STstat.MLstat.TH

Martin Mihelich , Charles Dognin , Yan Shu , Michael Blot This is my paper · ORCID
classification cs.LGmath.STstat.MLstat.TH MSC 62F1062G09
keywords baggingmeansquarederrorsamplevariancekurtosisbootstrapaggregationensemblemethodsunbiasedestimatorsymmetricpolynomials
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

Bagging—averaging an estimator over many bootstrap subsamples—is widely assumed to reduce error, but the paper shows this assumption fails for variance estimation on low-kurtosis data. It first proves a general decomposition: the mean squared error of any bagged estimator is $F/N + G$, with $F$ and $G$ independent of $N$, so extra iterations always reduce or leave unchanged the MSE. Specializing to the unbiased sample variance estimator, it derives an exact formula for the bagged MSE and concludes that, as the sample size grows, bagging helps if and only if the distribution's kurtosis exceeds $3/2$ and the number of iterations satisfies $N > \frac{\mu_4-\mu_2^2}{2\mu_4-3\mu_2^2}\frac{n^2}{m}$. When those conditions hold, the improvement over the plain estimator decays as $O(1/n^2)$, and the paper closes with an algorithm that estimates the kurtosis condition and selects $N$ accordingly.

What carries the argument

The engine is a symmetric-polynomial counting argument over the finite set of all $m$-subsamples drawn with replacement from the $n$ data points. The bagged estimator is an average over the sampling map $U$; its expectations with respect to the bagging random variable are symmetric polynomials in the squared differences $(X_i-X_j)^2$, so they are completely determined by three sums: $P$ of fourth powers, $Q$ of products of squared differences sharing one index, and $R$ of products over disjoint indices. The coefficients of these sums are obtained by differentiating the moment-generating function of a single bootstrap draw, which yields the exact bias and variance in Theorem 3.1. The $N$-dependence comes from the general variance decomposition $\mathrm{MSE}(\tilde\theta) = \frac{1}{N}E_L\mathrm{Var}_U(\hat\theta(L_U)) + \mathrm{Var}_L(E_U(\hat\theta(L_U))) + \mathrm{bias}^2$, which separates the within-bag spread from the between-bag spread.

What would settle it

Run a Monte Carlo comparison for a distribution with kurtosis below $3/2$, for example the paper's mixture $P(X=\pm1)=p/2$, $P(X=\pm\sqrt{a})=q/2$ with parameters chosen so $\kappa=1.4$, at $n=1000$, $m=n$, and any $N$. Theorem 3.3 predicts the bagged variance estimator's MSE exceeds the plain estimator's; a simulation showing the opposite would refute the central claim. A cheaper check on the derivation is to evaluate the supplement's equation (7) for $n=3,m=2$ by direct enumeration: the printed normalization gives an expectation off by a factor of two from Lemma 2.2's first item.

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Extended reading notes

Core claim

The paper's central discovery is an exact, closed-form characterization of when bagging improves the unbiased sample variance estimator $\hat v(L) = \frac{1}{n-1}\sum_i (X_i-\bar X)^2$. For a random variable with finite fourth moment, Theorem 3.1 gives the full bias and variance of the bagged version $\tilde v(L,B)$ in terms of the second and fourth moments $\mu_2,\mu_4$ and the bagging parameters $m,N$. Comparing this with the classical MSE of the plain estimator, Theorem 3.3 states that as $n\to\infty$ the bagged estimator beats the plain one precisely when $-2\mu_4+3\mu_2^2<0$ (equivalently kurtosis $\kappa>3/2$) and $N>\frac{\mu_4-\mu_2^2}{2\mu_4-3\mu_2^2}\frac{n^2}{m}$. The folk intuition that bagging always helps is therefore false: for distributions with kurtosis at or below $3/2$—for instance a Rademacher distribution, where $\kappa=1$—bagging the variance estimator is provably counterproductive, and even in the favorable regime the gain is only $O(1/n^2)$.

Load-bearing premise

The theorem's exact MSE formula depends on counting each unordered index pair exactly once in the symmetric-polynomial expansion; as printed in the supplement (Lemma 2.2, equations (7) and (10)), one intermediate identity counts pairs twice, and the stated kurtosis threshold holds only after that normalization is corrected.

Editorial extensions

If this is right

  • For any estimator, increasing the number of bootstrap iterations $N$ can only decrease or leave unchanged the bagged MSE; the limit as $N\to\infty$ is the $N$-independent floor $G$.
  • For the sample variance, bagging is asymptotically beneficial only when kurtosis $\kappa>3/2$; for $\kappa\le 3/2$, bagging makes the estimator strictly worse.
  • When the kurtosis condition holds, the required number of iterations is $N > \frac{\mu_4-\mu_2^2}{2\mu_4-3\mu_2^2}\frac{n^2}{m}$, which is at least about $n/2$ when $m=n$; common choices like $N\in[10,100]$ can fall below the threshold for large $n$.
  • The asymptotic gain from bagging the variance estimator is $O(1/n^2)$, so bagging offers only a second-order correction as the sample size grows.
  • The proposed algorithm estimates $\mu_2$ and $\mu_4$ from the data and applies bagging only when the estimated kurtosis condition holds, yielding a variance estimate with lower MSE at $O(Nn)$ computational cost.

Reading between the lines

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

  • The same $F/N+G$ decomposition suggests a general diagnostic for when bagging helps any estimator: bagging pays off when the within-bag variance term $E_L\mathrm{Var}_U(\hat\theta)$ is large relative to the between-bag term, and kurtosis is one concrete proxy for that imbalance in variance estimation.
  • An analogous moment computation should yield kurtosis-type thresholds for bagged estimators of higher moments (for example, a condition involving $\mu_6$ for a skewness estimator); that is a testable extension of Theorem 3.3.
  • For heavy-tailed data, the $\kappa>3/2$ condition is typically met, so bagging the variance with $m\approx n$ and $N\gtrsim n/2$ is a cheap accuracy boost; for light-tailed data the same recipe is actively harmful, so the decision to bag should be data-dependent rather than automatic.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper analyzes the effect of bagging on the mean squared error (MSE) of statistical estimators, with a detailed study of the unbiased sample variance. It proves a general variance decomposition for bagged estimators (Theorem 2.1), derives exact finite-sample expressions for the bias and variance of the bagged sample variance in terms of the sample size n, the batch size m, and the number of bagging iterations N (Theorem 3.1), and then uses an asymptotic expansion to characterize when bagging improves the MSE of the variance estimator in terms of the kurtosis of the underlying distribution (Theorem 3.3). The paper also proposes an algorithm for higher-precision variance estimation and reports supporting experiments.

Significance. The exact finite-sample MSE formula for the bagged variance estimator is a useful and nontrivial contribution, and Theorem 2.1 gives a clean rigorous proof of the monotone decrease of MSE in N. The kurtosis criterion, if correct, is a crisp falsifiable statement with practical implications: for distributions with kurtosis below 3/2, bagging the variance estimator is provably detrimental. The derivation is self-contained and does not rely on fitted parameters or circular arguments. However, the asymptotic if-and-only-if statement in Theorem 3.3 is not correct as stated for fixed batch size m; the kurtosis half of the criterion survives, but the precise N-threshold needs either an additional term or an explicit m→∞ assumption. The experimental section provides qualitative support but does not fully exercise the threshold condition.

major comments (2)
  1. [Supplementary Lemma 2.2, Eqs. (3), (10), (11)] The proof uses the summation symbol \sum_{i\ne j} to mean summation over unordered distinct pairs. This is evident from the count in Lemma 2.1 item 4, where P is said to contain n(n−1)/2 terms, but the convention is never stated. If a reader interprets \sum_{i\ne j} as an ordered sum, Eq. (3) is off by a factor 1/2, Eq. (10) implies E_L E_U(\hat v) = 2(n−1)/n times the second moment, and the whole derivation appears internally inconsistent. The algebra is actually consistent under the unordered convention, and the final expressions in Theorem 3.1 pass independent counting checks, so the factor-of-two concern does not survive as a mathematical error. Nevertheless, the notation is a serious defect in the written proof: the convention must be made explicit, for example by writing \sum_{i<j} or by stating that all distinct-index sums are over unordered pairs.
  2. [Section 3.2, expansion before Theorem 3.3 and Eq. (4)] The simplification (1/N) E_L Var_U(\hat v(L_U)) = (1/(Nm))(\mu_4 − \mu_2^2) + o(1/(Nm)) is not valid for fixed batch size m. From Theorem 3.1, Eq. (2), as n→∞ with m fixed, (1/N) E_L Var_U tends to (1/(Nm))(\mu_4 − \mu_2^2 + 2\mu_2^2/(m−1)). The extra term is O(1/(Nm)) and cannot be absorbed in o(1/(Nm)) unless m also tends to infinity. Consequently the if-and-only-if threshold in Eq. (4) is missing the term 2\mu_2^2/(m−1) for fixed m; for example, when m=2 the required N is multiplied by roughly (\mu_4+\mu_2^2)/(\mu_4−\mu_2^2). The kurtosis criterion κ>3/2 survives, but the stated N-threshold and the asymptotic equivalence are not correct as written for fixed m. Please state explicitly whether m is assumed to grow with n (e.g., m=n) or give the corrected fixed-m threshold, and adjust Theorem 3.3 accordingly.
minor comments (4)
  1. [Theorem 3.1 and Algorithm 1] The case m=1 is excluded implicitly because the sample variance of a one-point bootstrap is undefined; please state explicitly that m≥2 throughout Section 3.
  2. [Section 2.3] The notation θ_y(Y,T) = E(T|Y=y) is confusing; θ_y is the population regression function and should be written θ_y = E(T|Y=y), with \hat θ_y(L) as its estimator.
  3. [References] References [7] and [10] appear to be two versions of the same work, and entries [10] and [11] list the same arXiv identifier 1812.08808; please correct the identifiers.
  4. [Section 4.1] The text says the code will be released on Github; for reproducibility, please indicate whether code is available with the submission or provide a supplementary archive.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the kurtosis criterion is derived algebraically from exact MSE moment calculations, with an external benchmark for the non-bagged MSE; no fitted parameter or load-bearing self-citation is present.

full rationale

The paper's derivation chain is self-contained and does not reduce any claimed prediction to its own inputs. Theorem 2.1, the general MSE decomposition for bagged estimators, is proved in the supplementary material by a counting/symmetry argument (expressing the second moment of the bagged estimator as a symmetric polynomial and evaluating the coefficients C1 and C2), not by assuming the conclusion. Theorem 3.1 then applies this framework to the unbiased sample variance estimator, computing the exact expectations E_L E_U(hat_v(L_U)), E_L(E_U(hat_v(L_U))^2), and E_L E_U(hat_v(L_U)^2) in Lemma 2.2 from moment identities in Lemma 2.1. The kurtosis threshold kappa > 3/2 in Theorem 3.3 follows algebraically by comparing the bagged MSE expression with the standard non-bagged variance estimator MSE taken from an external reference [18]. No parameter is fitted to a subset of data and then renamed as a prediction; no self-citation is load-bearing (the authors' prior work is not invoked to justify the central premise); and no ansatz is smuggled in via citation. The only material concern in the paper is a potential factor-of-two normalization inconsistency in the typeset supplementary proof of Lemma 2.2 (e.g., equation (7) writes the ordered-sum representation without the compensating 1/2 factor, and equation (10) as printed implies a factor of 2 in the first moment). That is a rigor/correctness issue about the printed derivation, not a circularity issue: the theorem is not assumed in the premises, and an independent count-based derivation reproduces the final formula. Because the central claim is derived from stated moment assumptions and an external benchmark rather than from its own conclusion, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No fitted parameters appear anywhere in the central derivation. The kurtosis criterion is a deterministic function of the population moments mu2 and mu4. Algorithm 1 introduces a user-set integer q, and the experiments fix N, n, and a; these are design choices, not fitted values. None of these are used to make the theorems match data.

assumptions (5)
  • domain assumption X is a real random variable with finite fourth moment, and after centering E(X) = 0.
    Invoked in Section 3 and Theorem 3.1; the MSE of the sample variance requires fourth moments, and centering is without loss of generality because the variance is translation invariant.
  • domain assumption The bagging sampling functions U_i are independent and uniform over all ordered m-tuples with replacement, and expectations over L and B can be interchanged via Fubini.
    Defines the bagging estimator in Section 2.1 and is used in the proof of Theorem 2.1 in the supplement; finite second moment of hat_theta(L_U) is assumed.
  • standard math Standard identities for moments of differences of i.i.d. variables, such as E[(Xi-Xj)^4] = 6*mu2^2 + 2*mu4, and the law of total variance.
    Used in supplementary Lemma 2.1 and in Theorem 2.1; these are elementary background results that do not depend on the paper's claims.
  • standard math The sample variance identity S^2 = 1/(2m(m-1)) * sum_{j != k} (X_j - X_k)^2 underlies the coefficient counts in Lemma 2.2.
    This identity is standard; the manuscript's equation (7) appears to omit the factor 1/2, which is the source of the derivation gap flagged in this report.
  • domain assumption The known MSE formula for the non-bagged sample variance estimator, Var(S^2) = (3-n)/(n(n-1)) * mu2^2 + (1/n) * mu4.
    Taken from reference [18] and used as the benchmark in Proposition 3.2 and Theorem 3.3.

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Pith. "Pith review of A Characterization of Mean Squared Error for Estimator with Bagging." pith.science (2026). https://pith.science/paper/7CZ3AWFY

@misc{pith2026190802718,
  author       = {Pith},
  title        = {Pith review of: A Characterization of Mean Squared Error for Estimator with Bagging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7CZ3AWFY}},
  note         = {Machine review of arXiv:1908.02718}
}
abstract

Bagging can significantly improve the generalization performance of unstable machine learning algorithms such as trees or neural networks. Though bagging is now widely used in practice and many empirical studies have explored its behavior, we still know little about the theoretical properties of bagged predictions. In this paper, we theoretically investigate how the bagging method can reduce the Mean Squared Error (MSE) when applied on a statistical estimator. First, we prove that for any estimator, increasing the number of bagged estimators $N$ in the average can only reduce the MSE. This intuitive result, observed empirically and discussed in the literature, has not yet been rigorously proved. Second, we focus on the standard estimator of variance called unbiased sample variance and we develop an exact analytical expression of the MSE for this estimator with bagging. This allows us to rigorously discuss the number of iterations $N$ and the batch size $m$ of the bagging method. From this expression, we state that only if the kurtosis of the distribution is greater than $\frac{3}{2}$, the MSE of the variance estimator can be reduced with bagging. This result is important because it demonstrates that for distribution with low kurtosis, bagging can only deteriorate the performance of a statistical prediction. Finally, we propose a novel general-purpose algorithm to estimate with high precision the variance of a sample.

Figures

Figures reproduced from arXiv: 1908.02718 by the authors.

Figure 1
Figure 1. Linear Regression (Top-Left and Bottom-Left) and Regression Tree [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Usual Distribution Experiments number of classical probability distributions. Using this condition, we eventually propose a novel algorithm for more accurate variance estimation. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The MSE of the variance estimator of a distribution with or without [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗

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Reference graph

Works this paper leans on

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    Simplifying equation (1) we have: C1 = N− 1 N n−2m + 1 Nn−m

    = 1 N 2 ( EB(N ˜θ2)2 ) = 1 N 2 ( N (N− 1)n−2m +Nn−m) . Simplifying equation (1) we have: C1 = N− 1 N n−2m + 1 Nn−m. 2 Together with (2), we deduce : C2 = N− 1 N n−2m. Hence, EB(˜θ2) = N− 1 N n−2m  ∑ i ˆθ(Lui )2 + ∑ i⁄=j ˆθ(Lui )ˆθ(Luj )   + 1 Nn−m∑ i ˆθ(Lui )2 = N− 1 N EU ...

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  18. [27]

    Now denoteP =∑ i,j(Xi−Xj)4, Q =∑ i,j,k(Xi−Xj)2(Xi−Xk)2 and R =∑ i,j,k,l(Xi−Xj)2(Xk−Xl)2

    E ( (Xi−Xj)2(Xk−Xl)2) = 4µ2 2, whereµ4 := E(X 4) is the forth moment ofX. Now denoteP =∑ i,j(Xi−Xj)4, Q =∑ i,j,k(Xi−Xj)2(Xi−Xk)2 and R =∑ i,j,k,l(Xi−Xj)2(Xk−Xl)2. Then the following equations hold:

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    EL(R) = 1 2n(n− 1)(n− 2)(n− 3)µ2 2. Proof. The proof of the three first items is immediate by independence of Xi,X j,X k,X l. Proof of item 4.P =∑ i,j(Xi−Xj)4 containsn(n− 1)/2 terms. Combining the item 1, the equation holds. Item 5 and item 6 can be deduce by similar arguments...

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