Privacy noise may strengthen protection, but at high enough levels it can make reliable recovery of the underlying distribution mathematically impossible. In the model examined here, once the noise scale grows on the order of the square root of the sample size—or faster—no estimator can be uniformly consistent over all probability distributions on the bounded latent interval. In plain language, no method can guarantee that its Wasserstein-1 (W1) error, the gap between fitted and true distributions, will shrink across the entire class.
The setup is a one-dimensional bounded latent-distribution problem. Values θi are drawn from an unknown distribution g0, then released as θi plus independent Laplace noise with known scale b. The paper uses a nonparametric maximum likelihood estimator (NPMLE), a likelihood-based way to fit the unknown distribution without specifying its shape in advance.
The privacy dial is the noise scale. For latent values in [-a, a], the privacy budget is ε = 2a/b: increasing b lowers ε and gives stronger privacy, but also makes recovery harder. The study therefore treats privacy and accuracy as a linked trade-off.
Turning an unrestricted fit into a finite problem
One theoretical result makes the estimator much more manageable. Every support point of the fitted NPMLE—the locations where it places probability mass—is one of the observed values projected onto the latent interval. Instead of considering an unlimited range of possible locations, the calculation can work from this finite candidate set.
Once those candidate locations are fixed, the remaining task is to choose their weights. The reduced weight problem has a unique maximizer, and the paper computes it with an expectation-maximization (EM) algorithm. In that procedure, the E-step uses posterior responsibilities, the M-step averages them, and the observed log-likelihood rises monotonically.
The rates split recovery into two regimes
The first rate concerns the density of the privatized observations rather than the hidden distribution itself. Its Hellinger error—a way of comparing probability densities—is reported at the stochastic-order rate O_p((1 + a/b)^(1/4) exp(a/(2b)) n^(-3/8)). The authors present this as an asymptotic stochastic-order bound, not as a finite-sample confidence interval.
To move from the noisy density back to the latent distribution, the analysis uses a deconvolution inequality, meaning an inequality for statistically undoing the noise. It bounds latent W1 error with three ingredients: a smoothing term, a discrepancy between privatized distribution functions, and a high-frequency term that depends on the Laplace scale and multiplies the privatized density’s L1 error.
With the support radius a and noise scale b held fixed, the leading term in the latent W1 bound scales as n^(-3/16), multiplied by a logarithmic factor. This describes leading asymptotic behavior; the supplied analysis does not report the constants or finite-sample guarantees behind the expression.
The analysis then allows b to change with sample size. It says the NPMLE remains W1-consistent if b grows more slowly than n^(3/16)(log n)^(-1/2), meaning the W1 error tends to zero in that asymptotic regime. That is a sufficient condition, not a claim that the boundary is necessary.
At the other end, if b is on the order of n^(1/2) or larger, uniformly consistent recovery over all distributions on [-a, a] is impossible for any estimator. The lower-bound argument says the worst-case probability of W1 error at least a is no smaller than 1/4 exp(-2a^2/c^2), which is strictly positive; c is the constant in the stated noise-growth assumption. The result is a worst-case statement that rules out a guarantee across the whole class.
Finite simulations echo the trade-off
The numerical experiments fixed a = 3, used sample sizes n of 50, 100, 200, 500, 1,000 and 2,000, and varied ε across 0.1, 0.2, 0.5, 1, 2, 5, 10 and 20. Each configuration was repeated 50 times across three simulated latent distributions.
Across those scenarios, W1 error generally fell as n and ε increased. The improvement flattened at larger ε, and the discrete case did not have a uniform advantage. The fitted NPMLE’s thresholded active support—the number of locations carrying more than a chosen minimum weight—also grew with n and ε. Plots suggested faster-than-logarithmic growth over the displayed range, but the authors did not treat that as a precise asymptotic law.
The boundary is still unresolved
The paper has not found the exact point where recovery becomes impossible. It leaves a gap between the n^(-3/16) rate that is sufficient for consistency and the n^(-1/2) rate identified as necessary, and it calls for sharper theory on how many support points the fitted NPMLE uses.
The conclusions are tied to the bounded, one-dimensional model and the listed finite simulation settings. The manuscript is identified as arXiv:2608.25997v1 and dated 26 Aug 2026.
Paper data and sources
Original title: Statistical Properties of Nonparametric MLE under Laplace Noise
Authors: Yifei Xiong, Nianqiao Phyllis Ju, Vinayak Rao
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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