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The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting
Open accessAbstract
Summary: Simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate Markov chain Monte Carlo sampling. We propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution. The new methodology provides both a method for simulation and an efficient estimator to hitherto intractable Gaussian integrals. We prove that the estimator has a rare vanishing relative error asymptotic property. Numerical experiments suggest that the scheme proposed is accurate in a wide range of set-ups for which competing estimation schemes fail. We give an application to exact independently and identically distributed data simulation from the Bayesian posterior of the probit regression model.
Overview
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TL;DR
The paper proposes a minimax exponential tilting method for exact iid simulation and estimation of Gaussian integrals under linear restrictions, improving on the Genz separation-of-variables (SOV) estimator. It proves vanishing relative error in tail asymptotic regimes and demonstrates high accuracy and acceptance rates in numerical examples up to about 100 dimensions.
Key findings
- The minimax-tilted (MET) estimator is unbiased and can achieve vanishing relative error under conditions in Theorems 4.1 and 4.2; otherwise it has bounded relative error.
- In Example I, at \(d=25\) the SOV estimator gave \(2.371\times10^{-48}\) with 33% relative error, while MET gave \(2.6847\times10^{-53}\) with 0.02% relative error; at \(d=50\), MET gave \(2.1364\times10^{-153}\) with 0.06% relative error and acceptance probability 0.95.
- In Example IV with \(A=[1,\infty]^{100}\), MET had median relative error 0.077% versus SOV median 26%, and the MET variance was typically more than \(10^5\) times smaller than SOV.
- For probit posterior simulation, the accept-reject scheme had acceptance rate \(1/217\), compared with naive acceptance rejection of \(O(10^{-146})\) and without minimax tilting \(O(10^{-16})\).
Methods
- The method exponentially tilts the Genz SOV importance density and selects the tilting parameter by solving the minimax saddle-point problem \(\inf_\mu \sup_{x\in C}\psi(x;\mu)\), using log-concavity and Karush-Kuhn-Tucker conditions; the optimal pair is found via Powell’s Dogleg method or a convex solver.
- Exact simulation uses accept-reject with instrumental density \(g(x;\mu^*)\) and envelope \(c=\exp(\psi(x^*;\mu^*))\); the estimator is \(\hat\ell=\exp(\psi(X;\mu^*))\), \(X\sim P_{\mu^*}\).
- Numerical experiments compare MET with SOV using randomized quasi Monte Carlo (Richtmyer sequence), including structured covariance matrices, random correlation matrices, and an extramarital-affairs probit model with \(m=601\) observations, \(k=7\) predictors, and 8000 iid posterior simulations.
Limitations
- The paper states that the method is not a panacea against the curse of dimensionality: acceptance probability eventually becomes a rare-event probability as \(d\to\infty\), and the bounded or vanishing relative error properties do not hold in the \(d\uparrow\infty\) regime; it generally works reliably for \(d\le 100\), and may sometimes work higher only if \(\ell\) does not decay too fast.
- The paper notes that the probit acceptance rate \(1/217\) could have been better and will deteriorate with increasing dimensionality, and that MET is not immune to numerical underflow and loss of precision.
- Evident from the text: MET was about 20% more costly than SOV in Example III, and the numerical comparisons focus mainly on SOV because specialized alternatives are not competitive beyond low dimensions.
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“Note that under the conditions of Theorem 4.2, the minimax estimator enjoys the bounded normal approximation property (Tuffin, 1999).”
§ The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting > 4 Theoretical Properties of Minimax Tilting
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- Not checked yetcites Tuffin, 1999 source
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“In contrast, the MET estimator is guaranteed to have better relative error than the one given in column six (worst err.).”
§ The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting > Example I (Fernández et al., 2007).
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“Unlike existing algorithms, the accept-reject sampler proposed in this article enjoys high acceptance rates in over one hundred dimensions, and takes about the same time as one cycle of Gibbs sampling.”
§ The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting > 1 Introduction
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“In an appropriate asymptotic tail regime, the minimax tilting yields an estimator with vanishing relative error (VRE) property (Kroese et al., 2011).”
§ The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting > 1 Introduction
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- Not checked yetcites Kroese et al., 2011 source
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“One of the goals of this article is to propose a new methodology, which not only yields an unbiased estimator orders of magnitude less variable than the Genz estimator, but also works reliably in cases where the Genz estimator and other alternatives fail to deliver meaningful estimates (e.g., relative error close to 100%) 1 .”
§ The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting > 1 Introduction
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Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
In the citing paper
“MET is currently the state-of-the-art in terms of achieving the highest accuracy for estimating MVN probabilities and the highest acceptance rate for sampling TMVN distributions.”
From Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities (2026) · cites this paper as Botev 2017
In the citing paper
“Since the Vecchia approximation amounts to a valid MVN density with (slightly) different covariance, Algorithm 1 enjoys the same analytical properties and efficacy in the variance reduction of the integrand as the original MET method (Botev 2017).”
From Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities (2026) · cites this paper as Botev 2017
In the citing paper
“This results in substantially higher accuracy for tail MVN probabilities than SOV, but it requires iterative optimization of the proposal density in the pre-processing stage at a per-iteration complexity of O(n 3 ); in practice, this optimization can be more expensive than the MC sampling for moderately large n (e.g., n > 1,000).”
From Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities (2026) · cites this paper as Botev 2017
In the citing paper
“This results in substantially higher accuracy for tail MVN probabilities than SOV, but it requires iterative optimization of the proposal density in the pre-processing stage at a per-iteration complexity of O(n 3 ); in practice, this optimization can be more expensive than the MC sampling for moderately large n (e.g., n > 1,000).”
From Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities (2026) · cites this paper as Botev 2017
In the citing paper
“We proposed a linear-complexity method for estimating multivariate normal (MVN) probabilities and sampling truncated multivariate normal (TMVN) distributions at the same convergence and acceptance rate as the cubic-complexity state-of-the-art minimax exponential tilting (MET) method”
From Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities (2026) · cites this paper as Botev 2017
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In the paper
“10 8.5483 × 10 -15 8.4591 × 10 -15 (0.8%) 8.556 × 10 -15 (0.01%) 2.1046 × 10 -14 0.03% 0.97”
§ The Normal Law Under Linear Restrictions: Simulation and Estimation via Minimax Tilting > Example I (Fernández et al., 2007).
Reported 0.97 · recomputed 0.406538
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Metadata
- Authors
- Botev, Z. I.
- Affiliations
- UNSW Sydney (AU)
- Corresponding author
- Botev, Z. I.
- ORCID
- Botev, Z. I.
- Venue
- Journal of the Royal Statistical Society Series B: Statistical Methodology
- Publisher
- Oxford University Press (OUP)
- Published
- 2016-02-18
- Volume
- 79
- Issue
- 1
- Pages
- 125-148
- ISSN
- 1369-7412 (ISSN-L)
- 1467-9868
- OpenAlex
- W2288838258
- Semantic Scholar
- 533bf3ad398b299cc120f17ce2bd7674adb308ef
- Cited by
- 247
- References
- 41
- Citations by year
- 2026: 24 · 2025: 35 · 2024: 19 · 2023: 31 · 2022: 27 · 2021: 27 · 2020: 22 · 2019: 25 · 2018: 16 · 2017: 13 · 2016: 6
- FWCI
- 14.36
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model (vor)
- Locations
- Journal of the Royal Statistical Society Series B (Statistical Methodology) (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- UNSWorks (University of New South Wales, Sydney, Australia) (submittedVersion, cc-by-nc-nd)
- UNSWorks (UNSW Sydney) (submittedVersion, cc-by-nc-nd)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Markov Chains and Monte Carlo Methods
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Distribution Estimation and Applications
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Keywords
- truncated multivariate normal distribution, exact simulation, Markov chain Monte Carlo alternatives, probit model
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