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MCMC Importance Sampling via Moreau-Yosida Envelopes

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Pith's one-line read Using the Moreau-Yosida envelope density as the importance distribution makes self-normalized importance sampling for non-differentiable posteriors asymptotically normal with finite covariance, and it can substantially beat proximal MCMC…

desk verdict Solid new IS estimator for non-smooth targets; the CLT is real, but the finite-variance guarantee rests on unverified tail conditions the paper itself flags. read the letter →

arxiv 2501.02228 v2 pith:SWE3KBZR submitted 2025-01-04 stat.CO stat.ME

Apratim Shukla , Dootika Vats , Eric C. Chi This is my paper · ORCID
classification stat.COstat.ME MSC 62F1565C0560J22
keywords Moreau-Yosidaenvelopeimportancesamplingnon-differentiableposteriorgeometricergodicityMALAHamiltonianMonteCarloasymptoticvarianceproximalMCMC
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

This paper proposes a new way to estimate expectations under non-differentiable Bayesian posteriors: draw samples from the Moreau-Yosida envelope density $\pi_\lambda(x) \propto e^{-\psi_\lambda(x)}$ instead of the target $\pi(x) \propto e^{-\psi(x)}$, and correct with the self-normalized weights $w_\lambda(x) = e^{-(\psi(x)-\psi_\lambda(x))}$. The authors prove that the resulting estimator $\sqrt{n}(\hat{\theta}^{\mathrm{MY}}_n - \theta)$ is asymptotically normal with finite covariance whenever the $\pi_\lambda$-invariant MALA or HMC chain is geometrically ergodic; the envelope inequality $\psi_\lambda \le \psi$ makes the weights uniformly bounded, so the variance is finite. They give sufficient conditions for that geometric ergodicity and show in several simulation settings that the smoothed importance sampler can be substantially more efficient than proximal MCMC methods that target $\pi$ directly. If the argument is right, practitioners with $\ell_1$, fused-lasso, or nuclear-norm priors gain a principled finite-variance alternative to proximal MALA and HMC.

What carries the argument

The object carrying the argument is the Moreau-Yosida envelope $\psi_\lambda(x) = \inf_y\{\psi(y) + \tfrac{1}{2\lambda}\|y-x\|^2\}$ and its density $\pi_\lambda(x) \propto e^{-\psi_\lambda(x)}$; the proximal map $\mathrm{prox}^\lambda_\psi(x) = \arg\min_y\{\psi(y) + \tfrac{1}{2\lambda}\|y-x\|^2\}$ supplies the gradient $\nabla \log \pi_\lambda(x) = (\mathrm{prox}^\lambda_\psi(x) - x)/\lambda$, which is Lipschitz and computable whenever the proximal map is. The envelope inequality $\psi_\lambda \le \psi$ gives the uniformly bounded weights $w_\lambda(x) \le 1$ that turn the Markov chain central limit theorem into the finite asymptotic covariance $\Xi$ via the delta method on the augmented statistic $S(x) = (\xi(x)w_\lambda(x), w_\lambda(x))$. Assumption 1 is the tail-growth condition on the proximal map that, through Lemma 1, makes the drift conditions for MALA and HMC tractable: for MALA it becomes the simple step-size rule $h \le 2\lambda$, and for HMC it verifies three gradient conditions from Livingstone et al. (2019).

What would settle it

Run MY-IS with a MALA chain on a target where $\limsup_{\|x\|\to\infty} \|\mathrm{prox}^\lambda_\psi(x)\|/\|x\| = 1$ (for instance, a heavy-tailed log-concave $\psi$ whose proximal map grows along a ridge) and check whether the batch-means estimate of $\Xi$ in (40) stays finite as $n$ grows; if the drift condition (24) fails, the estimator may remain consistent but its asymptotic variance can diverge, which would show the geometric-ergodicity assumption is doing the work.

Watch

Extended reading notes

Core claim

The central claim is that the Moreau-Yosida envelope density $\pi_\lambda(x) \propto e^{-\psi_\lambda(x)}$ is a provably safe importance distribution for a log-concave target $\pi(x) \propto e^{-\psi(x)}$ whose proximal map of $\psi$ can be computed. Because $\psi_\lambda$ lies below $\psi$, the unnormalized weight $w_\lambda(x) = e^{-(\psi(x)-\psi_\lambda(x))}$ satisfies $0 < w_\lambda(x) \le 1$, and the smoothed density both bounds the importance ratio $\pi/\pi_\lambda$ and gives gradient samplers a well-conditioned target. Theorem 2 shows that the self-normalized estimator (12) is asymptotically normal with the explicit covariance $\Xi$ in (14), assuming only a second moment under $\pi$ and a geometrically ergodic $\pi_\lambda$-chain; Theorem 3 shows that for a Gaussian target the variance-minimizing $\lambda$ lies in $[s_1/d, s_d/d]$, so the optimal smoothing decreases with dimension. The paper further proves (Theorems 5 and 7) that under the tail-growth Assumption 1, $\limsup_{\|x\|\to\infty} \|\mathrm{prox}^\lambda_\psi(x)\|/\|x\| < 1$, both MALA and HMC sampling from $\pi_\lambda$ are geometrically ergodic for suitable step sizes, so the finite-variance guarantee applies to the algorithms practitioners would run.

Load-bearing premise

The finite-variance guarantee holds only when the chain targeting the smoothed density is geometrically ergodic, which the paper verifies under a tail-growth condition on the proximal map, plus an extra tail condition for Hamiltonian Monte Carlo that is not established for all log-concave targets.

Editorial extensions

If this is right

  • For any non-differentiable log-concave posterior whose proximal map satisfies Assumption 1, practitioners can replace proximal MALA/HMC with $\pi_\lambda$-sampling plus weights and obtain Monte Carlo error bars from the explicit covariance estimator in (40); the paper shows this is often many times more efficient.
  • The same pipeline applies to differentiable targets with heavy or non-Lipschitz tails, where standard MALA and HMC are unreliable; the paper demonstrates this on the Poisson random-effects model, where even the Barker proposal is outperformed.
  • The tuning guideline $n_e/n \in [0.4, 0.8]$ gives a simple, self-checking way to balance smoothing against weight degradation, and Theorem 3's inverse-dimension scaling tells users in high dimensions to start with small $\lambda$.
  • Marginal quantiles and credible intervals for each coordinate follow directly from the Chen–Shao weighted order-statistic estimator, so the method is usable as an end-to-end Bayesian inference routine, not just for means.

Reading between the lines

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

  • If the finite-variance guarantee extends to non-log-concave targets via weak convexity, the same estimator could apply to multi-modal posteriors by choosing $\lambda$ to make $\pi_\lambda$ log-concave; the paper leaves this conditional on $\pi_\lambda$ being integrable.
  • The paper's Theorem 3 suggests a default choice $\lambda \propto 1/d$ in high dimensions, since the optimal smoothing shrinks with dimension; this could be turned into a pilot-free recipe before any MCMC run.
  • The weight-boundedness mechanism is general: any approximating density that lies below the target pointwise and is smooth enough for gradient samplers inherits the same finite-variance asymptotic normality, so the envelope trick could be applied to other families like spline or kernel-smoothed potentials.
  • A testable extension is to estimate the asymptotic covariance $\Xi$ online and stop when the batch-means estimate stabilizes, giving a principled stopping rule for proximal MCMC that the paper's Theorem 8 would justify.
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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 / 5 minor

Summary. The paper proposes an importance sampling framework for targets of the form π(x) ∝ e^{−ψ(x)} with convex, possibly non-smooth ψ, using the Moreau-Yosida envelope density πλ(x) ∝ e^{−ψλ(x)} as the importance distribution. The main theoretical result (Theorem 2) establishes asymptotic normality of the self-normalized estimator θ̂^{MY}_n under geometric ergodicity of the πλ-invariant Markov chain, with an explicit covariance expression. The paper derives sufficient conditions for geometric ergodicity of πλ-MALA (Theorem 5) and πλ-HMC (Theorem 7), provides a closed-form optimal λ for Gaussian targets in the iid case (Theorem 3), and gives a practical tuning rule based on the importance sampling effective sample size. Numerical experiments cover a toy product target, Bayesian trend filtering, nuclear-norm matrix denoising, and a Poisson random effects model, comparing MY-IS with P-MALA, P-HMC, and Barker's algorithm, with 100 replications each and efficiency gains reported.

Significance. The central idea is simple and appealing: because ψλ envelopes ψ from below, the importance weights are uniformly bounded, and the smoothness of πλ makes gradient-based MCMC applicable. If the geometric ergodicity conditions hold, Theorem 2 delivers a genuinely useful finite-covariance CLT for an estimator that can have substantially lower variance than proximal MCMC alternatives. The Gaussian optimal-λ derivation is elegant, the batch-means variance estimator is standard, and the numerical studies are extensive, with code provided. The main weakness is that the advertised 'finite asymptotic variance' guarantee is conditional on geometric ergodicity of the πλ-chain, and the paper's sufficient conditions for that geometric ergodicity are not fully verified for the HMC case (condition (29)) and are assumed for MALA (condition (23)). These gaps are load-bearing for the central claim, especially for the β>2 targets used in the numerical evaluation.

major comments (2)
  1. [Section 5.2 / Proof E.2] The HMC geometric ergodicity guarantee is not fully established for the classes used in the numerical studies. Theorem 7 is stated conditionally on condition (29), but the proof in Section E.2 verifies only condition (28) (via (53), (54), and (56)); Section 5.2 explicitly acknowledges that demonstrating (29) is challenging. Item 3 of Section 5.2 nevertheless asserts that for β>2 targets πλ-HMC is geometrically ergodic for sufficiently small ε because πλ has Gaussian tails, without verifying (29). Since Theorem 2's finite covariance conclusion requires geometric ergodicity, the HMC efficiency gains reported in Sections 6.3 and 6.4 are not backed by the paper's theorems for the β=4 class of targets; either supply a proof of (29) for the targets considered or explicitly present the HMC results as conditional on (29).
  2. [Section 5.1, Theorem 5 and item 3] The MALA geometric ergodicity theorem assumes the inward-convergence condition (23) without verifying it for πλ or for the E(β,γ) class. Item 3 then states that for β>2, πλ-MALA is geometrically ergodic as long as h ≤ 4λ; this conclusion does not follow from Theorem 5 unless (23) holds. If (23) fails, the chain may not be geometrically ergodic and Theorem 2 is not applicable. Please either verify (23) for the examples in Section 6 or label the item as conditional on (23).
minor comments (5)
  1. [Supplement F.1, Algorithm 4] The displayed acceptance probability α(x,y) is missing the 'min' operator; as written, it is not a proper probability.
  2. [Equation (22)] The indicator notation 1(1 ≤ β < 2) is nonstandard and could be confused with the function value; writing 1_{[1,2)}(β) would be clearer.
  3. [Section 4.1, Remark 1] The phrase 'all choices of λ yield a finite variance estimator' is stronger than what is proved, since Theorem 2 is conditional on geometric ergodicity of the πλ-chain; qualify this statement.
  4. [Section 6.2] The sentence 'For HMC, estimation of components are at least 25 times more efficient' should specify that the comparison is relative to P-HMC, as is done for the MALA case.
  5. [Equation (17)] The notation w^{(i)}_{(l)} is easy to misread; consider defining the ordered-sample weights explicitly for each component before displaying the quantile estimator.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the estimator and variance analysis follow standard importance-sampling and Markov-chain CLT arguments, and all self-citations point to published external results that do not assume the paper's conclusions.

full rationale

The paper's central estimator in Eq. (12) is a self-normalized importance sampling estimator with weights w_λ(x)=exp(−ψ(x))/exp(−ψ_λ(x)) defined in Eq. (11). Its asymptotic normality in Theorem 2 is proved in Supplement B via the standard Markov chain CLT and the delta method; the key weight bound follows from the defining property ψ_λ≤ψ in Eq. (7) of the Moreau-Yosida envelope. This is not circular: the envelope property is the construction, not an imported conclusion. The geometric-ergodicity requirements are delegated to standard sufficient conditions of Roberts and Tweedie (1996) and Livingstone et al. (2019), which are external results, and Theorem 7 explicitly lists condition (29) as an assumption rather than deriving it. The self-citations to Agarwal et al. (2022), Vats et al. (2019), and Vats (2017) are used for computing asymptotic covariance estimates and for the iid Gaussian importance-sampling covariance; these are published results with independent standing and do not incorporate the present paper's target claims. The tuning recommendation that λ be chosen so that n_e/n lies in [0.40, 0.80] is a calibration rule, not a fitted result relabeled as a prediction. The skeptical concern about the unverified tail condition (29) for π_λ-HMC is a correctness gap or an unproven assertion, not a circular reduction: the paper does not claim to derive (29) from its own assumptions, and Section 5.2 openly acknowledges that demonstrating (29) can be challenging. No load-bearing step in the paper reduces, by construction or by self-citation, to its own inputs.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central method rests on standard convex analysis (MY envelope, proximal maps) plus standard MCMC/IS theory. No new entities are postulated. The main free choice is λ, with MCMC step sizes as secondary tuning. The most fragile assumptions are the geometric-ergodicity conditions (Assumption 1 and condition (29)) needed for the finite-variance guarantee.

free parameters (3)
  • λ (Moreau-Yosida smoothing parameter) = selected by practitioner; heuristic ne/n ∈ [0.4, 0.8]; examples use λ=0.001 or λ=1e-4
    Controls the approximation-vs-mixing tradeoff. Theorem 3 gives an optimal λ for the iid Gaussian case, but general targets require the heuristic.
  • MALA step size h = tuned to ~57% acceptance
    Standard MCMC tuning parameter; affects convergence of the πλ-MALA chain but not the central idea.
  • HMC step size ε and leapfrog steps L = tuned to ~65% acceptance; L chosen with Pr(L=1)>0
    Standard HMC tuning parameters; Assumption 2 restricts L. The central method inherits them.
assumptions (6)
  • domain assumption ψ belongs to Γ(R^d): proper, closed, convex, bounded below (log-concave target).
    The paper restricts attention to log-concave distributions (Section 3.1) so the Moreau-Yosida envelope and proximal map have the required uniqueness and inequality properties.
  • domain assumption The proximal mapping of ψ is computable efficiently.
    Efficient prox evaluation is required in practice (trendfiltering uses glmgen ADMM, nuclear norm uses SVT, Poisson model uses Newton-Raphson). Stated in Section 3.1 and used throughout Section 6.
  • domain assumption The πλ-invariant Markov chain is geometrically ergodic.
    Needed for the CLT in Theorem 2. The paper proves sufficient conditions in Theorems 5 and 7, but these are not necessary and their verification is problem-specific.
  • ad hoc to paper Assumption 1: lim sup_{||x||→∞} ||prox_ψ^λ(x)||/||x|| = l < 1.
    Introduced in Section 5 to prove Theorem 5 (MALA) and Theorem 7 (HMC). It holds for ℓ2+ℓ1 potentials but fails for the pure Laplace potential, which the paper acknowledges in Remark 3.
  • ad hoc to paper Condition (29): lim_{||x||→∞} ∫_{R(x)∩I(x)} q_H(x,y) dy = 0 for πλ.
    Required by Theorem 7 for HMC geometric ergodicity. The paper states this condition is challenging to verify for general targets and only provides guidance for the exponential family class E(β,γ) (Section 5.2).
  • standard math Self-normalized importance sampling weight wλ ≤ 1 (global underestimation bound ψλ ≤ ψ).
    Follows from the definition of the Moreau-Yosida envelope (Eq. 7) and is used to prove bounded weights and finite moments in Theorem 2.

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Pith. "Pith review of MCMC Importance Sampling via Moreau-Yosida Envelopes." pith.science (2026). https://pith.science/paper/SWE3KBZR

@misc{pith2026250102228,
  author       = {Pith},
  title        = {Pith review of: MCMC Importance Sampling via Moreau-Yosida Envelopes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SWE3KBZR}},
  note         = {Machine review of arXiv:2501.02228}
}
read the original abstract

Non-differentiable priors are standard in modern parsimonious Bayesian models. Lack of differentiability, however, precludes gradient-based Markov chain Monte Carlo (MCMC) for posterior sampling. Recently proposed proximal MCMC approaches can partially remedy this limitation by using a differentiable approximation, constructed via Moreau-Yosida (MY) envelopes, to make proposals. In this work, we build an importance sampling paradigm by using the MY envelope as an importance distribution. Leveraging properties of the envelope, we establish asymptotic normality of the importance sampling estimator with an explicit expression for the asymptotic covariance matrix. Since the MY envelope density is smooth, it is amenable to gradient-based samplers. We provide sufficient conditions for geometric ergodicity of Metropolis-adjusted Langevin and Hamiltonian Monte Carlo algorithms, sampling from this importance distribution. Our numerical studies show that the proposed scheme can yield lower variance estimators compared to existing proximal MCMC alternatives, and is effective in low and high dimensions.

Figures

Figures reproduced from arXiv: 2501.02228 by the authors.

Figure 1
Figure 1. (Left) Moreau-Yosida envelope for ψ(x) = |x| for different values of λ. (Right) The corresponding envelopes for the Laplace distribution. three different λ values. We see that the Moreau-Yosida envelope provides a differentiable approximation to a non-smooth function where the approximation improves as λ gets smaller. Having constructed ψ λ , consider the importance density π λ (x) ∝ e −ψ λ(x) , (8) that is integrab… view at source ↗
Figure 2
Figure 2. Top (Laplace) and bottom (Super-Gaussian) for [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Trend filtering fit using the marginal quantiles and posterior mean for importance [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Trendfiltering: Average relative efficiencies of MYIS-MALA over P-MALA (left) [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]
Figure 5
Figure 5. Figure 5: Trendfiltering: ACF difference plots computed using samples from [PITH_FULL_IMAGE:figures/full_fig_p028_5.png]
Figure 6
Figure 6. Figure 6: Nuclear-norm based matrix denoising: ACF difference plots computed using [PITH_FULL_IMAGE:figures/full_fig_p030_6.png]
Figure 7
Figure 7. Figure 7: Nuclear-norm based matrix denoising: Average relative efficiencies of MYIS [PITH_FULL_IMAGE:figures/full_fig_p031_7.png]
Figure 8
Figure 8. Figure 8: Poisson model: ACF difference plots computed using samples from [PITH_FULL_IMAGE:figures/full_fig_p032_8.png]
Figure 9
Figure 9. Figure 9: Poisson random effects model: Average relative efficiencies of MYIS-MALA over [PITH_FULL_IMAGE:figures/full_fig_p033_9.png]
Figure 10
Figure 10. Figure 10: (Laplace) From top to bottom d = 1, 10, 20. Left column has the importance sampling effective sample size ne/n for different λ, middle column has the MCMC effective sample size for the π λ -MALA chain for different λ, and the right column has the estimated importance …
Figure 11
Figure 11. Figure 11: (Super-Gaussian) From top to bottom d = 1, 10, 20. Left column has the importance sampling effective sample size ne/n for different λ, middle column has the MCMC effective sample size for the π λ -MALA chain for different λ, and the right column has the estimated impo…
Figure 12
Figure 12. Figure 12: Checkerboard image (left), singular values (right). F.4 Nuclear-norm based low rank matrix estimation We consider the 64 × 64 checkerboard image of MATLAB, shown in the left panel of [PITH_FULL_IMAGE:figures/full_fig_p059_12.png]

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Works this paper leans on

65 extracted references · 64 canonical work pages · cited by 1 Pith paper

  1. [1]

    Agarwal, M., Vats, D., and Elvira, V. (2022). A principled stopping rule for importance sampling. Electronic Journal of Statistics , 16(2):5570--5590

    work page 2022

  2. [2]

    Efficient Multivariate Initial Sequence Estimators for MCMC

    Banerjee, A. and Vats, D. (2024). Efficient multivariate initial sequence estimators for MCMC . arXiv preprint arXiv:2406.15874

    work page Pith review arXiv 2024

  3. [3]

    and Song, H

    Berg, S. and Song, H. (2023). Efficient shape-constrained inference for the autocovariance sequence from a reversible M arkov chain. The Annals of Statistics , 51(6):2440--2470

    work page 2023

  4. [4]

    S., Roberts, G

    Beskos, A., Pillai, N. S., Roberts, G. O., Sanz-Serna, J., and Stuart, A. (2013). Optimal tuning of the hybrid M onte C arlo algorithm. Bernoulli , 19(5a):1501--1534

    work page 2013

  5. [5]

    and Wright, S

    B\"ohm, A. and Wright, S. J. (2021). Variable smoothing for weakly convex composite functions. J. Optim. Theory Appl. , 188(3):628--649

    work page 2021

  6. [6]

    and Doss, H

    Buta, E. and Doss, H. (2011). Computational approaches for empirical B ayes methods and B ayesian sensitivity analysis. Ann. Statist. , pages 2658--2685

    work page 2011

  7. [7]

    J., and Shen, Z

    Cai, J.-F., Cand\` e s, E. J., and Shen, Z. (2010). A singular value thresholding algorithm for matrix completion. SIAM Journal on Optimization , 20(4):1956--1982

    work page 2010

  8. [8]

    Chaari, L., Tourneret, J.-Y., Chaux, C., and Batatia, H. (2016). A H amiltonian M onte C arlo method for non-smooth energy sampling. IEEE Trans. Signal Process. , 64(21):5585--5594

    work page 2016

Show all 65 references
  1. [9]

    Chen, D.-F. R. and Seila, A. F. (1987). Multivariate inference in stationary simulation using batch means. In Proceedings of the 19th Conference on Winter simulation , pages 302--304. ACM

    1987

  2. [10]

    and Shao, Q.-M

    Chen, M.-H. and Shao, Q.-M. (1998). Monte C arlo methods for B ayesian analysis of constrained parameter problems. Biometrika , 85:73--87

    1998

  3. [11]

    and Shao, Q.-M

    Chen, M.-H. and Shao, Q.-M. (1999). Monte Carlo estimation of Bayesian credible and HPD intervals . Journal of Computational and Graphical Statistics , pages 69--92

    1999

  4. [12]

    Combettes, P. L. and Pesquet, J.-C. (2011). Proximal Splitting Methods in Signal Processing , pages 185--212. Springer New York, New York, NY

    2011

  5. [13]

    and Jones, G

    Dai, N. and Jones, G. L. (2017). Multivariate initial sequence estimators in M arkov chain M onte C arlo. Journal of Multivariate Analysis , 159:184--199

    2017

  6. [14]

    Durmus, A., Moulines, \'E ., and Pereyra, M. (2022). A proximal M arkov chain M onte C arlo method for B ayesian inference in imaging inverse problems: When L angevin meets M oreau. SIAM Review , 64(4):991--1028

    2022

  7. [15]

    Elvira, V., Chouzenoux, \'E ., and Akyildiz, O. D. (2024). A proximal Newton adaptive importance sampler . arXiv preprint arXiv:2412.16558

    2024 arXiv

  8. [16]

    Faulkner, J. R. and Minin, V. N. (2018). Locally adaptive smoothing with M arkov random fields and shrinkage priors. Bayesian Analysis , 13(1):225

    2018

  9. [17]

    Fazel, M. (2002). Matrix rank minimization with applications . PhD thesis, Stanford University

    2002

  10. [18]

    Flegal, J. M. and Kurtz-Garcia, R. P. (2024). Implementing MCMC : Multivariate estimation with confidence. arXiv preprint arXiv:2408.15396

    2024 arXiv

  11. [19]

    and Gray, L

    Fristedt, B. and Gray, L. (1996). A Modern Approach to Probability Theory . Probability and Its Applications. Birkh \"a user Boston

    1996

  12. [20]

    Geyer, C. J. (1992). Practical M arkov chain M onte C arlo. Statistical Science , pages 473--483

    1992

  13. [21]

    and Calderhead, B

    Girolami, M. and Calderhead, B. (2011). Riemann manifold L angevin and H amiltonian M onte C arlo methods. J. R. Stat. Soc. Ser. B Stat. Method. , 73(2):123--214

    2011

  14. [22]

    Glynn, P. W. et al. (1996). Importance sampling for M onte C arlo estimation of quantiles. In Mathematical Methods in Stochastic Simulation and Experimental Design: Proceedings of the 2nd St. Petersburg Workshop on Simulation , pages 180--185. Citeseer

    1996

  15. [23]

    Glynn, P. W. and Whitt, W. (1991). Estimating the asymptotic variance with batch means. Operations Research Letters , 10:431--435

    1991

  16. [24]

    Heng, Q., Zhou, H., and Chi, E. C. (2023). Bayesian trend filtering via proximal M arkov chain M onte C arlo. Journal of Computational and Graphical Statistics , 32(3):938--949

    2023

  17. [25]

    Hesterberg, T. C. (1988). Advances in importance sampling . Stanford University

    1988

  18. [26]

    Hoheisel, T., Laborde, M., and Oberman, A. (2020). A regularization interpretation of the proximal point method for weakly convex functions. J. Dyn. Games , 7(1):79--96

    2020

  19. [27]

    Jones, G. L. (2004). On the M arkov chain central limit theorem. Probability Surveys , 1:299--320

    2004

  20. [28]

    Kim, S.-J., Koh, K., Boyd, S., and Gorinevsky, D. (2009). _1 trend filtering . SIAM Review , 51(2):339--360

    2009

  21. [29]

    and Varadhan, S

    Kipnis, C. and Varadhan, S. R. S. (1986). Central limit theorem for additive functionals of reversible M arkov processes and applications to simple exclusions. Comm. Math. Phys. , 104(1):1--19

    1986

  22. [30]

    Kong, A. (1992). A note on importance sampling using standardized weights. University of Chicago, Dept. of Statistics, Tech. Rep , 348:14

    1992

  23. [31]

    R., Matteson, D

    Kowal, D. R., Matteson, D. S., and Ruppert, D. (2019). Dynamic shrinkage processes. J. R. Stat. Soc. Ser. B Stat. Method. , 81(4):781--804

    2019

  24. [32]

    T., Shen, R., and Tian, K

    Lee, Y. T., Shen, R., and Tian, K. (2021). Structured logconcave sampling with a restricted G aussian oracle. In Belkin, M. and Kpotufe, S., editors, Proc. 34th Conf. on Learning Theory , volume 134 of Proc. Machine Learning Research , pages 2993--3050

    2021

  25. [33]

    and Chen, Y

    Liang, J. and Chen, Y. (2022). A proximal algorithm for sampling from non-smooth potentials. In 2022 Winter Simulation Conference (WSC) , pages 3229--3240

    2022

  26. [34]

    and Richard, J.-F

    Liesenfeld, R. and Richard, J.-F. (2008). Improving MCMC , using efficient importance sampling. Computational Statistics & Data Analysis , 53(2):272--288

    2008

  27. [35]

    Livingstone, S., Betancourt, M., Byrne, S., and Girolami, M. (2019). On the geometric ergodicity of H amiltonian M onte C arlo. Bernoulli , 25(4A):3109--3138

    2019

  28. [36]

    and Zanella, G

    Livingstone, S. and Zanella, G. (2022). The B arker proposal: C ombining robustness and efficiency in gradient-based MCMC . J. R. Stat. Soc. Ser. B Stat. Method. , 84(2):496--523

    2022

  29. [37]

    and Piccioni, M

    Madras, N. and Piccioni, M. (1999). Importance sampling for families of distributions. Ann. Appl. Probab. , 9(4):1202--1225

    1999

  30. [38]

    Martino, L., Elvira, V., and Camps-Valls, G. (2018). Group importance sampling for particle filtering and MCMC . Digital Signal Processing , 82:133--151

    2018

  31. [39]

    Mazumder, R., Hastie, T., and Tibshirani, R. (2010). Spectral regularization algorithms for learning large incomplete matrices. J. Mach. Learn. Res. , 11(80):2287--2322

    2010

  32. [40]

    Meyn, S. P. and Tweedie, R. L. (2012). Markov Chains and Stochastic Stability . Springer Science & Business Media

    2012

  33. [41]

    J., and Bartlett, P

    Mou, W., Flammarion, N., Wainwright, M. J., and Bartlett, P. L. (2022). An efficient sampling algorithm for non-smooth composite potentials. J. Mach. Learn. Res. , 23(233):1--50

    2022

  34. [42]

    Neal, R. M. (2011). MCMC Using Hamiltonian Dynamics . In Handbook of Markov Chain Monte Carlo , pages 113--162. CRC Press/Taylor & Francis

    2011

  35. [43]

    and Boyd, S

    Parikh, N. and Boyd, S. (2014). Proximal algorithms. Found. Trends Optim. , 1(3):127--239

    2014

  36. [44]

    Parikh, N., Boyd, S., et al. (2014). Proximal algorithms. Foundations and trends in Optimization , 1(3):127--239

    2014

  37. [45]

    Pereyra, M. (2016). Proximal M arkov chain M onte C arlo algorithms. Statist. Comput. , 26:745--760

    2016

  38. [46]

    G., Scott, J

    Polson, N. G., Scott, J. G., and Willard, B. T. (2015). Proximal algorithms in statistics and machine learning. Statistical Science , 30(4):559--581

    2015

  39. [47]

    Recht, B., Fazel, M., and Parrilo, P. A. (2010). Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM review , 52(3):471--501

    2010

  40. [48]

    Roberts, G. O. and Rosenthal, J. S. (1998). Optimal scaling of discrete approximations to L angevin diffusions. J. R. Stat. Soc. Ser. B Stat. Method. , 60(1):255--268

    1998

  41. [49]

    Roberts, G. O. and Rosenthal, J. S. (2001). Markov chains and de-initializing processes. Scandinavian Journal of Statistics , 28:489--504

    2001

  42. [50]

    Roberts, G. O. and Sahu, S. K. (1997). Updating schemes, correlation structure, blocking and parameterization for the G ibbs sampler. J. R. Stat. Soc. Ser. B Stat. Method. , 59(2):291--317

    1997

  43. [51]

    Roberts, G. O. and Tweedie, R. L. (1996). Exponential convergence of L angevin distributions and their discrete approximations. Bernoulli , pages 341--363

    1996

  44. [52]

    Roualdes, E. A. (2015). Bayesian trend filtering. arXiv preprint arXiv:1505.07710

    2015 arXiv

  45. [53]

    Roy, V. (2020). Convergence diagnostics for M arkov chain M onte C arlo. Annual Review of Statistics and Its Application , 7:387--412

    2020

  46. [54]

    and Klebanov, I

    Schuster, I. and Klebanov, I. (2020). Markov chain importance sampling—a highly efficient estimator for MCMC . Journal of Computational and Graphical Statistics , 30(2):260--268

    2020

  47. [55]

    Seila, A. F. (1982). Multivariate estimation in regenerative simulation. Operations Research Letters , 1:153--156

    1982

  48. [56]

    Silva, L. A. and Zanella, G. (2024). Robust leave-one-out cross-validation for high-dimensional B ayesian models. Journal of the American Statistical Association , 119(547):2369--2381

    2024

  49. [57]

    and Berg, S

    Song, H. and Berg, S. (2024). Multivariate moment least-squares variance estimators for reversible markov chains. Journal of Computational and Graphical Statistics , to appear

    2024

  50. [58]

    Tan, A., Doss, H., and Hobert, J. P. (2015). Honest importance sampling with multiple M arkov chains. Journal of Computational and Graphical Statistics , 24(3):792--826

    2015

  51. [59]

    Tibshirani, R., Saunders, M., Rosset, S., Zhu, J., and Knight, K. (2005). Sparsity and smoothness via the fused lasso. J. R. Stat. Soc. Ser. B Stat. Method. , 67(1):91--108

    2005

  52. [60]

    Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Ann. Statist. , 22(4):1701--1762

    1994

  53. [61]

    Van der Vaart, A. W. (2000). Asymptotic Statistics , volume 3. Cambridge University Press

    2000

  54. [62]

    Vats, D. (2017). Output Analysis for Markov Chain Monte Carlo . PhD thesis, University of Minnesota

    2017

  55. [63]

    and Flegal, J

    Vats, D. and Flegal, J. M. (2022). Lugsail lag windows for estimating time-average covariance matrices. Biometrika , 109(3):735--750

    2022

  56. [64]

    M., and Jones, G

    Vats, D., Flegal, J. M., and Jones, G. L. (2018). Strong consistency of multivariate spectral variance estimators in M arkov chain M onte C arlo. Bernoulli , 24:1860--1909

    2018

  57. [65]

    M., and Jones, G

    Vats, D., Flegal, J. M., and Jones, G. L. (2019). Multivariate output analysis for M arkov chain M onte C arlo. Biometrika , 106(2):321--337

    2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.