Nonasymptotic bounds on the estimation error of MCMC algorithms

Łatuszyński, Krzysztof; Miasojedow, Błażej; Niemiro, Wojciech

doi:10.3150/12-bej442

Cited by 47 publications

(62 citation statements)

References 85 publications

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“…For regenerative algorithms, alternative bounds established in [30] are typically tighter than those resulting from our Section 4. However, the algorithms proposed there are more difficult to implement in practically relevant examples.…”

Section: Applicability Of the Resultsmentioning

confidence: 93%

“…Tail inequalities valid in this setup have been established by Bertail and Clémençon in [8] by regenerative approach and using truncation arguments. However, they involve non-explicit constants and cannot be directly applied to derive lower bounds on t and n. In [30] a result analogous to (2) is established for a sequentialregenerative estimator (instead ofÎ t,n (f )). The approach of [30] requires identification of regeneration times.…”

Section: Introductionmentioning

confidence: 97%

“…However, they involve non-explicit constants and cannot be directly applied to derive lower bounds on t and n. In [30] a result analogous to (2) is established for a sequentialregenerative estimator (instead ofÎ t,n (f )). The approach of [30] requires identification of regeneration times. In many problems of practical interest, especially in high dimension, regeneration schemes are difficult to implement [19,49].…”

Section: Introductionmentioning

confidence: 97%

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Rigorous confidence bounds for MCMC under a geometric drift condition

Łatuszyński

Niemiro

2011

Journal of Complexity

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We assume a drift condition towards a small set and bound the mean square error of estimators obtained by taking averages along a single trajectory of a Markov chain Monte Carlo algorithm. We use these bounds to construct fixed-width nonasymptotic confidence intervals. For a possibly unbounded function $f:\stany \to R,$ let $I=\int_{\stany} f(x) \pi(x) dx$ be the value of interest and $\hat{I}_{t,n}=(1/n)\sum_{i=t}^{t+n-1}f(X_i)$ its MCMC estimate. Precisely, we derive lower bounds for the length of the trajectory $n$ and burn-in time $t$ which ensure that $$P(|\hat{I}_{t,n}-I|\leq \varepsilon)\geq 1-\alpha.$$ The bounds depend only and explicitly on drift parameters, on the $V-$norm of $f,$ where $V$ is the drift function and on precision and confidence parameters $\varepsilon, \alpha.$ Next we analyse an MCMC estimator based on the median of multiple shorter runs that allows for sharper bounds for the required total simulation cost. In particular the methodology can be applied for computing Bayesian estimators in practically relevant models. We illustrate our bounds numerically in a simple example

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Section: Applicability Of the Resultsmentioning

confidence: 93%

Section: Introductionmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 97%

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Rigorous confidence bounds for MCMC under a geometric drift condition

Łatuszyński

Niemiro

2011

Journal of Complexity

Self Cite

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“…Indeed, the existing papers [22,23,24,25,26] reported several difficulties to evaluate the variance of the sample mean in the continuous probability space even with the discrete time Markov chain. So, it is remained to extend the obtained results to the continuous case.…”

Section: Resultsmentioning

confidence: 99%

Information geometry approach to parameter estimation in Markov chains

Hayashi¹,

Watanabe²

2016

Ann. Statist.

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Abstract:We consider the parameter estimation of Markov chain when the unknown transition matrix belongs to an exponential family of transition matrices. Then, we show that the sample mean of the generator of the exponential family is an asymptotically efficient estimator. Further, we also define a curved exponential family of transition matrices. Using a transition matrix version of the Pythagorean theorem, we give an asymptotically efficient estimator for a curved exponential family. Primary 62M05.

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“…PR P-Rank consists of three phases: (a) For pre-processing (lines 1-6), PR P-Rank invokes the randomized algorithm [5] N (u, v) is the sample mean, and σ 2 the variance) , given any accuracy and confidence level 1 − α (α ∈ (0, 1)). This is because applying the Bernstein's Theorem [11] yields exp(− (a) An upper bound can be obtained from Bernstein's Theorem [11], which gives…”

Section: Else If There Exists a Positive Integermentioning

confidence: 99%

On the Efficiency of Estimating Penetrating Rank on Large Graphs

Lin

et al. 2012

Lecture Notes in Computer Science

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Abstract. P-Rank (Penetrating Rank) has been suggested as a useful measure of structural similarity that takes account of both incoming and outgoing edges in ubiquitous networks. Existing work often utilizes memoization to compute PRank similarity in an iterative fashion, which requires cubic time in the worst case. Besides, previous methods mainly focus on the deterministic computation of P-Rank, but lack the probabilistic framework that scales well for large graphs. In this paper, we propose two efficient algorithms for computing P-Rank on large graphs. The first observation is that a large body of objects in a real graph usually share similar neighborhood structures. By merging such objects with an explicit low-rank factorization, we devise a deterministic algorithm to compute P-Rank in quadratic time. The second observation is that by converting the iterative form of P-Rank into a matrix power series form, we can leverage the random sampling approach to probabilistically compute P-Rank in linear time with provable accuracy guarantees. The empirical results on both real and synthetic datasets show that our approaches achieve high time efficiency with controlled error and outperform the baseline algorithms by at least one order of magnitude.

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Nonasymptotic bounds on the estimation error of MCMC algorithms

Cited by 47 publications

References 85 publications

Rigorous confidence bounds for MCMC under a geometric drift condition

Rigorous confidence bounds for MCMC under a geometric drift condition

Information geometry approach to parameter estimation in Markov chains

On the Efficiency of Estimating Penetrating Rank on Large Graphs

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