Conditional convex orders and measurable martingale couplings

Leskelä, Lasse; Vihola, Matti

doi:10.3150/16-bej827

Cited by 5 publications

(4 citation statements)

References 40 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…T i,x,α has the same mean as T i,x , but is a worse estimator. In particular, Proposition 1.2 of Leskelä and Vihola (2014) implies T i,x ≤ cx T i,x,α for any 0 ≤ α ≤ 1 and any i ∈ {1, 2} (Equation (4) gives the coupling required by that proposition). We have:…”

Section: Efficiency Of Abc and Abc-mcmcmentioning

confidence: 99%

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The use of a single pseudo-sample in approximate Bayesian computation

et al. 2016

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We analyze the computational efficiency of approximate Bayesian computation (ABC), which approximates a likelihood function by drawing pseudo-samples from the associated model. For the rejection sampling version of ABC, it is known that multiple pseudo-samples cannot substantially increase (and can substantially decrease) the efficiency of the algorithm as compared to employing a high-variance estimate based on a single pseudo-sample. We show that this conclusion also holds for a Markov chain Monte Carlo version of ABC, implying that it is unnecessary to tune the number of pseudo-samples used in ABC-MCMC. This conclusion is in contrast to particle MCMC methods, for which increasing the number of particles can provide large gains in computational efficiency.

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Section: Efficiency Of Abc and Abc-mcmcmentioning

confidence: 99%

“…To obtain (5) of the main document, by Theorem 3 it is sufficient to take α = 1 − 1 M and show that T 1,θ ≤ cx T M,θ,α . By Proposition 2.2 of Leskelä and Vihola (2014), it is furthermore sufficient to show that, for all c ∈ R,…”

Section: A Proofsmentioning

confidence: 99%

The use of a single pseudo-sample in approximate Bayesian computation

et al. 2016

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“…Polynomial kernel function can promote the extrapolation, which has good overall properties, the lower the order, the stronger the promotion. And the multilayer perception kernel function including a hidden layer 17 can realize multilayer perception.…”

Section: Application Of Kernel Functionmentioning

confidence: 99%

Comprehensive evaluation of robotic global performance based on modified principal component analysis

Zhao

Wang

et al. 2020

International Journal of Advanced Robotic Systems

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The multivariate statistical method such as principal component analysis based on linear dimension reduction and kernel principal component analysis based on nonlinear dimension reduction as the modified principal component analysis method are commonly used. Because of the diversity and correlation of robotic global performance indexes, the two multivariate statistical methods principal component analysis and kernel principal component analysis methods can be used, respectively, to comprehensively evaluate the global performance of PUMA560 robot with different dimensions. When using the kernel principal component analysis method, the kernel function and parameters directly have an effect on the result of comprehensive performance evaluation. Because kernel principal component analysis with polynomial kernel function is time-consuming and inefficient, a new kernel function based on similarity degree is proposed for the big sample data. The new kernel function is proved according to Mercer’s theorem. By comparing different dimension reduction effects of principal component analysis method, the kernel principal component analysis method with polynomial kernel function, and the kernel principal component analysis method with the new kernel function, the kernel principal component analysis method with the new kernel function could deal more effectively with the nonlinear relationship among indexes, and its calculation result is more reasonable for containing more comprehensive information. The simulation shows that the kernel principal component analysis method with the new kernel function has the advantage of low time consuming, good real-time performance, and good ability of generalization.

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“…We preface the proof of Theorem 10 with a key result from [31], which ensures that the conditional convex order implies a conditional martingale coupling of the distributions involved. For the rest of this section, we assume that the conditions in Theorem 10 hold, and we denote byπ 1 ,π 2 the invariant distributions ofP 1 ,P 2 , respectively.…”

Section: Proofs By a Martingale Coupling Of Pseudo-marginal Kernelsmentioning

confidence: 99%

Establishing some order amongst exact approximations of MCMCs

Andrieu¹,

Vihola²

2016

Ann. Appl. Probab.

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Abstract. Exact approximations of Markov chain Monte Carlo (MCMC) algorithms are a general emerging class of sampling algorithms. One of the main ideas behind exact approximations consists of replacing intractable quantities required to run standard MCMC algorithms, such as the target probability density in a Metropolis-Hastings algorithm, with estimators. Perhaps surprisingly, such approximations lead to powerful algorithms which are exact in the sense that they are guaranteed to have correct limiting distributions. In this paper we discover a general framework which allows one to compare, or order, performance measures of two implementations of such algorithms. In particular, we establish an order with respect to the mean acceptance probability, the first autocorrelation coefficient, the asymptotic variance and the right spectral gap. The key notion to guarantee the ordering is that of the convex order between estimators used to implement the algorithms. We believe that our convex order condition is close to optimal, and this is supported by a counter-example which shows that a weaker variance order is not sufficient. The convex order plays a central role by allowing us to construct a martingale coupling which enables the comparison of performance measures of Markov chain with differing invariant distributions, contrary to existing results. We detail applications of our result by identifying extremal distributions within given classes of approximations, by showing that averaging replicas improves performance in a monotonic fashion and that stratification is guaranteed to improve performance for the standard implementation of the Approximate Bayesian Computation (ABC) MCMC method.

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Conditional convex orders and measurable martingale couplings

Cited by 5 publications

References 40 publications

The use of a single pseudo-sample in approximate Bayesian computation

The use of a single pseudo-sample in approximate Bayesian computation

Comprehensive evaluation of robotic global performance based on modified principal component analysis

Establishing some order amongst exact approximations of MCMCs

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