A simple variance inequality for -statistics of a Markov chain with applications

Fort, Gersende; Moulines, Éric; Priouret, Pierre; Vandekerkhove, Pierre

doi:10.1016/j.spl.2012.02.001

Cited by 8 publications

(7 citation statements)

References 10 publications

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“…The kernel (24) enjoys an increased sensitivity to single-dimensional perturbations over (22), which becomes especially apparent when the dimension d is large. For further discussion, see [61].…”

Section: ) and Somentioning

confidence: 99%

“…Now, we turn to proving the desired convergence. Our primary tool is the variance bound of [24] for U -statistics of non-stationary Markov chains. Since for each n, w (n) minimizes S n (w, x) := n i,j=1 w i w j k π (x i , x j ) over w, it suffices to find a sequence of normalized reference weights v (n) such that S n (v (n) , x) P → 0 in case (a) and S n (v (n) , x) = O P (n −1 ) in case (b).…”

Section: The Stein Correctionmentioning

confidence: 99%

“…For case (a), the results of [24] can no longer be applied directly. Instead, consider the truncated reference weights v…”

Section: The Stein Correctionmentioning

confidence: 99%

“…Since ϕ n ≤ τ n ϕ 1 and (x, u) → ϕ 1 ((x, u), (x, u)) is integrable with respect to the product of ν and the Lebesgue measure on [0, 1], the ergodic theorem implies τ −1 n V n 2 is bounded, and n −1 V n 2 → 0. Unlike part (b), ϕ n is no longer degenerate, however, by applying [24,Corollary 2.3] to the Hoeffding decomposition of U n , and recognising that sup…”

Section: The Stein Correctionmentioning

confidence: 99%

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The reproducing Stein kernel approach for post-hoc corrected sampling

Hodgkinson,

Salomone,

Roosta

2020

Preprint

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Stein importance sampling [42] is a widely applicable technique based on kernelized Stein discrepancy [43], which corrects the output of approximate sampling algorithms by reweighting the empirical distribution of the samples. A general analysis of this technique is conducted for the previously unconsidered setting where samples are obtained via the simulation of a Markov chain, and applies to an arbitrary underlying Polish space. We prove that Stein importance sampling yields consistent estimators for quantities related to a target distribution of interest by using samples obtained from a geometrically ergodic Markov chain with a possibly unknown invariant measure that differs from the desired target. The approach is shown to be valid under conditions that are satisfied for a large number of unadjusted samplers, and is capable of retaining consistency when data subsampling is used. Along the way, a universal theory of reproducing Stein kernels is established, which enables the construction of kernelized Stein discrepancy on general Polish spaces, and provides sufficient conditions for kernels to be convergence-determining on such spaces. These results are of independent interest for the development of future methodology based on kernelized Stein discrepancies.

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Section: ) and Somentioning

confidence: 99%

Section: The Stein Correctionmentioning

confidence: 99%

“…For case (a), the results of [24] can no longer be applied directly. Instead, consider the truncated reference weights v…”

Section: The Stein Correctionmentioning

confidence: 99%

Section: The Stein Correctionmentioning

confidence: 99%

See 2 more Smart Citations

The reproducing Stein kernel approach for post-hoc corrected sampling

Hodgkinson,

Salomone,

Roosta

2020

Preprint

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show abstract

“…We rather work with bounded kernels that are π-canonical. This latter notion was first introduced in Fort et al (2012) who proved a variance inequality for U-statistics of ergodic Markov chains. In Section B, we give a concrete connection between our result and Shen et al (2020).…”

Section: Introductionmentioning

confidence: 99%

Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains

Duchemin¹,

Castro²,

Lacour³

2020

Preprint

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We prove a new concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. Working with bounded π-canonical kernels, we show that we can recover the convergence rate of Arcones and Gine (1993) who proved a concentration result for U-statistics of independent random variables and canonical kernels. Our proof relies on an inductive analysis where we use martingale techniques, uniform ergodicity, Nummelin splitting and Bernstein's type inequality where the spectral gap of the chain emerges.Our result allows us to conduct three applications. First, we establish a new exponential inequality for the estimation of spectra of trace class integral operators with MCMC methods. The novelty is that this result holds for kernels with positive and negative eigenvalues, which is new as far as we know.In addition, we investigate generalization performance of online algorithms working with pairwise loss functions and Markov chain samples. We provide an online-to-batch conversion result by showing how we can extract a low risk hypothesis from the sequence of hypotheses generated by any online learner.We finally give a non-asymptotic analysis of a goodness-of-fit test on the density of the invariant measure of a Markov chain. We identify the classes of alternatives over which our test based on the L 2 distance has a prescribed power.

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Renewal type bootstrap for increasing degree U-process of a Markov chain

Soukarieh

Bouzebda

2023

Journal of Multivariate Analysis

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A simple variance inequality for -statistics of a Markov chain with applications

Cited by 8 publications

References 10 publications

The reproducing Stein kernel approach for post-hoc corrected sampling

The reproducing Stein kernel approach for post-hoc corrected sampling

Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains

Renewal type bootstrap for increasing degree U-process of a Markov chain

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