Directional Metropolis-Hastings

Mallik, Abhirup; Jones, Galin L.

doi:10.48550/arxiv.1710.09759

Cited by 2 publications

(4 citation statements)

References 4 publications

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Section: Hopmentioning

confidence: 99%

“…2 were omitted, the Hop algorithm would be a special case of the Directional Metropolis-Hastings algorithm of Mallik and Jones (2017), which also allows for a MALA-like offset of the proposal mean. However, unlike the algorithm in Mallik and Jones (2017), the Hop algorithm is specifically intended for jumping between contours. As we shall see in Theorem 2 below, which is proved in Appendix C, and the simulations in Section 3, the position-dependent scaling brings enormous (and, perhaps, unexpected) gains in efficiency for typical targets.…”

Section: Hopmentioning

confidence: 99%

“…There are a number of close analogies between Hug and HMC which we explore in more detail in Section 2.3, where we also point out that, unlike HMC, while evolving the dynamics, the standard version of Hug can be enhanced via an explicit scheme to use the local Hessian and account for local curvature or shape. Hop, described in Section 2.2, is a variation on the Directional Metropolis-Hastings algorithm of Mallik and Jones (2017), adjusted specifically cater to movements almost parallel with the gradient vector. As proved in Theorem 2 and demonstrated empirically in the simulation study that follows, this brings substantial benefits in terms of robustness to increasing dimension.…”

Section: Introductionmentioning

confidence: 99%

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Hug and Hop: a discrete-time, non-reversible Markov chain Monte-Carlo algorithm

Ludkin¹,

Sherlock²

2019

Preprint

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We introduced the Hug and Hop Markov chain Monte Carlo algorithm for estimating expectations with respect to an intractable distribution π. The algorithm alternates between two kernels: Hug and Hop. Hug is a non-reversible kernel that uses repeated applications of the bounce mechanism from the recently proposed Bouncy Particle Sampler to produce a proposal point far from the current position, yet on almost the same contour of the target density, leading to a high acceptance probability. Hug is complemented by Hop, which deliberately proposes jumps between contours and has an efficiency that degrades very slowly with increasing dimension. There are many parallels between Hug and Hamiltonian Monte Carlo (HMC) using a leapfrog intergator, including an O(δ 2 ) error in the integration scheme, however Hug is also able to make use of local Hessian information without requiring implicit numerical integration steps, improving efficiency when the gains in mixing outweigh the additional computational costs. We test Hug and Hop empirically on a variety of toy targets and real statistical models and find that it can, and often does, outperform HMC on the exploration of components of the target.

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Section: Hopmentioning

confidence: 99%

Section: Hopmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Hug and Hop: a discrete-time, non-reversible Markov chain Monte-Carlo algorithm

Ludkin¹,

Sherlock²

2019

Preprint

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“…This is in line with the intuition that the jump size should be smaller in directions orthogonal to that containing the signal. In Livingstone (2015) (see also Mallik and Jones (2017) for a similar algorithm and an adaptive version of it), the author studies the efficiency (in non-asymptotic regime) of RWMH using a position dependent covariance matrix (an approach which actually gathers a number of the aforementioned method under a generic framework). In particular, it is established that for sparse and filamentary distributions and under regulatory assumptions, the convergence of the position dependent RWMH occurs at a geometric rate, something which does not always hold for the standard RWMH.…”

Section: Introductionmentioning

confidence: 99%

Markov Kernels Local Aggregation for Noise Vanishing Distribution Sampling

Maire¹,

Vandekerkhove²

2018

Preprint

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A novel strategy that combines a given collection of π-reversible Markov kernels is proposed. It consists in a Markov chain that moves, at each iteration, according to one of the available Markov kernels selected via a state-dependent probability distribution which is thus dubbed locally informed. In contrast to random-scan approaches that assume a constant selection probability distribution, the state-dependent distribution is typically specified so as to privilege moving according to a kernel which is relevant for the local topology of the target distribution.The second contribution is to characterize situations where a locally informed strategy should be preferred to its random-scan counterpart. We find that for a specific class of target distribution, referred to as sparse and filamentary, that exhibits a strong correlation between some variables and/or which concentrates its probability mass on some low dimensional linear subspaces or on thinned curved manifolds, a locally informed strategy converges substantially faster and yields smaller asymptotic variances than an equivalent random-scan algorithm.The research is at this stage essentially speculative: this paper combines a series of observations on this topic, both theoretical and empirical, that could serve as a groundwork for further investigations.

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Directional Metropolis-Hastings

Cited by 2 publications

References 4 publications

Hug and Hop: a discrete-time, non-reversible Markov chain Monte-Carlo algorithm

Hug and Hop: a discrete-time, non-reversible Markov chain Monte-Carlo algorithm

Markov Kernels Local Aggregation for Noise Vanishing Distribution Sampling

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