(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

An Introduction to Probability Theory and its Applications, Vol. II.

Markov chain Monte Carlo (MCMC) methods are often deemed far too computationally intensive to be of any practical use for large datasets. This paper describes a methodology that aims to scale up the Metropolis-Hastings (MH) algorithm in this context. We propose an approximate implementation of the accept/reject step of MH that only requires evaluating the likelihood of a random subset of the data, yet is guaranteed to coincide with the accept/reject step based on the full dataset with a probability superior to a user-specified tolerance level. This adaptive subsampling technique is an alternative to the recent approach developed in (Korattikara et al., 2014), and it allows us to establish rigorously that the resulting approximate MH algorithm samples from a perturbed version of the target distribution of interest, whose total variation distance to this very target is controlled explicitly. We explore the benefits and limitations of this scheme on several examples.

/pdf/towards-scaling-up-markov-chain-monte-carlo-an-adaptive-2xywcym2oi.pdf

Towards scaling up Markov chain Monte Carlo: an adaptive subsampling approach

We consider ℝd-valued diffusion processes of type
dXt = b(Xt)dt+dBt.
Assuming a geometric drift condition, we establish contractions of the transitions kernels in Kantorovich (L1 Wasserstein) distances with explicit constants. Our results are in the spirit of Hairer and Mattingly's extension of Harris' Theorem. In particular, they do not rely on a small set condition. Instead we combine Lyapunov functions with reflection coupling and concave distance functions. We retrieve constants that are explicit in parameters which can be computed with little effort from one-sided Lipschitz conditions for the drift coefficient and the growth of a chosen Lyapunov function. Consequences include exponential convergence in weighted total variation norms, gradient bounds, bounds for ergodic averages, and Kantorovich contractions for nonlinear McKean-Vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities. We also establish quantitative bounds for sub-geometric ergodicity assuming a sub-geometric drift condition.

Quantitative Harris-type theorems for diffusions and McKean–Vlasov processes

We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis–Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler’s exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the exact posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this...

Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations

Let $(X_t, Y_t)_{t\in T}$ be a discrete or continuous-time Markov process with state space $X \times R^d$ where $X$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{t\in T}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{t\in T}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_t)_{t\in T}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup{t\in(0,1]\cap T : E{\pi,0}[|Y_t| ^{\alpha}] 0$ for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_t)_{t\in T}$ has an invariant probability distribution $\pi$, is stationary and has the $L^2(\pi)$-spectral gap property (that is, $(X_t)t\in N}$ is $\rho$-mixing in the discrete-time case). The case where $(X_t)_{t\in T}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $\rho$-mixing Markov chains.

Limit theorems for stationary Markov processes with L2-spectral gap

Let $\{X_{n}\}_{n\geq 0}$ be a $V$-geometrically ergodic Markov chain. Given some real-valued functional $F$, define $M_{n}(\alpha):=n^{-1}\sum^{n}_{k=1} F(\alpha, X_{k-1}, X_{k}), \alpha \in \mathcal A \subset \mathbb R$. Consider an $M$-estimator $\widehat{\alpha}_{n}$, that is as a measurable function of the observations satisfying $M_{n} (v)\leq min_{\alpha \in\mathcal A} M_{n}(\alpha)+ c_{n}$ with $\{c_{n}\}_{n\geq 1}$ some sequence of real numbers going to zero. Under some standard regularity and moment assumptions, close to those of the i.i.d. case, the estimator $\widehat{\alpha}_{n}$ satisfies a Berry-Esseen theorem uniformly with respect to the underlying probability distribution of the Markov chain.

/pdf/a-uniform-berry-esseen-theorem-on-m-estimators-for-23adaa7bxe.pdf

A uniform Berry-Esseen theorem on M¡estimators for geometrically ergodic Markov chains

Let $P$ be a Markov kernel on a measurable space $\X$ and let $V:\X\r[1,+\infty)$. We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space $(\cB_V,\|\cdot\|_V)$ of all the measurable functions $f : \X\r\C$ such that $\|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty$. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on $\cB_V$, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems are presented.

/pdf/quasi-compactness-of-markov-kernels-on-weighted-supremum-4di8qg9f8f.pdf

Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

Soit $(X_{t},Y_{t})_{t\in \mathbb {T}}$ un processus de Markov en temps discret ou continu et d’espace d’etat $\mathbb {X}\times \mathbb {R}^{d}$ ou $\mathbb {X}$ est un ensemble mesurable quelconque. Son semi-groupe de transition est suppose additif suivant la seconde composante, i.e. $(X_{t},Y_{t})_{t\in\mathbb {T}}$ est un processus additif Markovien. En particulier, ceci implique que la premiere composante $(X_{t})_{t\in \mathbb {T}}$ est egalement un processus de Markov. Les marches aleatoires Markoviennes ou les fonctionnelles additives d’un processus de Markov sont des exemples de processus additifs Markoviens. Dans cet article, on montre que le processus $(Y_{t})_{t\in \mathbb {T}}$ satisfait les theoremes limites classiques suivants : (a) le theoreme de la limite centrale, (b) le theoreme limite local, (c) le theoreme uniforme de Berry–Esseen en dimension un, (d) le developpement d’Edgeworth d’ordre un en dimension un, pourvu que la condition de moment $\sup_{t\in(0,1]\cap \mathbb {T}}\mathbb {E}_{\pi,0}[|Y_{t}|^{\alpha}]\textless\infty$ soit satisfaite, avec l’ordre attendu α du cas independant (a un e > 0 pres pour (c) et (d)). Pour les enonces (b) et (d), il faut ajouter une condition nonlattice comme dans le cas independant. Tous les resultats sont obtenus sous l’hypothese d’un processus de Markov $(X_{t})_{t\in \mathbb {T}}$ admettant une mesure de probabilite invariante π et possedant la propriete de trou spectral sur $\mathbb {L}^{2}(\pi)$ (c’est a dire, (Xt)t∈ℕ est ρ-melangeante dans le cas du temps discret). Le cas ou $(X_{t})_{t\in \mathbb {T}}$ est non-stationnaire est brievement aborde. Nous appliquons nos resultats pour obtenir une borne de Berry–Esseen pour les M-estimateurs associes aux chaines de Markov ρ-melangeantes.

/pdf/limit-theorems-for-stationary-markov-processes-with-l2-37dcqbib8k.pdf

In this paper, new conditions for the stability of V-geometrically ergodic Markov chains are introduced. The results are based on an extension of the standard perturbation theory formulated by Keller and Liverani. The continuity and higher regularity properties are investigated. As an illustration, an asymptotic expansion of the invariant probability measure for an autoregressive model with i.i.d. noises (with a non-standard probability density function) is obtained.

Regular perturbation of V -geometrically ergodic Markov chains

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Papers

Limit theorems for stationary Markov processes with L2-spectral gap

A uniform Berry-Esseen theorem on M¡estimators for geometrically ergodic Markov chains

Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

Limit theorems for stationary Markov processes with L2-spectral gap

Regular perturbation of V -geometrically ergodic Markov chains