126 research outputs found
On the ergodicity properties of some adaptive MCMC algorithms
In this paper we study the ergodicity properties of some adaptive Markov
chain Monte Carlo algorithms (MCMC) that have been recently proposed in the
literature. We prove that under a set of verifiable conditions, ergodic
averages calculated from the output of a so-called adaptive MCMC sampler
converge to the required value and can even, under more stringent assumptions,
satisfy a central limit theorem. We prove that the conditions required are
satisfied for the independent Metropolis--Hastings algorithm and the random
walk Metropolis algorithm with symmetric increments. Finally, we propose an
application of these results to the case where the proposal distribution of the
Metropolis--Hastings update is a mixture of distributions from a curved
exponential family.Comment: Published at http://dx.doi.org/10.1214/105051606000000286 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Sampling from a log-concave distribution with compact support with proximal Langevin Monte Carlo
This paper presents a detailed theoretical analysis of the Langevin Monte
Carlo sampling algorithm recently introduced in Durmus et al. (Efficient
Bayesian computation by proximal Markov chain Monte Carlo: when Langevin meets
Moreau, 2016) when applied to log-concave probability distributions that are
restricted to a convex body . This method relies on a
regularisation procedure involving the Moreau-Yosida envelope of the indicator
function associated with . Explicit convergence bounds in total
variation norm and in Wasserstein distance of order are established. In
particular, we show that the complexity of this algorithm given a first order
oracle is polynomial in the dimension of the state space. Finally, some
numerical experiments are presented to compare our method with competing MCMC
approaches from the literature
Balanced Training of Energy-Based Models with Adaptive Flow Sampling
Energy-based models (EBMs) are versatile density estimation models that
directly parameterize an unnormalized log density. Although very flexible, EBMs
lack a specified normalization constant of the model, making the likelihood of
the model computationally intractable. Several approximate samplers and
variational inference techniques have been proposed to estimate the likelihood
gradients for training. These techniques have shown promising results in
generating samples, but little attention has been paid to the statistical
accuracy of the estimated density, such as determining the relative importance
of different classes in a dataset. In this work, we propose a new maximum
likelihood training algorithm for EBMs that uses a different type of generative
model, normalizing flows (NF), which have recently been proposed to facilitate
sampling. Our method fits an NF to an EBM during training so that an
NF-assisted sampling scheme provides an accurate gradient for the EBMs at all
times, ultimately leading to a fast sampler for generating new data
On Riemannian Stochastic Approximation Schemes with Fixed Step-Size
This paper studies fixed step-size stochastic approximation (SA) schemes,
including stochastic gradient schemes, in a Riemannian framework. It is
motivated by several applications, where geodesics can be computed explicitly,
and their use accelerates crude Euclidean methods. A fixed step-size scheme
defines a family of time-homogeneous Markov chains, parametrized by the
step-size. Here, using this formulation, non-asymptotic performance bounds are
derived, under Lyapunov conditions. Then, for any step-size, the corresponding
Markov chain is proved to admit a unique stationary distribution, and to be
geometrically ergodic. This result gives rise to a family of stationary
distributions indexed by the step-size, which is further shown to converge to a
Dirac measure, concentrated at the solution of the problem at hand, as the
step-size goes to 0. Finally, the asymptotic rate of this convergence is
established, through an asymptotic expansion of the bias, and a central limit
theorem.Comment: 37 pages, 4 figures, to appear in AISTAT2
Nonparametric inference of photon energy distribution from indirect measurements
International audienceWe consider a density estimation problem arising in nuclear physics. Gamma photons are impinging on a semiconductor detector, producing pulses of current. The integral of this pulse is equal to the total amount of charge created by the photon in the detector, which is linearly related to the photon energy. Because the inter-arrival of photons can be shorter than the charge collection time, pulses corresponding to different photons may overlap leading to a phenomenon known as pileup. The distortions on the photon energy spectrum estimate due to pileup become worse when the photon rate increases, making pileup correction techniques a must for high counting rate experiments. In this paper, we present a novel technique to correct pileup, which extends a method introduced in \cite{hall:park:2004} for the estimation of the service time from the busy period in M/G/ models. It is based on a novel formula linking the joint distribution of the energy and duration of the cluster of pulses and the distribution of the energy of the photons. We then assess the performance of this estimator by providing an expression of its integrated square error. A Monte-Carlo experiment is presented to illustrate on practical examples the benefits of the pileup correction
Uniform minorization condition and convergence bounds for discretizations of kinetic Langevin dynamics
We study the convergence in total variation and -norm of discretization
schemes of the underdamped Langevin dynamics. Such algorithms are very popular
and commonly used in molecular dynamics and computational statistics to
approximatively sample from a target distribution of interest. We show first
that, for a very large class of schemes, a minorization condition uniform in
the stepsize holds. This class encompasses popular methods such as the
Euler-Maruyama scheme and the schemes based on splitting strategies. Second, we
provide mild conditions ensuring that the class of schemes that we consider
satisfies a geometric Foster--Lyapunov drift condition, again uniform in the
stepsize. This allows us to derive geometric convergence bounds, with a
convergence rate scaling linearly with the stepsize. This kind of result is of
prime interest to obtain estimates on norms of solutions to Poisson equations
associated with a given numerical method
On the two-filter approximations of marginal smoothing distributions in general state space models
International audienceA prevalent problem in general state space models is the approximation of the smoothing distribution of a state conditional on the observations from the past, the present, and the future. The aim of this paper is to provide a rigorous analysis of such approximations of smoothed distributions provided by the two-filter algorithms. We extend the results available for the approximation of smoothing distributions to these two-filter approaches which combine a forward filter approximating the filtering distributions with a backward information filter approximating a quantity proportional to the posterior distribution of the state given future observations
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