Alexandre H. Thiery

On the efficiency of pseudo-marginal random walk Metropolis algorithms

We examine the behaviour of the pseudo-marginal random walk Metropolis algorithm, where evaluations of the target density for the accept/reject probability are estimated rather than computed precisely. Under relatively general conditions on the target distribution, we obtain limiting formulae for the acceptance rate and for the expected squared jump distance, as the dimension of the target approaches infinity, under the assumption that the noise in the estimate of the log-target is additive and is independent of the position. For targets with independent and identically distributed components, we also obtain a limiting diffusion for the first component. We then consider the overall efficiency of the algorithm, in terms of both speed of mixing and computational time. Assuming the additive noise is Gaussian and is inversely proportional to the number of unbiased estimates that are used, we prove that the algorithm is optimally efficient when the variance of the noise is approximately 3.3 and the acceptance rate is approximately 7.0%. We also find that the optimal scaling is insensitive to the noise and that the optimal variance of the noise is insensitive to the scaling. The theory is illustrated with a simulation study using the particle random walk Metropolis.

Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an N-dimensional approximation of the target will take

On coupling particle filter trajectories

\mathcal{O}(N^{1/3})

A Deep Learning Approach to Digitally Stain Optical Coherence Tomography Images of the Optic Nerve Head

steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require

Consistency and fluctuations for stochastic gradient Langevin dynamics

\mathcal{O}(N)

ON THE CONVERGENCE OF ADAPTIVE SEQUENTIAL MONTE CARLO METHODS

steps when applied to the same class of problems.

On nonnegative unbiased estimators

Particle filters are a powerful and flexible tool for performing inference on state-space models. They involve a collection of samples evolving over time through a combination of sampling and re-sampling steps. The re-sampling step is necessary to ensure that weight degeneracy is avoided. In several situations of statistical interest, it is important to be able to compare the estimates produced by two different particle filters; consequently, being able to efficiently couple two particle filter trajectories is often of paramount importance. In this text, we propose several ways to do so. In particular, we leverage ideas from the optimal transportation literature. In general, though computing the optimal transport map is extremely computationally expensive, to deal with this, we introduce computationally tractable approximations to optimal transport couplings. We demonstrate that our resulting algorithms for coupling two particle filter trajectories often perform orders of magnitude more efficiently than more standard approaches.

Optimal Proposal Design for Random Walk Type Metropolis Algorithms with Gaussian Random Field Priors

Purpose To develop a deep learning approach to digitally stain optical coherence tomography (OCT) images of the optic nerve head (ONH). Methods A horizontal B-scan was acquired through the center of the ONH using OCT (Spectralis) for one eye of each of 100 subjects (40 healthy and 60 glaucoma). All images were enhanced using adaptive compensation. A custom deep learning network was then designed and trained with the compensated images to digitally stain (i.e., highlight) six tissue layers of the ONH. The accuracy of our algorithm was assessed (against manual segmentations) using the dice coefficient, sensitivity, specificity, intersection over union (IU), and accuracy. We studied the effect of compensation, number of training images, and performance comparison between glaucoma and healthy subjects. Results For images it had not yet assessed, our algorithm was able to digitally stain the retinal nerve fiber layer + prelamina, the RPE, all other retinal layers, the choroid, and the peripapillary sclera and lamina cribrosa. For all tissues, the dice coefficient, sensitivity, specificity, IU, and accuracy (mean) were 0.84 ± 0.03, 0.92 ± 0.03, 0.99 ± 0.00, 0.89 ± 0.03, and 0.94 ± 0.02, respectively. Our algorithm performed significantly better when compensated images were used for training (P < 0.001). Besides offering a good reliability, digital staining also performed well on OCT images of both glaucoma and healthy individuals. Conclusions Our deep learning algorithm can simultaneously stain the neural and connective tissues of the ONH, offering a framework to automatically measure multiple key structural parameters of the ONH that may be critical to improve glaucoma management.