Free Energy Methods for Bayesian Inference: Efficient Exploration of Univariate Gaussian Mixture Posteriors

Because of their multimodality, mixture posterior distributions are difficult to sample with standard Markov chain Monte Carlo (MCMC) methods. We propose a strategy to enhance the sampling of MCMC in this context, using a biasing procedure which originates from computational Statistical Physics. The principle is first to choose a "reaction coordinate", that is, a "direction" in which the target distribution is multimodal. In a second step, the marginal log-density of the reaction coordinate with respect to the posterior distribution is estimated; minus this quantity is called "free energy" in the computational Statistical Physics literature. To this end, we use adaptive biasing Markov chain algorithms which adapt their targeted invariant distribution on the fly, in order to overcome sampling barriers along the chosen reaction coordinate. Finally, we perform an importance sampling step in order to remove the bias and recover the true posterior. The efficiency factor of the importance sampling step can easily be estimated \emph{a priori} once the bias is known, and appears to be rather large for the test cases we considered. A crucial point is the choice of the reaction coordinate. One standard choice (used for example in the classical Wang-Landau algorithm) is minus the log-posterior density. We discuss other choices. We show in particular that the hyper-parameter that determines the order of magnitude of the variance of each component is both a convenient and an efficient reaction coordinate. We also show how to adapt the method to compute the evidence (marginal likelihood) of a mixture model. We illustrate our approach by analyzing two real data sets

Long-time convergence of an Adaptive Biasing Force method

We propose a proof of convergence of an adaptive method used in molecular dynamics to compute free energy profiles. Mathematically, it amounts to studying the long-time behavior of a stochastic process which satisfies a non-linear stochastic differential equation, where the drift depends on conditional expectations of some functionals of the process. We use entropy techniques to prove exponential convergence to the stationary state

Derivation of Langevin Dynamics in a Nonzero Background Flow Field

We propose a derivation of a nonequilibrium Langevin dynamics for a large particle immersed in a background flow field. A single large particle is placed in an ideal gas heat bath composed of point particles that are distributed consistently with the background flow field and that interact with the large particle through elastic collisions. In the limit of small bath atom mass, the large particle dynamics converges in law to a stochastic dynamics. This derivation follows the ideas of [D. D\"urr, S. Goldstein, and J. L. Lebowitz, 1981 and 1983; P. Calderoni, D. D\"urr, and S. Kusuoka, 1989] and provides extensions to handle the nonzero background flow. The derived nonequilibrium Langevin dynamics is similar to the dynamics in [M. McPhie, et al., 2001]. Some numerical experiments illustrate the use of the obtained dynamic to simulate homogeneous liquid materials under flow.Comment: Minor revisions, refined discussion of the laminar bath approach and non-Hamiltonian dynamics approach in Section 2. 41 pages, 8 figure

Computation of free energy profiles with parallel adaptive dynamics

We propose a formulation of adaptive computation of free energy differences, in the ABF or nonequilibrium metadynamics spirit, using conditional distributions of samples of configurations which evolve in time. This allows to present a truly unifying framework for these methods, and to prove convergence results for certain classes of algorithms. From a numerical viewpoint, a parallel implementation of these methods is very natural, the replicas interacting through the reconstructed free energy. We show how to improve this parallel implementation by resorting to some selection mechanism on the replicas. This is illustrated by computations on a model system of conformational changes.Comment: 4 pages, 1 Figur

An efficient sampling algorithm for Variational Monte Carlo

We propose a new algorithm for sampling the $N$-body density $|\Psi({\bf R})|^2/\int_{\mathbb{R}^{3N}} |\Psi|^2$ in the Variational Monte Carlo (VMC) framework. This algorithm is based upon a modified Ricci-Ciccotti discretization of the Langevin dynamics in the phase space $({\bf R},{\bf P})$ improved by a Metropolis acceptation/rejection step. We show through some representative numerical examples (Lithium, Fluorine and Copper atoms, and phenol molecule), that this algorithm is superior to the standard sampling algorithm based on the biased random walk (importance sampling).Comment: 23 page

Concurrent Geometric Multicasting

We present MCFR, a multicasting concurrent face routing algorithm that uses geometric routing to deliver a message from source to multiple targets. We describe the algorithm's operation, prove it correct, estimate its performance bounds and evaluate its performance using simulation. Our estimate shows that MCFR is the first geometric multicast routing algorithm whose message delivery latency is independent of network size and only proportional to the distance between the source and the targets. Our simulation indicates that MCFR has significantly better reliability than existing algorithms

Immune Reactivity and Pseudoprogression or Tumor Flare in a Serially Biopsied Neuroendocrine Patient Treated with the Epigenetic Agent RRx-001.

Neuroendocrine tumors (NETs) are grouped together as a single class on the basis of histologic appearance, immunoreactivity for the neuroendocrine markers chromogranin A and synaptophysin, and potential secretion of hormones, neurotransmitters, neuromodulators and neuropeptides. Nevertheless, despite these common characteristics, NETs differ widely in terms of their natural histories: high-grade NETs are clinically aggressive and, like small cell lung cancer, which they most closely resemble, tend to respond to cisplatin and etoposide. In contrast, low-grade NETs, which as a rule progress and behave indolently, do not. In either case, the treatment strategy, apart from potentially curative surgical resection, is very poorly defined. This report describes the case of a 28-year-old white male with a diagnosis of high-grade NET of undetermined primary site metastatic to the lymph nodes, skin and paraspinal soft tissues, treated with the experimental anticancer agent RRx-001, in the context of a phase II clinical trial called TRIPLE THREAT (NCT02489903); serial sampling of tumor material through repeat biopsies demonstrated an intratumoral inflammatory response, including the amplification of infiltrating T cells, which correlated with clinical and symptomatic benefit. This case suggests that pseudoprogression or RRx-001-induced enlargement of tumor lesions, which has been previously described for several RRx-001-treated patients, is the result of tumoral lymphocyte infiltration

An objective frequency domain method for quantifying confined aquifer compressible storage using Earth and atmospheric tides

The groundwater hydraulic head response to the worldwide and ubiquitous atmospheric tide at 2 cycles per day (cpd) is a direct function of confined aquifer compressible storage. The ratio of the responses of hydraulic head to the atmospheric pressure change is a measure of aquifer barometric efficiency, from which formation compressibility and aquifer specific storage can be determined in situ rather than resorting to laboratory or aquifer pumping tests. The Earth tide also impacts the hydraulic head response at the same frequency, and a method is developed here to quantify and remove this interference. As a result, the barometric efficiency can be routinely calculated from 6-hourly hydraulic head, atmospheric pressure, and modeled Earth tide records where available for a minimum of 15 days duration. This new approach will be of critical importance in assessing worldwide problems of land subsidence or groundwater resource evaluation that both occur due to groundwater abstractio

Reviews

Europe In the Round CD‐ROM, Guildford, Vocational Technologies, 1994