We study sampling algorithms for $β$-ensembles with time complexity less than cubic in the cardinality of the ensemble. Following Dumitriu & Edelman (2002), we see the ensemble as the eigenvalues of a random tridiagonal matrix, namely a random Jacobi matrix. First, we provide a unifying and elementary treatment of the tridiagonal models associated to the three classical Hermite, Laguerre and Jacobi ensembles. For this purpose, we use simple changes of variables between successive reparametrizations of the coefficients defining the tridiagonal matrix. Second, we derive an approximate sampler for the simulation of $β$-ensembles, and illustrate how fast it can be for polynomial potentials. This method combines a Gibbs sampler on Jacobi matrices and the diagonalization of these matrices. In practice, even for large ensembles, only a few Gibbs passes suffice for the marginal distribution of the eigenvalues to fit the expected theoretical distribution. When the conditionals in the Gibbs sampler can be simulated exactly, the same fast empirical convergence is observed for the fluctuations of the largest eigenvalue. Our experimental results support a conjecture by Krishnapur et al. (2016), that the Gibbs chain on Jacobi matrices of size $N$ mixes in $\mathcal{O}(\log(N))$.
We introduce Ensemble Rejection Sampling, a scheme for exact simulation from the posterior distribution of the latent states of a class of non-linear non-Gaussian state-space models. Ensemble Rejection Sampling relies on a proposal for the high-dimensional state sequence built using ensembles of state samples. Although this algorithm can be interpreted as a rejection sampling scheme acting on an extended space, we show under regularity conditions that the expected computational cost to obtain an exact sample increases cubically with the length of the state sequence instead of exponentially for…
Institute for Astronomy, University of Edinburgh（爱丁堡大学天文研究所）
Slice Sampling has emerged as a powerful Markov Chain Monte Carlo algorithm that adapts to the characteristics of the target distribution with minimal hand-tuning. However, Slice Sampling's performance is highly sensitive to the user-specified initial length scale hyperparameter and the method generally struggles with poorly scaled or strongly correlated distributions. This paper introduces Ensemble Slice Sampling (ESS), a new class of algorithms that bypasses such difficulties by adaptively tuning the initial length scale and utilising an ensemble of parallel walkers in order to efficiently h…
The recent growth in multi-fidelity uncertainty quantification has given rise to a large set of variance reduction techniques that leverage information from model ensembles to provide variance reduction for estimates of the statistics of a high-fidelity model. In this paper we provide two contributions: (1) we utilize an ensemble estimator to account for uncertainties in the optimal weights of approximate control variate (ACV) approaches and derive lower bounds on the number of samples required to guarantee variance reduction; and (2) we extend an existing multi-fidelity importance sampling (M…
The embedded ensemble propagation approach introduced in [49] has been demonstrated to be a powerful means of reducing the computational cost of sampling-based uncertainty quantification methods, particularly on emerging computational architectures. A substantial challenge with this method however is ensemble-divergence, whereby different samples within an ensemble choose different code paths. This can reduce the effectiveness of the method and increase computational cost. Therefore grouping samples together to minimize this divergence is paramount in making the method effective for challengin…
Bayesian inference for factorial hidden Markov models is challenging due to the exponentially sized latent variable space. Standard Monte Carlo samplers can have difficulties effectively exploring the posterior landscape and are often restricted to exploration around localised regions that depend on initialisation. We introduce a general purpose ensemble Markov Chain Monte Carlo (MCMC) technique to improve on existing poorly mixing samplers. This is achieved by combining parallel tempering and an auxiliary variable scheme to exchange information between the chains in an efficient way. The latt…
The choice of the prior model can have a large impact on the ability to assimilate data. In standard applications of ensemble-based data assimilation, all realizations in the initial ensemble are generated from the same covariance matrix with the implicit assumption that this covariance is appropriate for the problem. In a hierarchical approach, the parameters of the covariance function, for example the variance, the orientation of the anisotropy and the ranges in two principal directions, may all be uncertain. Thus, the hierarchical approach is much more robust against model misspecification.…
I introduce a Markov chain Monte Carlo (MCMC) scheme in which sampling from a distribution with density pi(x) is done using updates operating on an "ensemble" of states. The current state x is first stochastically mapped to an ensemble, x^{(1)},...,x^{(K)}. This ensemble is then updated using MCMC updates that leave invariant a suitable ensemble density, rho(x^{(1)},...,x^{(K)}), defined in terms of pi(x^{(i)}) for i=1,...,K. Finally a single state is stochastically selected from the ensemble after these updates. Such ensemble MCMC updates can be useful when characteristics of pi and the ensem…
Ensembles of networks arise in various fields where multiple independent networks are observed on the same set of nodes, for example, a collection of brain networks constructed on the same brain regions for different individuals. However, there are few models that describe both the variations and characteristics of networks in an ensemble at the same time. In this paper, we propose to model the ensemble of networks using a Dirichlet Process Mixture of Exponential Random Graph Models (DPM-ERGMs), which divides the ensemble into different clusters and models each cluster of networks using a sepa…
Sampling of sharp posteriors in high dimensions is a challenging problem, especially when gradients of the likelihood are unavailable. In low to moderate dimensions, affine-invariant methods, a class of ensemble-based gradient-free methods, have found success in sampling concentrated posteriors. However, the number of ensemble members must exceed the dimension of the unknown state in order for the correct distribution to be targeted. Conversely, the preconditioned Crank-Nicolson (pCN) algorithm succeeds at sampling in high dimensions, but samples become highly correlated when the posterior dif…
The FFT EnKF data assimilation method is proposed and applied to a stochastic cell simulation of an epidemic, based on the S-I-R spread model. The FFT EnKF combines spatial statistics and ensemble filtering methodologies into a localized and computationally inexpensive version of EnKF with a very small ensemble, and it is further combined with the morphing EnKF to assimilate changes in the position of the epidemic.
Data assimilation combines information from models, measurements, and priors to estimate the state of a dynamical system such as the atmosphere. The Ensemble Kalman filter (EnKF) is a family of ensemble-based data assimilation approaches that has gained wide popularity due its simple formulation, ease of implementation, and good practical results. Most EnKF algorithms assume that the underlying probability distributions are Gaussian. Although this assumption is well accepted, it is too restrictive when applied to large nonlinear models, nonlinear observation operators, and large levels of unce…
emcee is a Python library implementing a class of affine-invariant ensemble samplers for Markov chain Monte Carlo (MCMC). This package has been widely applied to probabilistic modeling problems in astrophysics where it was originally published, with some applications in other fields. When it was first released in 2012, the interface implemented in emcee was fundamentally different from the MCMC libraries that were popular at the time, such as PyMC, because it was specifically designed to work with "black box" models instead of structured graphical models. This has been a popular interface for …
