Particle Efficient Importance Sampling

The efficient importance sampling (EIS) method is a general principle for the numerical evaluation of high-dimensional integrals that uses the sequential structure of target integrands to build variance minimising importance samplers. Despite a number of successful applications in high dimensions, it is well known that importance sampling strategies are subject to an exponential growth in variance as the dimension of the integration increases. We solve this problem by recognising that the EIS framework has an offline sequential Monte Carlo interpretation. The particle EIS method is based on non-standard resampling weights that take into account the look-ahead construction of the importance sampler. We apply the method for a range of univariate and bivariate stochastic volatility specifications. We also develop a new application of the EIS approach to state space models with Student's t state innovations. Our results show that the particle EIS method strongly outperforms both the standard EIS method and particle filters for likelihood evaluation in high dimensions. Moreover, the ratio between the variances of the particle EIS and particle filter methods remains stable as the time series dimension increases. We illustrate the efficiency of the method for Bayesian inference using the particle marginal Metropolis-Hastings and importance sampling squared algorithms

Blister patterns and energy minimization in compressed thin films on compliant substrates

This paper is motivated by the complex blister patterns sometimes seen in thin elastic films on thick, compliant substrates. These patterns are often induced by an elastic misfit which compresses the film. Blistering permits the film to expand locally, reducing the elastic energy of the system. It is natural to ask: what is the minimum elastic energy achievable by blistering on a fixed area fraction of the substrate? This is a variational problem involving both the {\it elastic deformation} of the film and substrate and the {\it geometry} of the blistered region. It involves three small parameters: the {\it nondimensionalized thickness} of the film, the {\it compliance ratio} of the film/substrate pair and the {\it mismatch strain}. In formulating the problem, we use a small-slope (F\"oppl-von K\'arm\'an) approximation for the elastic energy of the film, and a local approximation for the elastic energy of the substrate. For a 1D version of the problem, we obtain "matching" upper and lower bounds on the minimum energy, in the sense that both bounds have the same scaling behavior with respect to the small parameters. For a 2D version of the problem, our results are less complete. Our upper and lower bounds only "match" in their scaling with respect to the nondimensionalized thickness, not in the dependence on the compliance ratio and the mismatch strain. The upper bound considers a 2D lattice of blisters, and uses ideas from the literature on the folding or "crumpling" of a confined elastic sheet. Our main 2D result is that in a certain parameter regime, the elastic energy of this lattice is significantly lower than that of a few large blisters

The coarsening of folds in hanging drapes

We consider the elastic energy of a hanging drape -- a thin elastic sheet, pulled down by the force of gravity, with fine-scale folding at the top that achieves approximately uniform confinement. This example of energy-driven pattern formation in a thin elastic sheet is of particular interest because the length scale of folding varies with height. We focus on how the minimum elastic energy depends on the physical parameters. As the sheet thickness vanishes, the limiting energy is due to the gravitational force and is relatively easy to understand. Our main accomplishment is to identify the "scaling law" of the correction due to positive thickness. We do this by (i) proving an upper bound, by considering the energies of several constructions and taking the best; (ii) proving an ansatz-free lower bound, which agrees with the upper bound up to a parameter-independent prefactor. The coarsening of folds in hanging drapes has also been considered in the recent physics literature, using a self-similar construction whose basic cell has been called a "wrinklon." Our results complement and extend that work, by showing that self-similar coarsening achieves the optimal scaling law in a certain parameter regime, and by showing that other constructions (involving lateral spreading of the sheet) do better in other regions of parameter space. Our analysis uses a geometrically linear F\"{o}ppl-von K\'{a}rm\'{a}n model for the elastic energy, and is restricted to the case when Poisson's ratio is zero.Comment: 34 page

On approximating copulas by finite mixtures

Copulas are now frequently used to approximate or estimate multivariate distributions because of their ability to take into account the multivariate dependence of the variables while controlling the approximation properties of the marginal densities. Copula based multivariate models can often also be more parsimonious than fitting a flexible multivariate model, such as a mixture of normals model, directly to the data. However, to be effective, it is imperative that the family of copula models considered is sufficiently flexible. Although finite mixtures of copulas have been used to construct flexible families of copulas, their approximation properties are not well understood and we show that natural candidates such as mixtures of elliptical copulas and mixtures of Archimedean copulas cannot approximate a general copula arbitrarily well. Our article develops fundamental tools for approximating a general copula arbitrarily well by a mixture and proposes a family of finite mixtures that can do so. We illustrate empirically on a financial data set that our approach for estimating a copula can be much more parsimonious and results in a better fit than approximating the copula by a mixture of normal copulas.Comment: 26 pages and 1 figure and 2 table

On Scalable Particle Markov Chain Monte Carlo

Particle Markov Chain Monte Carlo (PMCMC) is a general approach to carry out Bayesian inference in non-linear and non-Gaussian state space models. Our article shows how to scale up PMCMC in terms of the number of observations and parameters by expressing the target density of the PMCMC in terms of the basic uniform or standard normal random numbers, instead of the particles, used in the sequential Monte Carlo algorithm. Parameters that can be drawn efficiently conditional on the particles are generated by particle Gibbs. All the other parameters are drawn by conditioning on the basic uniform or standard normal random variables; e.g. parameters that are highly correlated with the states, or parameters whose generation is expensive when conditioning on the states. The performance of this hybrid sampler is investigated empirically by applying it to univariate and multivariate stochastic volatility models having both a large number of parameters and a large number of latent states and shows that it is much more efficient than competing PMCMC methods. We also show that the proposed hybrid sampler is ergodic