The Alive Particle Filter and Its Use in Particle Markov Chain Monte Carlo

In the following article, we investigate a particle filter for approximating Feynman–Kac models with indicator potentials and we use this algorithm within Markov chain Monte Carlo (MCMC) to learn static parameters of the model. Examples of such models include approximate Bayesian computation (ABC) posteriors associated with hidden Markov models (HMMs) or rare-event problems. Such models require the use of advanced particle filter or MCMC algorithms to perform estimation. One of the drawbacks of existing particle filters is that they may “collapse,” in that the algorithm may terminate early, due to the indicator potentials. In this article, using a newly developed special case of the locally adaptive particle filter, we use an algorithm that can deal with this latter problem, while introducing a random cost per-time step. In particular, we show how this algorithm can be used within MCMC, using particle MCMC. It is established that, when not taking into account computational time, when the new MCMC algorithm is applied to a simplified model it has a lower asymptotic variance in comparison to a standard particle MCMC algorithm. Numerical examples are presented for ABC approximations of HMMs.     

Conditional Monte Carlo method for dynamic systems with random properties

A method is developed for calculating moments and other properties of states X(t) of dynamic systems with random coefficients depending on semi-Markov processes ξ(t) and subjected to Gaussian white noise. Random vibration theory is used to find probability laws of conditional processes X(t)∣ξ(·). Unconditional properties of X(t) are estimated by averaging conditional statistics of this process corresponding to samples of ξ(t). The method is particularly efficient for linear systems since X(t)∣ξ(·) is Gaussian during periods of constant values of ξ(t), so that and its probability law is completely defined by the process mean and covariance functions that can be obtained simply from equations of linear random vibration. The method is applied to find statistics of an Ornstein-Uhlenbeck process X(t) whose decay parameter is a semi-Markov process ξ(t). Numerical results show that X(t) is not Gaussian and that the law of this process depends essentially on features of ξ(t). A version of the method is used to calculate the failure probability for an oscillator with degrading stiffness subjected to Gaussian white noise.     

Monte Carlo and quasi-Monte Carlo sampling methods for a class of stochastic mathematical programs with equilibrium constraints

In this paper, we consider a class of stochastic mathematical programs with equilibrium constraints introduced by Birbil et al. (Math Oper Res 31:739–760, 2006). Firstly, by means of a Monte Carlo method, we obtain a nonsmooth discrete approximation of the original problem. Then, we propose a smoothing method together with a penalty technique to get a standard nonlinear programming problem. Some convergence results are established. Moreover, since quasi-Monte Carlo methods are generally faster than Monte Carlo methods, we discuss a quasi-Monte Carlo sampling approach as well. Furthermore, we give an example in economics to illustrate the model and show some numerical results with this example. The first author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science and SRF for ROCS, SEM. The second author’s work was supported in part by the United Kingdom Engineering and Physical Sciences Research Council grant. The third author’s work was supported in part by the Scientific Research Grant-in-Aid from Japan Society for the Promotion of Science.     

Nonparametric adaptive detection in fading channels based on sequential Monte Carlo and Bayesian model averaging

Recently, a Bayesian receiver for blind detection in fading channels has been proposed by Chen, Wang and Liu (200,IEEE Trans. Inform. Theory,46, 2079–2094), based on the sequential Monte Carlo methodology. That work is built on a parametric modelling of the fading process in the form of a state-space model, and assumes the knowledge of the second-order statistics of the fading channel. In this paper, we develop a nonparametric approach to the problem of blind detection in fading channels, without assuming any knowledge of the channel statistics. The basic idea is to decompose the fading process using a wavelet basis, and to use the sequential Monte Carlo technique to track both the wavelet coefficients and the transmitted symbols. Moreover, the algorithm is adaptive to time varying speed/smoothness in the fading process and the uncertainty on the number of wavelet coefficients (shrinkage order) needed. Simulation results are provided to demonstrate the excellent performance of the proposed blind adaptive receivers. This work was supported in part by the U.S. National Science Foundation (NSF) under grants CCR-9875314, CCR-9980599, DMS-9982846, DMS-0073651 and DMS-0073601.