Perfect Sampling using MCMC and its application in Signal P rocessing

1. Perfect Sampling using MCMC andits application in Signal Processing Presented by Sridivya Gajjarapu Florida State University sg12k@my.fsu.edu

2. Overview • Perfect Sampling • Coupling from the Past technique • Gibbs coupler • Applications in signal processing

3. Perfect sampling • MCMC methods which include Metropolis-Hastings algorithm and Gibbs sampler generate samples from the desired distribution only approximately. And the generated samples are dependent . • So, to overcome the drawbacks, Propp and Wilson proposeda new markov chain method which produces independent and identically distributed samples(i.i.d). And the samples are guaranteed to come from the desired distribution.

4. Perfect sampling methods • The proposed method was Coupling from the past(CFTP). Initially, it was developed for discrete distributions on finite number of states but later it was extended to allow for sampling from continuous state spaces. • An alternative approach to CFTP method was proposed called as Fill’s perfect rejection sampling algorithm. It is based on rejection sampling.

5. Coupling From The Past(CFTP) • Exploits an important tool in probability known as Coupling. • Coupling involves running coupled markov chains that start from the initial states. • With CFTP, the markov chains are run from the past(t<0) to the present(t=0) and the samples are always drawn at fixed time t=0,provided convergence of the chains occur. • N copies of the chain are run with different initial states and after sometime they coalesce and follow same path.

6. All the chains are started at time t=-1 and checked for convergence at t=0;If convergence occurs ,then the state of chain at t=0 is taken as sample from the desired distribution. If there is no convergence, then the starting time is moved back to t=-2 and the process is repeated.

7. Gibbs Coupler • An implementation of CFTP on binary state spaces and combines the CFTP scheme with the Gibbs sampling. • Important when sampling from high dimensional state spaces. • Gibbs coupler is designed for problems with large number of variables, and full conditional distributions are required for its implementation. • One critical issue of Gibbs coupler is the efficient determination of the support content S for every component at every time instant t. • Efficient updates can be achieved by introducing the concept of sandwich distributions.

8. At any time instant t, sandwich distributions are required which are defined as • It is proved that the sandwich distributions achieve the largest probability of convergence and hence their use leads to the fastest convergence of the chains. • Gibbs coupler has same rate of convergence as CFTP ,but it is computationally much more efficient especially for high dimensional state spaces.

9. Applications in signal processing • MCMC sampling methods are widely used in signal processing problems. • They are used to solve signal processing problems by producing samples which extract relevant information about a signal unknown. • They are used in wavelet reconstruction of signals, in binary restoration of the images, for implementing multiuser detectors in CDMA etc.

10. Multiuser detection using Gibbs coupler • In CDMA systems which have a large number of users, the transmitted signals from the users overlap in time and frequency. • So in order to recover the transmitted signals at the receiver, multiuser detection techniques are used. • A new multi user detector is proposed which is developed under the Bayesian framework and implemented by Gibbs coupler.

11. Consider K-user synchronous CDMA channel • It is assumed that all the parameters are known except b. • The objective is to estimate b. Various Bayesian estimators can be used for this and here Maximum a posteriori (MAP) is used to estimate b. • Gibbs coupler can be used to calculate the output of MAP detector. • The optimum estimate of b from Bayesian’s perspective is the set that maximizes the posterior distribution.

12. The implementation of the Gibbs coupler here is to determine the sandwich distributions on the full conditional distributions. • After the samples are obtained, the posterior probability of each perfect sample is determined and the MAP estimate will be the one which has the largest posterior probability. • When experimented, the numerical simulation results showed that MAP detectors using Gibbs coupler performed better than the single-user matched filter or the decorrelating detector which were earlier used for multiuser detection problem.

13. THANK YOU