## Dynamic Weighting In Markov Chain Monte Carlo (1998)

Citations: | 1 - 1 self |

### BibTeX

@TECHREPORT{Liu98dynamicweighting,

author = {Jun S. Liu and Faming Liang and Wing Hung Wong},

title = {Dynamic Weighting In Markov Chain Monte Carlo},

institution = {},

year = {1998}

}

### OpenURL

### Abstract

This article provides a first theoretical analysis on a new Monte Carlo approach, the dynamic weighting, proposed recently by Wong and Liang. In dynamic weighting, one augments the original state space of interest by a weighting factor, which allows the resulting Markov chain to move more freely and to escape from local modes. It uses a new invariance principle to guide the construction of transition rules. We analyze the behaviors of the weights resulting from such a process and provide detailed recommendations on how to use these weights properly. Our recommendations are supported by a renewal theory-type analysis. Our theoretical investigations are further demonstrated by a simulation study and applications in the neural network training and the Ising model simulations. Keywords: Gibbs Sampling; Importance Sampling; Ising Model, Metropolis algorithm, Neural Network, Renewal Theory, Simulated Annealing, Simulated Tempering, 1 Jun S. Liu is Assistant Professor, Department of Statisti...

### Citations

3719 |
Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images
- Geman, Geman
- 1984
(Show Context)
Citation Context ...e in these endeavors. In recent decades, Monte Carlo algorithms have received a lot of attention from researchers in engineering and computer science [e.g., Kirkpatrick, Gelatt, and Vecchi (1983) and =-=Geman and Geman 1984-=-)], statistical physics [e.g., Goodmand and Sokal (1989); Marinari and Parisi (1987); Swendsen and Wang (1987)], computational biology [e.g., Lawrence et al. (1993); Liu, Neuwald and Lawrence (1999); ... |

3530 | Optimization by simulated annealing - Kirkpatrick, Gellatt, et al. - 1983 |

2243 | Equation of state calculations by fast computing machines - Metropolis, Rosenbluth, et al. - 1953 |

1578 |
Neural networks and physical systems with emergent collective computational abilities
- Hopfield
- 1982
(Show Context)
Citation Context ...me. 7.2. Training a neural network. The artificial neural network is a simple mathematical model motivated by neuron functions and has been a widely used tool in learning and classification problems (=-=Hopfield, 1982; Rumelhar-=-t and McClelland 1986). The most popular among these networks is the so-called "multi-layer perceptrons (MLP)," which is a type of feedforward network. Our stochastic learning algorithm will... |

1217 | Monte carlo sampling methods using markov chains and their applications - Hastings |

1113 |
Pattern recognition and neural networks
- Ripley
- 1996
(Show Context)
Citation Context ... grouped into layers (typically there are three layers). The layers are ordered (i.e., input-hidden-output) so that the units in lower layers (input) only connects with the units in the higher layer (=-=Ripley 1996). E-=-ach node in a higher layer independently processes the values fed to it by nodes in the lower layer in the form y k = f k (ff k + X j��k w jk x j ); where the x j are inputs, and then present the ... |

825 | Reversible jump Markov chain Monte Carlo computation and Bayesian model determination - Green - 1995 |

745 |
Learning representations by back-propagating errors
- Rumelhart, Hinton, et al.
- 1986
(Show Context)
Citation Context ...t-decent and conjugate gradient. More details of the method are given by Liang (1997). We now illustrate this method in the encoder problem (Ackley, Hinton and Sejnowski 1985) and the parity problem (=-=Rumelhart et al. 1986-=-). These two problems have been regarded as classic benchmarks to test new methods in neural network community. Their difficulties stem from the stringent noiseless output requirement. The input in th... |

744 | Sampling-based approaches to calculating marginal densities - Gelfand, Smith - 1990 |

608 | Bayesian Learning for Neural Networks
- Neal
- 1996
(Show Context)
Citation Context ...lta ? t L . Wong and Liang chose L = 4 for the two-spiral problem. Conditional on T = t l (i.e., within each level), we use a standard Metropolis move to do local changes on the connection strengths (=-=Neal 1996-=-), whereas conditional on the w jk , we use a Q-type move to jump across level. After we have obtained reasonable configurations of the connection strengths from the lowest-temperature level, we condu... |

549 | The calculation of posterior distributions by data augmentation - Tanner, Wong - 1987 |

527 | Applied probability and queues - Asmussen - 2003 |

506 | Detecting subtle sequence signals: a Gibbs sampling strategy for multiple alignment - Lawrence, Altschul, et al. - 1993 |

431 | A learning algorithm for Boltzmann machines - Ackley, Hinton, et al. - 1985 |

301 | A Course in Probability Theory - Chung - 1968 |

288 |
Self-Organization and Associative
- Kohonen
- 1984
(Show Context)
Citation Context ...14-4-1 networks have been fitted and the results were close to be perfect, whereas the error rate for BP is generally greater than 40%). In training programs such as back propagation, LVQ algorithms (=-=Kohonen 1989-=-), the total mean squared error E p = X p kO p \Gamma T p k 2 ; where T p is the p-th training case's ideal output and O p is the output of the network, is used as the cost function. We use the same c... |

277 | Understanding molecular simulation: from algorithms to applications - Frenkel, Smit - 1996 |

255 | Nonuniversal critical dynamics in Monte Carlo simulations,” Phys - Swendsen, Wang - 1987 |

239 | General irreducible Markov chains and non-negative operators - Nummelin - 1984 |

187 | Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes - Liu, Wong, et al. - 1994 |

159 |
Annealing Markov chain Monte Carlo with applications to ancestral inference
- Geyer, Thompson
- 1995
(Show Context)
Citation Context ...method can fail badly in some cases, one of which is the two-spiral problem (Lang and Witbrock 1989). By using the dynamic weighting method together with the tempering idea (Marinari and Parisi 1992; =-=Geyer and Thompson 1995-=-), Wong and Liang (1997) treated the two-spiral problem with considerable success (both the 2-25-1 and 2-14-4-1 networks have been fitted and the results were close to be perfect, whereas the error ra... |

152 | Simulated tempering: a new Monte Carlo scheme
- Marinari, Parisi
- 1992
(Show Context)
Citation Context ...However, back-propagation method can fail badly in some cases, one of which is the two-spiral problem (Lang and Witbrock 1989). By using the dynamic weighting method together with the tempering idea (=-=Marinari and Parisi 1992-=-; Geyer and Thompson 1995), Wong and Liang (1997) treated the two-spiral problem with considerable success (both the 2-25-1 and 2-14-4-1 networks have been fitted and the results were close to be perf... |

142 | Molecular Modelling. Principles and Applications - Leach - 1996 |

129 |
Learning to tell two spirals apart
- Lang, Witbrock
- 1988
(Show Context)
Citation Context ... algorithm is the back-propagation (BP) and its variants (Rumelhart, Hinton, and Williams 1986). However, back-propagation method can fail badly in some cases, one of which is the two-spiral problem (=-=Lang and Witbrock 1989-=-). By using the dynamic weighting method together with the tempering idea (Marinari and Parisi 1992; Geyer and Thompson 1995), Wong and Liang (1997) treated the two-spiral problem with considerable su... |

38 |
Multicanonical algorithms for first order phase transitions
- Berg, Neuhaus
- 1991
(Show Context)
Citation Context ...eneralize to other systems, e.g., random field Ising models (Marinari and Parisi, 1992). Other successful methods include the simulated tempering (Marinari and Parisi, 1992) and muticanonical method (=-=Berg and Neuhaus, 1991-=-). But the methods may encounter difficulties when simulating an Ising system at a temperature below the critical point (where the energy variation is huge). The multigrid Monte Carlo method of Goodma... |

27 | Nuovo Cimento Suppl - Onsager - 1949 |

18 |
Dynamic weighting in Monte Carlo and optimization
- Wong, Liang
- 1997
(Show Context)
Citation Context ...on of the importance weight as a dynamic variable for the control of the Markov chain simulation in each step. They tested this method on many large scale simulation and global optimization problems (=-=Wong and Liang 1997-=-) and the results were promising. Some of these problems will be reviewed in Section 7. The purpose of the present paper is to provide a first theoretical analysis of the properties of the dynamic wei... |

11 | Multigrid Monte Carlo Method - Goodman, Sokal - 1989 |

5 |
Dynamic weighting in simulations of spin systems
- Liang, Wong
- 1999
(Show Context)
Citation Context ...and Sokal (1989) can be successful for some other models but is not suitable for the Ising model. We now review the results obtained on Ising model simulations by dynamic weighting with R-type moves (=-=Liang and Wong 1998-=-). The simulations were done on lattices of size 32 2 , 64 2 and 128 2 . Similar to simulated tempering, we treat the inverse temperature K as a dynamic variable taking values in a ladder of suitable ... |

4 |
Weighted Markov Chain Monte Carlo and Optimization,” unpublished doctorial thesis, The Chinese University of Hong Kong
- Liang
- 1997
(Show Context)
Citation Context ... importance weight as a dynamic variable for the control of the Markov chain simulation in each step. They tested this method on many large scale simulation and global optimization problems (Wong and =-=Liang 1997-=-) and the results were promising. Some of these problems will be reviewed in Section 7. The purpose of the present paper is to provide a first theoretical analysis of the properties of the dynamic wei... |

2 | Renewal theory for Markov chains - Kesten - 1974 |