scholarly journals Empirical Bayes Method for Boltzmann Machines

2021 ◽  
pp. 277-293
Author(s):  
Muneki Yasuda
Keyword(s):  
Partition Function ◽  
Empirical Bayes ◽  
Likelihood Function ◽  
Replica Method ◽  
Boltzmann Machine ◽  
Bayes Method ◽  
Boltzmann Machines ◽  
Computational Problem ◽  
Empirical Bayes Method ◽  
Iterative Procedures

AbstractThe framework of the empirical Bayes method allows the estimation of the values of the hyperparameters in the Boltzmann machine by maximizing a specific likelihood function referred to as the empirical Bayes likelihood function. However, the maximization is computationally difficult because the empirical Bayes likelihood function involves intractable integrations of the partition function. The method presented in this chapter avoids this computational problem by using the replica method and the Plefka expansion, which is quite simple and fast because it does not require any iterative procedures and gives reasonable estimates under certain conditions.

scholarly journals Safety Effects of Freeway Hard Shoulder Running

2019 ◽  
Vol 9 (17) ◽  
pp. 3614
Author(s):  
Jaisung Choi ◽  
Richard Tay ◽  
Sangyoup Kim ◽  
Seungwon Jeong ◽  
Jeongmin Kim ◽  
...  
Keyword(s):  
South Korea ◽  
Empirical Bayes ◽  
Positive Relationship ◽  
Crash Severity ◽  
Bayes Method ◽  
Road Capacity ◽  
Empirical Bayes Method ◽  
Safety Effect

Hard shoulder running (HSR) has been increasingly used as a sustainable and viable way to increase road capacity. This study investigated the safety effect of HSR on freeways in South Korea using the empirical Bayes method. This study found an increase in the total number of crashes. In terms of crash severity, a higher proportion of crashes (25.3%) on 2(3)-lane sections were found to be serious (involving injuries and/or fatalities) compared to those on 4(5)-lane sections (3.6%). Also, a positive relationship was found between the length of the hard shoulder running and changes in crash frequencies. Thus, hard shoulder running on lengthy 2(3)-lane freeways should be avoided.

AN EMPIRICAL BAYES METHOD FOR STUDYING VARIATION IN KNEE REPLACEMENT RATES

1996 ◽  
Vol 15 (17) ◽  
pp. 1875-1884 ◽  
Author(s):  
XIAO-HUA ZHOU ◽  
B. P. KATZ ◽  
E. HOLLEMAN ◽  
C. A. MELFI ◽  
R. DITTUS
Keyword(s):  
Knee Replacement ◽  
Empirical Bayes ◽  
Bayes Method ◽  
Empirical Bayes Method

Weighted Analysis of Paired Microarray Experiments

2005 ◽  
Vol 4 (1) ◽  
Author(s):  
Erik Kristiansson ◽  
Anders Sjögren ◽  
Mats Rudemo ◽  
Olle Nerman
Keyword(s):  
Empirical Bayes ◽  
Mixed Type ◽  
Simulated Data ◽  
Analysis Method ◽  
Bayes Method ◽  
Microarray Experiments ◽  
The Third ◽  
Weighted Analysis ◽  
Empirical Bayes Method ◽  
Before And After

In microarray experiments quality often varies, for example between samples and between arrays. The need for quality control is therefore strong. A statistical model and a corresponding analysis method is suggested for experiments with pairing, including designs with individuals observed before and after treatment and many experiments with two-colour spotted arrays. The model is of mixed type with some parameters estimated by an empirical Bayes method. Differences in quality are modelled by individual variances and correlations between repetitions. The method is applied to three real and several simulated datasets. Two of the real datasets are of Affymetrix type with patients profiled before and after treatment, and the third dataset is of two-colour spotted cDNA type. In all cases, the patients or arrays had different estimated variances, leading to distinctly unequal weights in the analysis. We suggest also plots which illustrate the variances and correlations that affect the weights computed by our analysis method. For simulated data the improvement relative to previously published methods without weighting is shown to be substantial.

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