Markov Chain Monte Carlo: The Gibbs Sampler and the Metropolis Algorithm

1993 ◽  
pp. 102-146 ◽  
Author(s):  
Martin A. Tanner
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Gibbs Sampler ◽  
Metropolis Algorithm

ON THE ADAN–WEISS LOSS MODEL HAVING SKILL-BASED SERVERS AND LONGEST IDLE ASSIGNMENT RULE

2015 ◽  
Vol 29 (2) ◽  
pp. 181-189 ◽  
Author(s):  
Babak Haji
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Monte Carlo Method ◽  
Gibbs Sampler ◽  
Service Time ◽  
Loss System ◽  
Multiplicative Constant ◽  
Assignment Rule ◽  
Poisson Arrivals

We consider a queueing loss system with heterogeneous skill based servers with arbitrary distributions. We assume Poisson arrivals, with each arrival having a vector indicating which of the servers are eligible to serve it. Arrivals can only be assigned to a server that is both idle and eligible. We assume arrivals are assigned to the idle eligible server that has been idle the longest and derive, up to a multiplicative constant, the limiting distribution for this system. We show that the limiting probabilities of the ordered list of idle servers depend on the service time distributions only through their means. Moreover, conditional on the ordered list of idle servers, the remaining service times of the busy servers are independent and have their respective equilibrium service distributions. We also provide an algorithm using Gibbs sampler Markov Chain Monte Carlo method for estimating the limiting probabilities and other desired quantities of this system.

scholarly journals PERBANDINGAN METODE MAXIMUM LIKELIHOOD DAN METODE BAYES DALAM MENGESTIMASI PARAMETER MODEL REGRESI LINIER BERGANDA UNTUK DATA BERDISTRIBUSI NORMAL

2019 ◽  
Vol 4 (2) ◽  
pp. 100
Author(s):  
Catrin Muharisa ◽  
Ferra Yanuar ◽  
Hazmira Yozza
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Maximum Likelihood ◽  
Gibbs Sampler

Analisis regresi merupakan salah satu metode untuk melihat hubungan antara variabel bebas (independent) dengan variabel terikat (dependent) yang dinyatakan dalam model regresi. Beberapa metode yang bisa digunakan untuk mengestimasi parameter model regresi, diantaranya adalah metode klasik dan metode Bayes. Salah satu metode klasik adalah metode maximum likelihood. Penelitian ini membahas tentang perbandingan metode maximum likelihood dan metode Bayes dalam mengestimasi parameter model regresi linear berganda untuk data berdistribusi normal. Adapun rumus untuk mengestimasi parameter dengan metode maximum likelihood adalah βˆ=(XTX)-1XTY dan ˆσ2 = 1 n P∞ k=1 ei. Sedangkan untuk mengestimasi parameter dengan metode Bayes adalah dengan menggunakan distribusi prior dan fungsi likelihood. Distribusi prior yag dipilih pada kajian ini adalah f(β, σ2 ) = Qn i=1 f(βj |σ 2 )f(σ 2 ) dengan βj ∼ N(µβj , σ2 ) dan σ 2 ∼ IG(a, b). Distribusi prior konjugat tersebut kemudian dikalikan dengan fungsi likelihood L(β, σ2 ) sehingga membentuk distribusi posterior f(β|σ 2 ). Distribusi posterior inilah yang digunakan untuk mengestimasi parameter model melalui proses Markov Chain Monte Carlo (MCMC). Algoritma MCMC yang digunakan adalah algoritma Gibbs Sampler. Model regresi linear berganda yang diperoleh dengan metode maximum likelihood adalahyˆ = −27, 8210000 + 0, 0307430X1 + 0, 0039211X2 + 0, 0034631X3 + 0, 6537000X4dengan kecocokan modelnya adalah sebesar 95,7 %. Sedangkan model regresi linear berganda yang diperoleh dengan metode Bayes adalahyˆ = −26, 620000 + 0, 029380X1 + 0, 004204X2 + 0, 003321X3 + 0, 656200X4dengan kecocokan modelnya adalah sebesar 99,99 %. Dengan demikian dapat disimpulkan bahwa metode Bayes lebih baik dari pada metode maximum likelihood.Kata Kunci: Model Regresi Linear Berganda, metode Maximum Likelihood, dan metode Bayes

Markov Chain Monte Carlo Simulation Methods in Econometrics

1996 ◽  
Vol 12 (3) ◽  
pp. 409-431 ◽  
Author(s):  
Siddhartha Chib ◽  
Edward Greenberg
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Em Algorithm ◽  
Gibbs Sampler ◽  
Model Specification ◽  
Simulation Methods ◽  
Theoretical Literature ◽  
The Em Algorithm

We present several Markov chain Monte Carlo simulation methods that have been widely used in recent years in econometrics and statistics. Among these is the Gibbs sampler, which has been of particular interest to econometricians. Although the paper summarizes some of the relevant theoretical literature, its emphasis is on the presentation and explanation of applications to important models that are studied in econometrics. We include a discussion of some implementation issues, the use of the methods in connection with the EM algorithm, and how the methods can be helpful in model specification questions. Many of the applications of these methods are of particular interest to Bayesians, but we also point out ways in which frequentist statisticians may find the techniques useful.

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