APPENDIX A
Basic Statistical Distributions
and Stochastic Processes
In this appendix, we list some useful distributions including abbreviation,
support, density, mean, variance, covariance, properties, relationship with
other distributions and the random

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Chapter 3
Integration
In this chapter, we introduce ve basic tools for evaluating integrals often encountered in Bayesian computation. The Laplace approximation is
an analytic approach to approximate integral based on the Taylor expansion around the mode

Chapter 4
Markov Chain Monte Carlo
Methods
4.1
Bayes Formula and Inverse Bayes Formulae
The Bayes formula, Bayes rule, or Bayes theorem, was published posthumously and named after Reverend Thomas Bayes (1763) and has been the
foundation of Bayesian infere

Chapter 5
Bootstrap Methods
The bootstrap is a data-based method for statistical inference. Its introduction into statistics is relatively recent because the method is computationally
intensive. In essence, the bootstrap approach provides a general method

Chapter 2
Optimization
Let Yobs denote the observed (or observable) data and the parameter vector
of interest. A central subject of statistics is to make inference on based
on Yobs . Frequentist/classical method arrives its inferential statements by
combi

Chapter 1
Generation of Random Variables
In this chapter, we describe basic Monte Carlo simulation techniques for
generating random samples from univariate and multivariate distributions.
These techniques also play a critical role in Monte Carlo integrati

APPENDIX C
How to Introduce Latent Variables
in EM-Type or DA-Type Algorithms?
C.1
MLEs of Parameters in t Distribution
iid
The issue and aim. Let X1 , . . . , Xn t(, 2 , ), where < <
and 2 > 0 are two unknown parameters and > 0 is known. The aim is
to n

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