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In Bayesian analysis, we seek a balance between prior information in a form of expert knowledge
or belief and evidence from data at hand. Achieving the right balance is one of the difficulties in
Bayesian modeling and inference. In general, we should not allow the prior information to overwhelm
the evidence from the data, especially when we have a large data sample. A famous theoretical
result, the Bernstein–von Mises theorem, states that in large data samples, the posterior distribution is
independent of the prior distribution and, therefore, Bayesian and likelihood-based inferences should
yield essentially the same results. On the other hand, we need a strong enough prior to support weak
evidence that usually comes from insufficient data. It is always good practice to perform sensitivity
analysis to check the dependence of the results on the choice of a prior.

We consider two types of CRIs. The first one is based on quantiles. The second one is the highest
posterior density (HPD) interval.
An f(1 �� ) 100g% quantile-based, or also known as an equal-tailed CRI, is defined as
(q=2; q1��=2), where qa denotes the ath quantile of the posterior distribution. A commonly reported
equal-tailed CRI is (q0:025; q0:975).
HPD interval is defined as an f(1 �� ) 100g% CRI of the shortest width. As its name implies,
this interval corresponds to the region of the posterior density with the highest concentration. For a
unimodal posterior distribution, HPD is unique, but for a multimodal distribution it may not be unique.
Computational approaches for calculating HPD are described in Chen and Shao (1999) and Eberly
and Casella (2003).

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主讲老师
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参与编写书籍《中国金融发展的收入分配效应》,参与课题包括:1)扩大中等收入群体路径研究2)冀中南地区农村金融促进农民增收的机制究3)河北省环境管制与环境效率——地区间差异与影响因素研究4)基于工业-能源-环境DEA分析的河北省环境规制效率评价研究5)代农民弃农问题与农业:理论探讨和之道,产业结构调整过程中结构性失业的预防和治理。
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Remarks and examples
Remarks are presented under the following headings:
What is Bayesian analysis?
Bayesian versus frequentist analysis, or why Bayesian analysis?
How to do Bayesian analysis
Advantages and disadvantages of Bayesian analysis
Brief background and literature review
Bayesian statistics
Posterior distribution
Selecting priors
Point and interval estimation
Comparing Bayesian models
Posterior prediction
Bayesian computation
Markov chain Monte Carlo methods
Metropolis–Hastings algorithm
Adaptive random-walk Metropolis–Hastings
Blocking of parameters
Metropolis–Hastings with Gibbs updates
Convergence diagnostics of MCMC
Summary
The first five sections provide a general introduction to Bayesian analysis. The remaining sections
provide a more technical discussion of the concepts of Bayesian analysis.
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