# 有序数据的贝叶斯分位数回归Bayesian Quantile Regression Associated with the Ordinal Data

• 全文下载: PDF(528KB)    PP.565-575   DOI: 10.12677/SA.2017.65064
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In this paper, we introduce an ordinal Bayesian quantile regression model associated with the ordinal data based on asymmetric Laplace distribution. We show that the posterior distributions of estimated parameters always proper when the prior distributions are given, and we also give an efficient Gibbs sampling algorithm for fitting the model to such data. To illustrate this approach, we give a simulation and a real data example.

1. 引言

1978年，Koenker and Bassett [1] 首次提出了用分位数回归方法来描述因变量的条件分位数与自变量之间的关系。分位数回归的提出引起了广泛的关注。分位数回归被广泛应用于农业、基因芯片技术、生存研究、经济学、医疗卫生、环境科学等领域。

2. 贝叶斯分位数回归

$f\left(x;\mu ,\sigma ,p\right)=\frac{p\left(1-p\right)}{p}\mathrm{exp}\left\{-\frac{x-\mu }{\sigma }\left(p-{1}_{\left\{x\le \mu \right\}}\right)\right\},\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\in R$ (1)

$E\left(x\right)={\int }_{-\infty }^{\infty }xf\left(x;\mu ,\sigma ,p\right)\text{d}x=\mu +\sigma \frac{1-2p}{p\left(1-p\right)}$

$\Psi \left(\sigma ,p\right)={\int }_{-\infty }^{\infty }{\left(x-E\left(\xi \right)\right)}^{2}f\left(x;\mu ,\sigma ,p\right)\text{d}x={\sigma }^{2}\frac{1-2p+2{p}^{2}}{{p}^{2}{\left(1-p\right)}^{2}}.$

$g\left(p\right)=\frac{1-2p+2{p}^{2}}{{p}^{2}{\left(1-p\right)}^{2}}=\frac{1}{{p}^{2}}+\frac{1}{{\left(1-p\right)}^{2}}$

$\underset{p\in \left[0,1\right]}{\mathrm{min}}g\left(p\right)=g\left(\frac{1}{2}\right)=8$ 证毕。

${F}^{-1}\left(x;\mu ,\sigma ,p\right)=\left\{\begin{array}{ll}\mu +\frac{\sigma }{1-p}\mathrm{log}\frac{x}{p},\hfill & \text{if}\text{\hspace{0.17em}}0\le x\le p\hfill \\ \mu -\frac{\sigma }{p}\mathrm{log}\frac{1-x}{1-p},\hfill & \text{if}\text{\hspace{0.17em}}p (2)

${{F}^{-1}\left(x;\mu ,\sigma ,p\right)|}_{x=p}=\mu$ (3)

 服从如下的线性回归模型，

${r}_{t}={Q}_{p}\left({r}_{t}|{x}_{t},\beta \right)+{u}_{t},\text{\hspace{0.17em}}t=1,2,\cdots ,n$ (4)

${r}_{t}~ALD\left({Q}_{p}\left({r}_{t}|{x}_{t},\beta \right),\sigma ,p\right)$

${r}_{t}$ 的密度函数为：

$f\left({r}_{t};{Q}_{p}\left({r}_{t}|{x}_{t},\beta \right),\sigma ,p\right)=\frac{p\left(1-p\right)}{p}\mathrm{exp}\left\{-\frac{{Q}_{p}\left({r}_{t}|{x}_{t},\beta \right)}{\sigma }\left[p-I\left({r}_{t}\le {Q}_{p}\left({r}_{t}|{x}_{t},\beta \right)\right)\right]\right\}$ (5)

$t=1,2,\cdots ,n$ 。样本的似然函数可以表示为

$L\left(r|\beta ,\sigma \right)=\frac{{p}^{n}{\left(1-p\right)}^{n}}{{\sigma }^{n}}\mathrm{exp}\left[-\frac{1}{\sigma }{\sum }_{t=1}^{n}{\rho }_{p}\left({r}_{t}-{Q}_{p}\left({r}_{t}|{x}_{t},\beta \right)\right)\right]$ (6)

${\rho }_{p}\left(u\right)=u\left(p-I\left(u<0\right)\right)>0$ .

$\pi \left(\beta ,\sigma |r\right)\propto L\left(r|\beta ,\sigma \right)f\left(\beta \right)f\left(\sigma \right)$ (7)

${\int }_{0}^{\infty }f\left(\sigma \right)\frac{1}{{\sigma }^{n}}\text{d}\sigma <\infty$

$0<{\int }_{-\infty }^{\infty }{\int }_{0}^{\infty }\pi \left(\beta ,\sigma |r\right)\text{d}\beta \text{d}\sigma <\infty$

$\begin{array}{c}\pi \left(\beta ,\sigma |r\right)\propto L\left(r|\beta ,\sigma \right)f\left(\beta \right)f\left(\sigma \right)\\ =f\left(\beta \right)f\left(\sigma \right)\frac{{p}^{n}{\left(1-p\right)}^{n}}{{\sigma }^{n}}\left[-\frac{1}{\sigma }\underset{t=1}{\overset{n}{\sum }}\text{ }\text{ }{\rho }_{p}\left({r}_{t}-{Q}_{p}\left({r}_{t}|{x}_{t},\beta \right)\right)\right]\end{array}$

$0<\mathrm{exp}\left[-\frac{1}{\sigma }\underset{t=1}{\overset{n}{\sum }}\text{ }\text{ }{\rho }_{p}\left({r}_{t}-{Q}_{p}\left({r}_{t}|{x}_{t},\beta \right)\right)\right]<1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}0<{\int }_{0}^{\infty }f\left(\sigma \right)\frac{1}{{\sigma }^{n}}\text{d}\sigma <\infty ,$

$0

${\int }_{-\infty }^{\infty }f\left(\beta \right)\text{d}\beta =\infty$

$\begin{array}{c}0={\int }_{-\infty }^{\infty }{\int }_{0}^{\infty }0\text{d}\beta \text{d}\sigma \\ <{\int }_{-\infty }^{\infty }{\int }_{0}^{\infty }L\left(r|\beta ,\sigma \right)f\left(\beta \right)f\left(\sigma \right)\text{d}\beta \text{d}\sigma \\ =\underset{i=1}{\overset{n}{\prod }}\underset{k=1}{\overset{K}{\prod }}{\left[{F}_{AL}\left(\frac{{\gamma }_{p,j}-{{x}^{\prime }}_{i}{\beta }_{p}}{\sigma }\right)-{F}_{AL}\left(\frac{{\gamma }_{p,j-1}-{{x}^{\prime }}_{i}{\beta }_{p}}{\sigma }\right)\right]}^{I\left({y}_{i}=k\right)}\\ ={p}^{n}{\left(1-p\right)}^{n}{\int }_{-\infty }^{\infty }f\left(\beta \right)\text{d}\beta {\int }_{0}^{\infty }f\left(\sigma \right)\frac{1}{{\sigma }^{n}}\text{d}\sigma =\infty \end{array}$

3. 有序数据模型

${t}_{i}={{x}^{\prime }}_{i}{\beta }_{p}+{ϵ}_{i},\text{\hspace{0.17em}}i=1,\cdots ,n$ (8)

${\gamma }_{p,k-1}<{t}_{i}<{\gamma }_{p,k}⇒{y}_{i}=k,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n;\text{\hspace{0.17em}}k=1,\cdots ,K$

$\begin{array}{c}f\left({\beta }_{p},{\gamma }_{p},\sigma ;y\right)=\underset{i=1}{\overset{n}{\prod }}\underset{k=1}{\overset{K}{\prod }}P{\left({y}_{i}=k|{\beta }_{p},{\gamma }_{p},\sigma \right)}^{I\left({y}_{i}=k\right)}\\ =\underset{i=1}{\overset{n}{\prod }}\underset{k=1}{\overset{K}{\prod }}{\left[{F}_{AL}\left(\frac{{\gamma }_{p,j}-{{x}^{\prime }}_{i}{\beta }_{p}}{\sigma }\right)-{F}_{AL}\left(\frac{{\gamma }_{p,j-1}-{{x}^{\prime }}_{i}{\beta }_{p}}{\sigma }\right)\right]}^{I\left({y}_{i}=k\right)}\end{array}$

${ϵ}_{i}=\theta {\omega }_{i}+\tau \sqrt{{\omega }_{i}}{u}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n$ (9)

$\theta =\frac{1-2p}{p\left(1-p\right)},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\tau =\sqrt{\frac{2}{p\left(1-p\right)}}.$

${t}_{i}={{x}^{\prime }}_{i}{\beta }_{p}+\theta {\omega }_{i}+\tau \sqrt{{\omega }_{i}}{u}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n$ (10)

$\left\{\begin{array}{l}{t}_{i}={{x}^{\prime }}_{i}{\beta }_{p}+{\sigma }_{p}{ϵ}_{i}={{x}^{\prime }}_{i}{\beta }_{p}+\theta {\omega }_{i}+\tau \sqrt{{\omega }_{i}}{u}_{i},\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n\hfill \\ {\gamma }_{k-1}<{t}_{i}\le {\gamma }_{k}{y}_{i}=k,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,\cdots ,n\hfill \end{array}$ (11)

${t}_{i}={{x}^{\prime }}_{i}{\beta }_{p}+\theta {\nu }_{i}+\tau \sqrt{{\sigma }_{p}{\nu }_{i}}{u}_{i}$ (12)

$\left\{\begin{array}{l}{\beta }_{p}\sim N\left({\beta }_{p0},{B}_{p0}\right)\\ {\sigma }_{p}\sim IG\left({n}_{0}/2,{d}_{0}/2\right)\\ {\nu }_{i}\sim E\left( \sigma p \right)\end{array}$

$\begin{array}{c}\pi \left(t,{\beta }_{p},\nu ,{\sigma }_{p}|y\right)\propto f\left(y|t,{\beta }_{p},\nu ,{\sigma }_{p}\right)\pi \left(t|{\beta }_{p},\nu ,{\sigma }_{p}\right)\pi \left(\nu |{\sigma }_{p}\right)\pi \left({\beta }_{p}\right)\pi \left({\sigma }_{p}\right)\\ \propto \left\{\underset{i=1}{\overset{n}{\prod }}f\left({y}_{i}|{t}_{i},{\sigma }_{p}\right)\right\}\pi \left(t|{\beta }_{p},\nu ,{\sigma }_{p}\right)\pi \left(\nu |{\sigma }_{p}\right)\pi \left({\beta }_{p}\right)\pi \left({\sigma }_{p}\right).\end{array}$

$\begin{array}{c}\pi \left(t,{\beta }_{p},\nu ,{\sigma }_{p}|y\right)\propto \left\{\underset{i=1}{\overset{n}{\prod }}\underset{k=1}{\overset{3}{\prod }}\text{ }\text{ }I\left({\gamma }_{k-1}<{t}_{i}\le {\gamma }_{k}\right)N\left({t}_{i}|{{x}^{\prime }}_{i}{\beta }_{p}+\theta {\nu }_{i},{\tau }^{2}{\sigma }_{p}{\nu }_{i}\right)E\left({\nu }_{i}|{\sigma }_{p}\right)\right\}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×N\left({\beta }_{p0},{B}_{p0}\right)IG\left({n}_{0}/2,{d}_{0}/2\right),\end{array}$

${\beta }_{p}$ 的条件后验密度 $\pi \left({\beta }_{p}|t,{\sigma }_{p},\nu \right)$ 正比于 $\pi \left({\beta }_{p}\right)×f\left(t|{\beta }_{p},{\sigma }_{p},\nu \right)$

$\begin{array}{c}\pi \left({\beta }_{p}|t,{\sigma }_{p},\nu \right)\propto \mathrm{exp}\left[-\frac{1}{2}\left\{\underset{i=1}{\overset{n}{\sum }}{\left(\frac{{t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}-\theta {\nu }_{i}}{\tau \sqrt{{\sigma }_{p}{\nu }_{i}}}\right)}^{2}+{\left({\beta }_{p}-{\beta }_{p0}\right)}^{\prime }{B}_{p0}^{-1}\left({\beta }_{p}-{\beta }_{p0}\right)\right\}\right]\\ =\mathrm{exp}\left[-\frac{1}{2}\left\{{{\beta }^{\prime }}_{p}\left(\underset{i=1}{\overset{n}{\sum }}\frac{{x}_{i}{{x}^{\prime }}_{i}}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}\right){\beta }_{p}-{{\beta }^{\prime }}_{p}\left(\underset{i=1}{\overset{n}{\sum }}\frac{{x}_{i}\left({t}_{i}-\theta {\nu }_{i}\right)}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}\right)-\left(\underset{i=1}{\overset{n}{\sum }}\frac{{{x}^{\prime }}_{i}\left({t}_{i}-\theta {\nu }_{i}\right)}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\left(\underset{i=1}{\overset{n}{\sum }}\frac{{\left({t}_{i}-\theta {\nu }_{i}\right)}^{2}}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}\right)+{\left({\beta }_{p}-{\beta }_{p0}\right)}^{\prime }{B}_{p0}^{-1}\left({\beta }_{p}-{\beta }_{p0}\right)\right\}\right]\\ \propto \mathrm{exp}\left[-\frac{1}{2}\left\{{{\beta }^{\prime }}_{p}\left(\underset{i=1}{\overset{n}{\sum }}\frac{{x}_{i}{{x}^{\prime }}_{i}}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}+{B}_{p0}^{-1}\right){\beta }_{p}-{{\beta }^{\prime }}_{p}\left(\underset{i=1}{\overset{n}{\sum }}\frac{{x}_{i}\left({t}_{i}-\theta {\nu }_{i}\right)}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}+{B}_{p0}^{-1}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}-\left(\underset{i=1}{\overset{n}{\sum }}\frac{{{x}^{\prime }}_{i}\left({t}_{i}-\theta {\nu }_{i}\right)}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}+{{\beta }^{\prime }}_{p}{B}_{p0}^{-1}\right){\beta }_{p}\right\}\right]\end{array}$

${\stackrel{˜}{B}}_{p}^{-1}=\left({\sum }_{i=1}^{n}\frac{{x}_{i}{{x}^{\prime }}_{i}}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}+{B}_{p0}^{-1}\right)$${\stackrel{˜}{\beta }}_{p}={\stackrel{˜}{B}}_{p}\left({\sum }_{i=1}^{n}\frac{{x}_{i}\left({t}_{i}-\theta {\nu }_{i}\right)}{{\tau }^{2}{\sigma }_{p}{\nu }_{i}}+{B}_{p0}^{-1}{\beta }_{p0}\right)$ 则有

$\pi \left({\beta }_{p}|t,{\sigma }_{p},\nu \right)\propto \mathrm{exp}\left[-\frac{1}{2}\left\{{{\beta }^{\prime }}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\beta }_{p}-{{\beta }^{\prime }}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\stackrel{˜}{\beta }}_{p}-{\stackrel{˜}{{\beta }^{\prime }}}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\beta }_{p}\right\}\right]$

$\pi \left({\beta }_{p}|t,{\sigma }_{p},\nu \right)\propto \mathrm{exp}\left[-\frac{1}{2}\left\{{{\beta }^{\prime }}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\beta }_{p}-{{\beta }^{\prime }}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\stackrel{˜}{\beta }}_{p}-{{\stackrel{˜}{\beta }}^{\prime }}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\beta }_{p}+{{\stackrel{˜}{\beta }}^{\prime }}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\stackrel{˜}{\beta }}_{p}-{{\stackrel{˜}{\beta }}^{\prime }}_{p}{\stackrel{˜}{B}}_{p}^{-1}{\stackrel{˜}{\beta }}_{p}\right\}\right]$

$\pi \left({\beta }_{p}|t,{\sigma }_{p},\nu \right)\propto \mathrm{exp}\left[-\frac{1}{2}\left\{{\left({\beta }_{p}-{\stackrel{˜}{\beta }}_{p}\right)}^{\prime }{\stackrel{˜}{B}}_{p}^{-1}\left({\beta }_{p}-{\stackrel{˜}{\beta }}_{p}\right)\right\}\right]$

$\begin{array}{c}\pi \left({\sigma }_{p}|t,{\beta }_{p},\nu \right)\propto \underset{i=1}{\overset{n}{\prod }}\left\{{\sigma }_{p}^{-1/2}\mathrm{exp}\left[-\frac{1}{2}{\left(\frac{{t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}-\theta {\nu }_{i}}{\tau \sqrt{{\sigma }_{p}{\nu }_{i}}}\right)}^{2}\right]×{\sigma }_{p}^{-1}\mathrm{exp}\left(-\frac{{\nu }_{i}}{{\sigma }_{p}}\right)\right\}\mathrm{exp}\left[-\frac{{d}_{0}}{2{\sigma }_{p}}\right]{\sigma }_{p}^{-\left(\frac{{n}_{0}}{2}+1\right)}\\ \propto {\sigma }_{p}^{-\left(\frac{{n}_{0}}{2}++\frac{3n}{2}+1\right)}\mathrm{exp}\left[-\frac{1}{{\sigma }_{p}}\left\{\underset{i=1}{\overset{n}{\sum }}\frac{{\left({t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}-\theta {\nu }_{i}\right)}^{2}}{2{\tau }^{2}{\nu }_{i}}+\frac{{d}_{0}}{2}+\underset{i=1}{\overset{n}{\sum }}\text{ }\text{ }{\nu }_{i}\right\}\right]\end{array}$

$\nu$ 的条件后验密度 $\pi \left(\nu |t,{\beta }_{p},{\sigma }_{p}\right)$ 正比于 $f\left(t|{\beta }_{p},\nu ,{\sigma }_{p}\right)\pi \left(\nu \right)$${\nu }_{i}$ 有如下形式

$\begin{array}{c}\pi \left({\nu }_{i}|w,{\beta }_{p},{\sigma }_{p}\right)\propto {\nu }_{i}^{-1/2}\mathrm{exp}\left[-\frac{1}{2}{\left(\frac{{w}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}-\theta {\nu }_{i}}{\tau \sqrt{{\sigma }_{p}{\nu }_{i}}}\right)}^{2}-\frac{{\nu }_{i}}{{\sigma }_{p}}\right]\\ \propto {\nu }_{i}^{-1/2}\mathrm{exp}\left[-\frac{1}{2{\sigma }_{p}}\left(\frac{{\left({w}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}\right)}^{2}+{\theta }^{2}{\nu }_{i}^{2}-2\theta {\nu }_{i}\left({t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}\right)}{{\tau }^{2}{\nu }_{i}}+2{\nu }_{i}\right)\right]\\ ={\nu }_{i}^{-1/2}\mathrm{exp}\left[-\frac{1}{2}\left\{\frac{{\left({t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}\right)}^{2}}{{\tau }^{2}{\sigma }_{p}}{\nu }_{i}^{-1}+\left(\frac{{\theta }^{2}}{{\tau }^{2}{\sigma }_{p}}+\frac{2}{{\sigma }_{p}}\right){\nu }_{i}-\frac{2\theta \left({t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}\right)}{{\tau }^{2}{\sigma }_{p}}\right\}\right]\\ \propto {\nu }_{i}^{-1/2}\mathrm{exp}\left[-\frac{1}{2}\left\{\frac{{\left({t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}\right)}^{2}}{{\tau }^{2}{\sigma }_{p}}{\nu }_{i}^{-1}+\left(\frac{{\theta }^{2}}{{\tau }^{2}{\sigma }_{p}}+\frac{2}{{\sigma }_{p}}\right){\nu }_{i}\right\}\right]\end{array}$

${\stackrel{˜}{\lambda }}_{i}=\frac{{\left({t}_{i}-{{x}^{\prime }}_{i}{\beta }_{p}\right)}^{2}}{{\tau }^{2}{\sigma }_{p}}$$\stackrel{˜}{\eta }=\left(\frac{{\theta }^{2}}{{\tau }^{2}{\sigma }_{p}}+\frac{2}{{\sigma }_{p}}\right)$ 则有

$\pi \left({\nu }_{i}|t,{\beta }_{p},{\sigma }_{p}\right)\propto {\nu }_{i}^{-1/2}\mathrm{exp}\left[-\frac{1}{2}\left\{{\stackrel{˜}{\lambda }}_{i}{\nu }_{i}^{-1}+\stackrel{˜}{\eta }{\nu }_{i}\right\}\right]$

${\nu }_{i}$ 的条件后验分布为广义逆伽玛分布，即 ${\nu }_{i}|t,{\beta }_{p},{\sigma }_{p}\sim GIG\left(0.5,{\stackrel{˜}{\lambda }}_{i},\stackrel{˜}{\eta }\right)$

4. 数据模拟分析

$\left\{\begin{array}{l}{Y}_{i}={\beta }_{0}+{\beta }_{1}x+\nu ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}i=1,2,\cdots ,n,\hfill \\ x\sim unif\left(0,10\right),\hfill \\ {\beta }_{0}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\beta }_{1}=2.\hfill \end{array}$

${\beta }_{0}$${\beta }_{1}$ 的先验分布为 $N\left(0,10\right)$ ，尺度参数 $\sigma$ 的先验分布设定自由度为3的卡方分布。采用Gibbs抽样法共模拟5000次，为消除初值对抽样分布的影响，去掉前2000个抽样值。在应用Gibbs抽样算法对参数进行抽样时，特定参数的Gibbs模拟值是否是真实后验分布的合理近似，这对参数估计值的正确性意义重大，因此在统计推断之前需要对抽样分布进行检验。

Figure 1. The MCMC trace plots and the density of ${\beta }_{0},{\beta }_{1},\sigma$

Figure 2. The autocorrelation of ${\beta }_{0},{\beta }_{1},\sigma$

Table 1. Estimation of β 0 in different situation

Table 2. Estimation of β 1 in different situation

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