含混合高斯项的AR-GJR-GARCH模型的贝叶斯估计

含混合高斯项的AR-GJR-GARCH模型的贝叶斯估计 Bayesian Estimation of AR-GJR-GARCH Model with Mixed Gaussian Terms first-author 王祥赛，男，硕士生，现从事随机微分方程方面的研究。E-mail： 019202010074@whu.edu.cn 王祥赛，男，硕士生，现从事随机微分方程方面的研究。E-mail： 019202010074@whu.edu.cn 019202010074@whu.edu.cn 武汉大学 数学与统计学院，湖北 武汉 430072 武汉大学 数学与统计学院，湖北 武汉 430072 School of Mathematics and Statistics， Wuhan University， Wuhan 430072， Hubei， China School of Mathematics and Statistics， Wuhan University， Wuhan 430072， Hubei， China 基于不需要后验密度解析形式的随机梯度哈密尔顿蒙特卡洛（stochastic gradient Hamiltonian Monte Carlo,SGHMC）方法对AR-GJR-GARCH模型的参数进行了贝叶斯估计。以2019.3.13—2020.1.2和2020.1.3—2020.11.3两个时间段的中证医药指数的数据为例，对本文提出的方法进行了检验。结果显示，所得的参数估计值反映了与该指数的波动性相关的市场背景信息。 Based on the stochastic gradient Hamiltonian Monte Carlo(SGHMC) method which does not require a specific posterior density form, the parameters of AR（autoregressive）-GJR（Glosten-Jagannanthan-Runkle）-GARCH(generalized autoregressive conditional heteroscedasticity) model were estimated by Bayesian method. The method proposed in this paper was tested by taking the data of Chinese Medicine Index from March 13, 2019 to January 2, 2020 and January 3, 2020 to November 3, 2020 as examples. The results show that the estimated parameters reflect the market background information related to the volatility of the index. AR-GJR-GARCH模型 贝叶斯估计 哈密尔顿系统 随机梯度哈密尔顿蒙特卡洛 AR-GJR-GARCH model Bayesian estimation Hamiltonian system SGHMC(stochastic gradient Hamiltonian Monte Carlo) 0　 引 言 1 在对市场的风险研究中，波动率作为其中重要的一个度量，一直被认为是风险管理等方面的一个重点研究对象。在波动率研究方面，Engle ［ 1 1 ］ 提出了ARCH（autoregressive conditional heteroscedasticity）模型，Bollerslev ［ 2 2 ］ 在其研究的基础上又提出了GARCH（generalized ARCH）模型，该模型一定程度上包含了ARCH模型所不能涵盖的市场信息。但随着研究的深入，很多学者发现，现存的模型对一些金融数据的集聚性、尖峰厚尾性等不能做出解释。为解释这些特性，Nelson ［ 3 3 ］ 提出了EARCH（exponential ARCH）模型，Glosten等 ［ 4 4 ］ 提出了GJR（Glosten-Jagannanthan-Runkle）-GARCH模型。 AR（autoregressive）-GJR-GARCH模型是在GJR-GARCH模型的基础上，进一步添加了自回归项发展而来，对此类模型的参数估计通常是采用贝叶斯估计法 ［ 5 5 ］ 。但在贝叶斯估计的后验分布中通常存在一个未知的数值积分项，这使得直接获取后验分布的解析形式比较困难。为解决这个问题，Alexander等 ［ 6 6 ］ 用Griddy-Gibbs方法对后验分布进行抽样，但是这种方法在每次迭代前均需对数值积分做一个估计，导致计算成本比较高。 Chen等 ［ 7 7 ］ 和Talay ［ 8 8 ］ 在引入物理学中哈密尔顿系统的基础上，提出了一种随机梯度哈密尔顿蒙特卡洛（stochastic gradient Hamiltonian Monte Carlo，SGHMC）方法，该方法在未知数值积分项时，也可以获取一系列服从目标后验分布的样本，而且不需要引入M-H（Metropolis-Hasting）步骤，因而该方法的计算复杂度大大降低。基于此方法，Kim等 ［ 9 9 ］ 在贝叶斯神经网络的应用上取得了进展。 本文在张新星等 ［ 10 10 ］ 给出的含混合高斯项的AR-GJR-GARCH模型基础上，采用SGHMC方法对后验分布进行采样，给出了最终的参数估计结果，并结合结果对原始数据所包含的信息进行了解释。 1　 含混合高斯项的AR-GJR-GARCH模型及贝叶斯后验密度 1 AR（ s ）-GJR-GARCH（ p ， q ）模型如下（其中， s ， p ， q 表示相应的滞后阶数） <math display="block" id="M1"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>μ</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mo>+</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>t</mi></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>λ</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></munderover></mstyle><mrow><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mo stretchy="false">[</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>j</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo stretchy="false">]</mo></mrow></msub><mo>+</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>j</mi></mrow></msub><msub><mrow><mi>I</mi></mrow><mrow><mo stretchy="false">[</mo><msub><mrow><mi>ε</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>j</mi></mrow></msub><mo><</mo><mn mathvariant="normal">0</mn><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">)</mo><msubsup><mrow><mi>ε</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>j</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>k</mi></mrow></msub></mrow></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mi>t</mi><mo>=</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778489&type= 118.53332520 20.48933220 其中， <math id="M2"><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778784&type= 2.62466669 4.57200003 为观察样本， <math id="M3"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778529&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778514&type= 1.94733346 3.64066648 为观察变量的长期稳定均值， <math id="M4"><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778534&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778532&type= 2.53999996 2.79399991 为观察样本总数， <math id="M5"><mi>I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778540&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778536&type= 1.26999998 2.79399991 为示性函数， <math id="M6"><mi>r</mi><mo>=</mo><mi mathvariant="normal">m</mi><mi mathvariant="normal">a</mi><mi mathvariant="normal">x</mi><mo stretchy="false">{</mo><mi>s</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778555&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778554&type= 26.41600037 3.97933316 。实际情况中，为了确保AR（ s ）过程的平稳性，滞后算子 <math id="M7"><mi mathvariant="normal">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778548&type= 2.53999996 2.79399991 的多项式 <math id="M8"><mn mathvariant="normal">1</mn><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><msup><mrow><mi mathvariant="normal">B</mi></mrow><mrow><mi>i</mi></mrow></msup></mrow></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778569&type= 17.35666656 8.55133343 的零点在单位圆外。 <math id="M9"><msub><mrow><mi>ε</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778572&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778567&type= 2.20133328 3.72533321 为一混合高斯分布项，即 <math display="block" id="M10"><msub><mrow><mi>ε</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>~</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">概</mi><mi mathvariant="normal">率</mi><mi mathvariant="normal">为</mi><mi mathvariant="normal">ρ</mi></mtd></mtr><mtr><mtd><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mfrac><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>l</mi></mrow></mfrac><mo stretchy="false">)</mo><mo>,</mo><mi mathvariant="normal">概</mi><mi mathvariant="normal">率</mi><mi mathvariant="normal">为</mi><mn mathvariant="normal">1</mn><mo>-</mo><mi>ρ</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mi>t</mi><mo>=</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778576&type= 82.88867188 14.81666565 并假定 <math id="M11"><msub><mrow><mi>ε</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">(</mo><mi>t</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778604&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778601&type= 32.68133163 4.48733330 ， <math id="M12"><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mi>ρ</mi><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn><mo>-</mo><mi>ρ</mi></mrow><mrow><mi>l</mi></mrow></mfrac></mrow></mfrac><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo><</mo><mi>l</mi><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi mathvariant="normal">为</mi><mi mathvariant="normal">待</mi><mi mathvariant="normal">估</mi><mi mathvariant="normal">参</mi><mi mathvariant="normal">数</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778593&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778608&type= 64.51599884 13.03866673 。为了辅助参数估计，引入一个服从两点分布的辅助变量 <math id="M13"><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778777&type= 2.45533323 3.72533321 ，具体形式如下 <math display="block" id="M14"><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="normal">概</mi><mi mathvariant="normal">率</mi><mi mathvariant="normal">为</mi><mi mathvariant="normal">ρ</mi></mtd></mtr><mtr><mtd><mn mathvariant="normal">1</mn><mo>,</mo><mi mathvariant="normal">概</mi><mi mathvariant="normal">率</mi><mi mathvariant="normal">为</mi><mn mathvariant="normal">1</mn><mo>-</mo><mi>ρ</mi></mtd></mtr></mtable></mrow></mfenced><mo>,</mo><mi>t</mi><mo>=</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">1</mn><mo>,</mo><mi>s</mi><mo>+</mo><mn mathvariant="normal">2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>T</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778631&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778628&type= 67.64866638 10.49866676 并假定 <math id="M15"><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">(</mo><mi>t</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778634&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778622&type= 32.85066605 4.48733330 ，长期平均方差项 <math id="M16"><mi>λ</mi><mo>></mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778636&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778643&type= 8.46666718 2.79399991 ， <math id="M17"><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">(</mo><mi>t</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778651&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778649&type= 33.61266708 4.48733330 ， <math id="M18"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>j</mi></mrow></msub><mo>≥</mo><mn mathvariant="normal">0</mn><mo stretchy="false">(</mo><mi>j</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>p</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778673&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778642&type= 47.49800110 4.82600021 （ <math id="M19"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>j</mi></mrow></msub><mo>></mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778661&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778675&type= 12.19200039 3.72533321 时表明负面信息对观察变量的影响高于正面信息的影响， <math id="M20"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>j</mi></mrow></msub><mo><</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778665&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778679&type= 12.19200039 3.72533321 时则反之， <math id="M21"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>j</mi></mrow></msub><mo>=</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778684&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778669&type= 12.19200039 3.72533321 时则表明二者对观察变量的影响相同）， <math id="M22"><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>></mo><mn mathvariant="normal">0</mn><mo stretchy="false">(</mo><mi>k</mi><mo>=</mo><mn mathvariant="normal">1,2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>q</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778704&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778686&type= 34.71333313 4.91066647 。其中，为了确保GARCH模型是稳定的，通常要求 <math id="M23"><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>j</mi></mrow></msub></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>j</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>p</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>j</mi></mrow></msub></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>k</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>q</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>β</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778695&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778706&type= 57.40399933 9.39799976 。为方便接下来的推导，记 <math id="M24"><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mover accent="true"><mi>γ</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>γ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>γ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>T</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>⃗</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">12</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">1</mn><mi>p</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>⃗</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">22</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">2</mn><mi>p</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mover accent="true"><mi>β</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>β</mi></mrow><mrow><mi>q</mi></mrow></msub><mo stretchy="false">)</mo><mo>,</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo>=</mo><mo stretchy="false">(</mo><mi>ρ</mi><mo>,</mo><mi>l</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mover accent="true"><mi>γ</mi><mo>⃗</mo></mover><mo>,</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>⃗</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>⃗</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mover accent="true"><mi>β</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778714&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778711&type= 231.98667908 5.84200001 。 接下来给出参数的贝叶斯后验分布表达式。给定观察样本 <math id="M25"><mover accent="true"><mi>y</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779468&type= 1.86266661 3.97933316 以及辅助变量 <math id="M26"><mover accent="true"><mi>z</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778736&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778721&type= 1.69333339 3.21733332 ，参数 <math id="M27"><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778769&type= 1.77800000 4.06400013 的后验分布密度 <math id="M28"><mi>π</mi><mo stretchy="false">(</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778729&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778740&type= 13.29266548 4.91066647 由贝叶斯定理有如下形式 <math display="block" id="M29"><mi>π</mi><mo stretchy="false">(</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>∝</mo><mi>L</mi><mo stretchy="false">(</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mi>p</mi><mo stretchy="false">(</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778747&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778744&type= 39.96266937 4.91066647 其中， <math id="M30"><mi>L</mi><mo stretchy="false">(</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778753&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778751&type= 14.05466747 4.91066647 为似然函数， <math id="M31"><mi>p</mi><mo stretchy="false">(</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778766&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778764&type= 7.11199999 4.82600021 为 <math id="M32"><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778770&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778769&type= 1.77800000 4.06400013 的先验概率密度。 对于似然函数 <math id="M33"><mi>L</mi><mo stretchy="false">(</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778774&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778761&type= 14.05466747 4.91066647 ，在给定辅助变量 <math id="M34"><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778783&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778777&type= 2.45533323 3.72533321 时，可得 <math id="M35"><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778785&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778784&type= 2.62466669 4.57200003 的条件分布如下 <math display="block" id="M36"><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">|</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>~</mo><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mi>N</mi><mo stretchy="false">(</mo><mi>μ</mi><mo>+</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mo>,</mo><mfrac><mrow><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>l</mi></mrow></mfrac><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778790&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778787&type= 101.93866730 20.06599998 由此可得似然函数 <math id="M37"><mi>L</mi><mo stretchy="false">(</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778803&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778808&type= 14.05466747 4.91066647 如下 <math display="block" id="M38"><mtable columnalign="left"><mtr><mtd><mi>L</mi><mo stretchy="false">(</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>∝</mo><mi>L</mi><mo stretchy="false">(</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo>,</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>Z</mi></mrow></msub><mo stretchy="false">(</mo><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>∝</mo><munder><mo movablelimits="false">∏</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></mrow></munder><mrow><mo stretchy="false">[</mo><mi>ρ</mi><mo stretchy="false">(</mo><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mi>e</mi><mi>x</mi><mi>p</mi><mo stretchy="false">{</mo><mo>-</mo><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mo>-</mo><mi>μ</mi><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy="false">}</mo><mo stretchy="false">]</mo></mrow><mo>×</mo></mtd></mtr><mtr><mtd><munder><mo movablelimits="false">∏</mo><mrow><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></mrow></munder><mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>ρ</mi><mo stretchy="false">)</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow><mrow><mi>l</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup></mrow><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><mo>-</mo><mfrac><mrow><mi>l</mi><mo stretchy="false">[</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mo>-</mo><mi>μ</mi><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy="false">}</mo><mo stretchy="false">]</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778813&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778810&type= 135.46667480 30.56466866 各参数的先验分布按照下面形式选取 <math display="block" id="M39"><mtable columnalign="left"><mtr><mtd><mi>ρ</mi><mo>~</mo><mi>U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0.5,1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>l</mi><mo>~</mo><mi>U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">)</mo><mo>,</mo><mi>λ</mi><mo>~</mo><mi>U</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><msubsup><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msubsup><mo stretchy="false">)</mo><mo>,</mo><mi>μ</mi><mo>~</mo><mi>U</mi><mo stretchy="false">(</mo><mover><mrow><mi>y</mi></mrow><mo>¯</mo></mover><mo>-</mo><mfrac><mrow><mn mathvariant="normal">5</mn><msub><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></mrow><mrow><mroot><mrow><mi>T</mi></mrow><mrow/></mroot></mrow></mfrac><mo>,</mo><mover><mrow><mi>y</mi></mrow><mo>¯</mo></mover><mo>+</mo><mfrac><mrow><mn mathvariant="normal">5</mn><msub><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></mrow><mrow><mroot><mrow><mi>T</mi></mrow><mrow/></mroot></mrow></mfrac><mo stretchy="false">)</mo><mo>,</mo><mi>γ</mi><mo>~</mo><mi>U</mi><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1,1</mn><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mover accent="true"><mi>α</mi><mo>⃗</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>~</mo><mi>U</mi><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>⃗</mo></mover></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>~</mo><mi>U</mi><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>p</mi></mrow></msup><mo>,</mo><mover accent="true"><mi>β</mi><mo>⃗</mo></mover><mo>~</mo><mi>U</mi><msup><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">0,1</mn><mo stretchy="false">)</mo></mrow><mrow><mi>q</mi></mrow></msup></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778816&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778824&type= 123.95200348 16.84866714 其中， <math id="M40"><mover><mrow><mi>y</mi></mrow><mo>¯</mo></mover><mo>,</mo><msub><mrow><mover accent="true"><mi>σ</mi><mo>̂</mo></mover></mrow><mrow><mi>y</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778828&type= 7.02733374 4.31799984 分别为样本观察值的均值与标准差，并且假定各参数相互独立。将以上式子组合在一起即得 <math id="M41"><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778822&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778820&type= 1.77800000 4.06400013 的后验分布概率密度。 对于 <math id="M42"><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778841&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778840&type= 2.45533323 3.72533321 的后验分布密度，由其先验分布以及关于 <math id="M43"><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778844&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778843&type= 2.62466669 4.57200003 和 <math id="M44"><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778857&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778855&type= 1.77800000 4.06400013 的条件分布可得 <math display="block" id="M45"><mtable columnalign="left"><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn><mo stretchy="false">|</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>t</mi></mrow></msub><mo>,</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mtd></mtr><mtr><mtd><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>z</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn><mo stretchy="false">|</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn><mo>:</mo><mi>t</mi></mrow></msub><mo>,</mo><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778860&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778858&type= 38.86199951 19.55800056 其中， <math id="M46"><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mi>ρ</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><mo>-</mo><mfrac><mrow><mo stretchy="false">[</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mo>-</mo><mi>μ</mi><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy="false">}</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo>-</mo><mi>ρ</mi><mo stretchy="false">)</mo><msup><mrow><mi>l</mi></mrow><mrow><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">{</mo><mo>-</mo><mfrac><mrow><mi>l</mi><mo stretchy="false">[</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi></mrow></msub><msup><mrow><mo>-</mo><mi>μ</mi><mo>-</mo><mstyle displaystyle="true"><munderover><mo largeop="true">∑</mo><mrow><mi>i</mi><mo>=</mo><mn mathvariant="normal">1</mn></mrow><mrow><mi>s</mi></mrow></munderover></mstyle><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>t</mi><mo>-</mo><mi>i</mi></mrow></msub><mo>-</mo><mi>μ</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">]</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mn mathvariant="normal">2</mn><msup><mrow><mi>σ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>h</mi></mrow><mrow><mi>t</mi></mrow></msub></mrow></mfrac><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778851&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778862&type= 153.41600037 13.88533401 。 2　 随机梯度哈密尔顿蒙特卡洛法 1 随机梯度哈密尔顿蒙特卡洛法是在哈密尔顿系统的基础上发展而来。假定观察样本的集合为 <math id="M47"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779434&type= 2.45533323 2.79399991 ，我们感兴趣的参数 <math id="M48"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779183&type= 1.77800000 2.79399991 ，其基于样本观察值 <math id="M49"><mi>x</mi><mo>∈</mo><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778888&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778887&type= 8.29733276 3.13266683 的后验密度 <math id="M50"><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">|</mo><mi>Ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778879&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778890&type= 10.41399956 3.97933316 形式如下 <math display="block" id="M51"><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">|</mo><mi>Ω</mi><mo stretchy="false">)</mo><mo>∝</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778898&type= 23.02933502 4.23333359 这里将 <math id="M52"><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779005&type= 8.63599968 3.55599999 称为势能函数，根据贝叶斯定理，该函数有如下形式 <math display="block" id="M53"><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>-</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi mathvariant="normal">Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778915&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778906&type= 76.03067017 4.82600021 其中， <math id="M54"><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy="false">|</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778919&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778911&type= 27.34733391 4.82600021 为在给定 <math id="M55"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779183&type= 1.77800000 2.79399991 时样本的条件概率密度， <math id="M56"><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778926&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778934&type= 7.11199999 3.97933316 为 <math id="M57"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779183&type= 1.77800000 2.79399991 的先验概率密度。 为了使用哈密尔顿系统，还需要一个关于动能的函数，这里记动量变量为 <math id="M58"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779188&type= 1.52400005 2.79399991 ， <math id="M59"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779188&type= 1.52400005 2.79399991 一般为人为设置的辅助变量且服从正态分布 <math id="M60"><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="bold-italic">M</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778965&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778963&type= 14.30866718 3.55599999 ， <math id="M61"><mi mathvariant="bold-italic">M</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778954&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778966&type= 3.47133350 2.96333337 称之为质量矩阵且一般为单位矩阵 <math id="M62"><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778972&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778970&type= 1.43933344 2.79399991 。由 <math id="M63"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779183&type= 1.77800000 2.79399991 和 <math id="M64"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779188&type= 1.52400005 2.79399991 一起构造如下形式的微分方程组 <math display="block" id="M65"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">d</mi><mi>θ</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">d</mi><mi>r</mi><mo>=</mo><mo>-</mo><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778986&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29778994&type= 28.44799995 9.22866726 其中， <math id="M66"><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779034&type= 11.43000031 3.55599999 为 <math id="M67"><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779021&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779005&type= 8.63599968 3.55599999 的梯度，这样的微分方程组形成的系统称为哈密尔顿系统。去掉采集到的样本 <math id="M68"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779188&type= 1.52400005 2.79399991 那一部分，剩下的 <math id="M69"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779183&type= 1.77800000 2.79399991 那一部分即为我们需要的满足后验分布 <math id="M70"><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">|</mo><mi>Ω</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779013&type= 10.41399956 3.97933316 的样本。 而随机梯度汉密尔顿蒙特卡洛法是对原有使用的 <math id="M71"><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779036&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779034&type= 11.43000031 3.55599999 进行调整，不再选用全部的样本集计算，而是从 <math id="M72"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779434&type= 2.45533323 2.79399991 中随机抽取一个子集 <math id="M73"><mover><mrow><mi>Ω</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779050&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779049&type= 2.45533323 3.55599999 出来，并定义如下形式的随机梯度 <math display="block" id="M74"><mo>∇</mo><mover><mrow><mi>U</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mfrac><mrow><mfenced open="|" close="|" separators="|"><mrow><mi>Ω</mi></mrow></mfenced></mrow><mrow><mfenced open="|" close="|" separators="|"><mrow><mover><mrow><mi>Ω</mi></mrow><mrow><mi>˜</mi></mrow></mover></mrow></mfenced></mrow></mfrac><mo>∇</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mi>p</mi><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>m</mi></mrow></msub><mo stretchy="false">|</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>-</mo><mo>∇</mo><mi mathvariant="normal">l</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">g</mi><mi>p</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mover><mrow><mi>Ω</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779064&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779047&type= 92.62533569 10.92199993 由中心极限定理可知 <math display="block" id="M75"><mo>∇</mo><mover><mrow><mi>U</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>≈</mo><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779071&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779068&type= 51.13866425 3.97933316 其中， <math id="M76"><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779075&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779055&type= 8.72066593 3.55599999 为一未知的协方差矩阵。在实际应用哈密尔顿系统时通常会设置一个迭代步长 <math id="M77"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779412&type= 1.94733346 2.79399991 ，由此可知辅助变量 <math id="M78"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779188&type= 1.52400005 2.79399991 的变化量 <math id="M79"><mi mathvariant="normal">Δ</mi><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779095&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779087&type= 3.97933316 2.79399991 在 <math id="M80"><mo>∇</mo><mover><mrow><mi>U</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779101&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779098&type= 11.43000031 3.97933316 作用时有 <math id="M81"><mi mathvariant="normal">Δ</mi><mi>r</mi><mo>=</mo><mo>-</mo><mi>δ</mi><mo>∇</mo><mover><mrow><mi>U</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mi>δ</mi><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><msup><mrow><mi>δ</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779090&type= 73.82933807 4.23333359 ，并记 <math id="M82"><mi>B</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><mi>δ</mi><mi mathvariant="bold-italic">V</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779106&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779112&type= 26.33133125 7.87400007 ，则原微分方程组变为如下形式 <math display="block" id="M83"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">d</mi><mi>θ</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">d</mi><mi>r</mi><mo>=</mo><mo>-</mo><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mi>B</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779120&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779124&type= 56.98066711 9.22866726 但该形式并不能保证对应的哈密尔顿系统收敛到我们想要的平稳分布 <math id="M84"><mi>π</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779331&type= 10.83733368 3.55599999 ，需要添加一项进行修正，因此得到如下形式的微分方程组 <math display="block" id="M85"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">d</mi><mi>θ</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">d</mi><mi>r</mi><mo>=</mo><mo>-</mo><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>-</mo><mi>B</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mi>B</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779136&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779132&type= 81.87266541 9.22866726 为方便讨论，这里假定 <math id="M86"><mi>B</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779143&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779139&type= 7.78933382 3.55599999 和 <math id="M87"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779183&type= 1.77800000 2.79399991 无关，将其简记为 <math id="M88"><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779148&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779158&type= 2.53999996 2.79399991 。下面给出证明，该方程组可以收敛到我们想要的平稳分布 <math id="M89"><mi>π</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779331&type= 10.83733368 3.55599999 。 定理1 微分方程组 <math display="block" id="M90"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">d</mi><mi>θ</mi><mo>=</mo><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">d</mi><mi>r</mi><mo>=</mo><mo>-</mo><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>-</mo><mi>B</mi><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mi>B</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779170&type= 69.76532745 9.22866726 当其满足：任意 t 时刻的联合概率密度 <math id="M91"><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779238&type= 11.59933376 4.82600021 关于 <math id="M92"><mi>θ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779186&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779183&type= 1.77800000 2.79399991 ， <math id="M93"><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779190&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779188&type= 1.52400005 2.79399991 的二阶混合偏导数连续时，可以收敛到 <math id="M94"><mi>π</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>∝</mo><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi><mi mathvariant="normal">p</mi><mo stretchy="false">(</mo><mo>-</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>-</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779203&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779193&type= 55.20266724 7.87400007 的平稳分布。 证 记 <math id="M95"><mi>H</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac><msup><mrow><mi>r</mi></mrow><mrow><mi mathvariant="normal">'</mi></mrow></msup><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779207&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779204&type= 44.28066635 7.87400007 ， <math id="M96"><mi mathvariant="bold-italic">G</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mi mathvariant="bold-italic">I</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">I</mi></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779211&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779209&type= 20.99733353 8.55133343 ， <math id="M97"><mi mathvariant="bold-italic">D</mi><mo>=</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mn mathvariant="normal">0</mn></mtd></mtr><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779228&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779219&type= 18.79599953 8.55133343 ， <math id="M98"><mi mathvariant="bold-italic">I</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779230&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779225&type= 1.43933344 2.79399991 为单位矩阵，则（11）式可记为以下形式 <math display="block" id="M99"><mi mathvariant="normal">d</mi><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mi>θ</mi></mtd></mtr><mtr><mtd><mi>r</mi></mtd></mtr></mtable></mrow></mfenced><mo>=</mo><mo>-</mo><mfenced open="[" close="]" separators="|"><mrow><mtable><mtr><mtd><mn mathvariant="normal">0</mn></mtd><mtd><mo>-</mo><mi mathvariant="bold-italic">I</mi></mtd></mtr><mtr><mtd><mi mathvariant="bold-italic">I</mi></mtd><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mfenced><mfenced open="[" close="]" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi></mtd></mtr></mtable></mrow></mfenced><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>-</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">D</mi><mo>+</mo><mi mathvariant="bold-italic">G</mi><mo stretchy="false">]</mo><mo>∇</mo><mi>H</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mi mathvariant="bold-italic">D</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779234&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779232&type= 135.38200378 9.22866726 由Fokker-Planck方程易知，在 t 时刻 <math id="M100"><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779247&type= 8.29733276 3.55599999 的联合概率密度 <math id="M101"><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779238&type= 11.59933376 4.82600021 有如下等式 <math display="block" id="M102"><mfrac><mrow><mo>∂</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msup><mrow><mo>∇</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">D</mi><mo>+</mo><mi mathvariant="bold-italic">G</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mo>∇</mo><mi>H</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo><mo>+</mo><msup><mrow><mo>∇</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">[</mo><mi mathvariant="bold-italic">D</mi><mo>∇</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779258&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779256&type= 105.24067688 8.29733276 而由定理中的条件，可以得到 <math id="M103"><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>θ</mi><mo>∂</mo><mi>r</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>r</mi><mo>∂</mo><mi>θ</mi></mrow></mfrac></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779261&type= 36.66066742 8.29733276 ，从而有 <math id="M104"><msup><mrow><mo>∇</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">[</mo><mi>G</mi><mo>∇</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo>=</mo><mo>-</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>θ</mi><mo>∂</mo><mi>r</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><msup><mrow><mo>∂</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>r</mi><mo>∂</mo><mi>θ</mi></mrow></mfrac><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779268&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779279&type= 77.38533020 8.29733276 ，因此（13）式可化为 <math display="block" id="M105"><mfrac><mrow><mo>∂</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mo>=</mo><msup><mrow><mo>∇</mo></mrow><mrow><mi mathvariant="normal">T</mi></mrow></msup><mo stretchy="false">{</mo><mo stretchy="false">[</mo><mi mathvariant="bold-italic">D</mi><mo>+</mo><mi mathvariant="bold-italic">G</mi><mo stretchy="false">]</mo><mo stretchy="false">[</mo><mo>∇</mo><mi>H</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>+</mo><mo>∇</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo><mo stretchy="false">}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779284&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779272&type= 93.30266571 8.29733276 当 <math id="M106"><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779287&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779294&type= 27.00866890 4.82600021 时，（14）式右边有 <math id="M107"><mo>∇</mo><mi>H</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>H</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msup><mo>+</mo><mo>∇</mo><msup><mrow><mi mathvariant="normal">e</mi></mrow><mrow><mo>-</mo><mi>H</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msup><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779307&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779305&type= 49.44533539 3.80999994 ，因此有 <math id="M108"><mfrac><mrow><mo>∂</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><mrow><mo>∂</mo><mi>t</mi></mrow></mfrac><mfenced open="|" close="" separators="|"><mrow><msub><mrow/><mrow><msub><mrow><mi>p</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>π</mi></mrow></msub></mrow></mfenced><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779309&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779302&type= 27.77066803 8.29733276 ，这表明 <math id="M109"><mi>π</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779331&type= 10.83733368 3.55599999 和时间 <math id="M110"><mi>t</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779316&type= 1.10066664 2.79399991 无关，因而 <math id="M111"><mi>π</mi><mo stretchy="false">(</mo><mi>θ</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779337&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779331&type= 10.83733368 3.55599999 就是该系统下的平稳分布，证明完毕。 在实际应用中，通常不知道 <math id="M112"><mi mathvariant="bold-italic">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779354&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779349&type= 2.53999996 2.79399991 的具体形式，因此要先利用样本对 <math id="M113"><mi mathvariant="bold-italic">B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779358&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779356&type= 2.53999996 2.79399991 做一个估计，记为 <math id="M114"><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779363&type= 2.53999996 3.72533321 ，然后人为指定一项 <math id="M115"><mi mathvariant="bold-italic">C</mi><mo>≥</mo><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779378&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779370&type= 9.90600014 3.97933316 ，从而（11）式的第二个微分方程变为如下形式 <math display="block" id="M116"><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">d</mi><mi>r</mi><mo>=</mo><mo>-</mo><mo>∇</mo><mi>U</mi><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>-</mo><mi mathvariant="bold-italic">C</mi><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><mo>-</mo><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>̂</mo></mover><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mi mathvariant="bold-italic">B</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo></mtd></mtr><mtr><mtd><mo>-</mo><mo>∇</mo><mover><mrow><mi>U</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><mi>θ</mi><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo>-</mo><mi mathvariant="bold-italic">C</mi><msup><mrow><mi>M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mi>r</mi><mi mathvariant="normal">d</mi><mi>t</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><mo>-</mo><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>̂</mo></mover><mo stretchy="false">)</mo><mi mathvariant="normal">d</mi><mi>t</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779372&type= 105.74867249 10.41399956 为了能尽可能多地利用观察样本的信息，在每构建一个哈密尔顿系统前随机抽取一个样本子集进行随机梯度的计算，这样就克服了仅使用单一样本子集信息量不足的缺陷，具体算法如下： 步骤1 给定初值 <math id="M117"><msup><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779384&type= 4.40266657 4.23333359 ，要采集的样本数 <math id="M118"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779545&type= 1.86266661 2.79399991 ，内循环次数 <math id="M119"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779522&type= 2.79399991 2.79399991 ，步长 <math id="M120"><mi>δ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779412&type= 1.94733346 2.79399991 ，矩阵 <math id="M121"><mi mathvariant="bold-italic">C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779426&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779424&type= 2.53999996 2.79399991 和 <math id="M122"><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>̂</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779442&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779430&type= 2.53999996 3.72533321 步骤2 从观察样本全集 <math id="M123"><mi>Ω</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779435&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779434&type= 2.45533323 2.79399991 中抽取一个样本子集 <math id="M124"><mover><mrow><mi>Ω</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779451&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779437&type= 2.45533323 3.55599999 。后面所用到的样本 <math id="M125"><mover accent="true"><mi>y</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779456&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779468&type= 1.86266661 3.97933316 均为 <math id="M126"><mover><mrow><mi>Ω</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779474&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779473&type= 2.45533323 3.55599999 内的样本 步骤3 抽取样本 <math id="M127"><mi>r</mi><mo>~</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>,</mo><mi>M</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779477&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779464&type= 18.88066673 3.55599999 ，取 <math id="M128"><msub><mrow><mi>θ</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779480&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779479&type= 2.96333337 3.72533321 为前一次采集得到的样本值， <math id="M129"><msub><mrow><mi>r</mi></mrow><mrow><mn mathvariant="normal">0</mn></mrow></msub><mo>=</mo><mi>r</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779499&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779481&type= 8.97466660 3.72533321 步骤4 <math id="M130"><msub><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>δ</mi><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779493&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779502&type= 33.10466385 5.24933338 步骤5 抽取 <math id="M131"><mover accent="true"><mi>z</mi><mo>⃗</mo></mover><mo stretchy="false">|</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><mover accent="true"><mi>y</mi><mo>⃗</mo></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779515&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779496&type= 8.63599968 5.84200001 步骤6 <math id="M132"><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mo>-</mo><mo>∇</mo><mover><mrow><mi>U</mi></mrow><mrow><mi>˜</mi></mrow></mover><mo stretchy="false">(</mo><msub><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mi>δ</mi><mo>-</mo><mi mathvariant="bold-italic">C</mi><msup><mrow><mi mathvariant="bold-italic">M</mi></mrow><mrow><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><msub><mrow><mi>r</mi></mrow><mrow><mi>i</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msub><mi>δ</mi><mo>+</mo><mi>N</mi><mo stretchy="false">(</mo><mn mathvariant="normal">0,2</mn><mo stretchy="false">(</mo><mi mathvariant="bold-italic">C</mi><mo>-</mo><mover accent="true"><mi mathvariant="bold-italic">B</mi><mo>̂</mo></mover><mo stretchy="false">)</mo><mi>δ</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779520&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779518&type= 87.71466827 5.58799982 步骤7 重复步骤4~6 <math id="M133"><mi>m</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779522&type= 2.79399991 2.79399991 次，得到 <math id="M134"><msub><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mi>m</mi></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779533&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779532&type= 3.47133350 4.91066647 ，将其作为该次采样过程中取得的样本。 重复步骤2~7 <math id="M135"><mi>n</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779536&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779545&type= 1.86266661 2.79399991 次，即取得样本 <math id="M136"><msup><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">1</mn><mo stretchy="false">)</mo></mrow></msup><mo>,</mo><msup><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mo stretchy="false">(</mo><mn mathvariant="normal">2</mn><mo stretchy="false">)</mo></mrow></msup><mo>,</mo><mo>⋯</mo><mo>,</mo><msup><mrow><mover accent="true"><mi>θ</mi><mo>⃗</mo></mover></mrow><mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779554&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779538&type= 23.11400032 4.65666676 。 3　 实例分析 1 在实例分析中采用中证医药2019年3月13日—2020年11月3日的指数收盘价数据的变化率，数据集总计有398个数据，将其拆分为2019年3月13日—2020年1月2日199个数据以及2020年1月3日—2020年11月3日199个数据分别利用上述模型进行分析。这两部分的数据集的时间序列分别如 图1 图1 和 图2 图2 。 从 图1 图1 和 图2 图2 中可以发现这两部分数据的波动幅度随时间变化，因此适合用异方差模型进行估计，进一步绘制了两组数据的偏自相关函数（autocorrelation function，ACF）图，如 图3 图3 和 图4 图4 。 可以发现两组数据的偏自相关函数都是1阶截尾的，因此选用AR（1）-GJR-GARCH（1，1）模型比较合适。利用MATLAB软件使用SGHMC算法来对该模型的参数进行贝叶斯估计，每次内循环开始前从数据集中随机选择99个数据作为子集 <math id="M137"><mover><mrow><mi>Ω</mi></mrow><mrow><mi>˜</mi></mrow></mover></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779614&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779624&type= 2.45533323 3.55599999 ，并设置 <math id="M138"><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779632&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779630&type= 9.73666668 4.57200003 。利用该算法获取1 000个样本，分别得到这两组数据的参数估计结果如 表1 表1 和 表2 表2 。 参数 均值 标准差 中位数 95%置信区间 <math id="M139"><mi>ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779737&type= 1.60866666 3.13266683 0.647 126 0.027 466 0.646 838 (0.593 292,0.700 959) <math id="M140"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779752&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779750&type= 1.52400005 2.37066650 0.004 542 0.006 148 0.004 545 ( <math id="M141"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779692&type= 2.53999996 2.37066650 0.007 51,0.016 593) <math id="M142"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779763&type= 1.69333339 3.13266683 0.195 053 0.348 665 0.234 757 ( <math id="M143"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779692&type= 2.53999996 2.37066650 0.488 33,0.878 437) <math id="M144"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779787&type= 1.69333339 3.13266683 0.026 127 0.014 912 0.026 340 ( <math id="M145"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779698&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779692&type= 2.53999996 2.37066650 0.003 1,0.055 355) <math id="M146"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779794&type= 1.01600003 2.37066650 0.547 681 0.027 298 0.547 284 (0.494 178,0.601 185) <math id="M147"><msub><mrow><mi>α</mi></mrow><mrow><mn>11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779811&type= 3.55599999 3.21733332 0.248 535 0.027 912 0.248 801 (0.193 826,0.303 243) <math id="M148"><msub><mrow><mi>α</mi></mrow><mrow><mn>21</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779817&type= 3.55599999 3.21733332 0.248 330 0.028 072 0.248 115 (0.193 308,0.303 351) <math id="M149"><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779832&type= 2.70933342 3.89466691 0.247 328 0.027 143 0.246 775 (0.194 127,0.300 528) 参数 均值 标准差 中位数 95%置信区间 <math id="M150"><mi>ρ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779749&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779737&type= 1.60866666 3.13266683 0.697 204 0.027 550 0.697 494 (0.643 205,0.7512 03) <math id="M151"><mi>λ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779752&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779750&type= 1.52400005 2.37066650 0.007 687 0.007 129 0.007 062 ( <math id="M152"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779791&type= 2.53999996 2.37066650 0.006 29,0.021 66) <math id="M153"><mi>γ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779763&type= 1.69333339 3.13266683 <math id="M154"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779791&type= 2.53999996 2.37066650 0.170 110 0.344 651 0.011 953 ( <math id="M155"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779791&type= 2.53999996 2.37066650 0.845 63,0.505 403) <math id="M156"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779801&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779787&type= 1.69333339 3.13266683 0.026 327 0.015 099 0.026 263 ( <math id="M157"><mo>-</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779806&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779791&type= 2.53999996 2.37066650 0.003 27,0.055 92) <math id="M158"><mi>l</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779809&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779794&type= 1.01600003 2.37066650 0.447 679 0.027 722 0.447 676 (0.393 345,0.502 013) <math id="M159"><msub><mrow><mi>α</mi></mrow><mrow><mn>11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779811&type= 3.55599999 3.21733332 0.295 355 0.026 125 0.295 147 (0.244 15,0.346 56) <math id="M160"><msub><mrow><mi>α</mi></mrow><mrow><mn>21</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779819&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779817&type= 3.55599999 3.21733332 0.247 108 0.026 744 0.247 972 (0.194 69,0.299 526) <math id="M161"><msub><mrow><mi>β</mi></mrow><mrow><mn>1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779821&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779832&type= 2.70933342 3.89466691 0.266 454 0.027 284 0.266 250 (0.212 978,0.319 93) 选择样本均值作为参数估计值。 表1 表1 和 表2 表2 的参数 <math id="M162"><mover accent="true"><mi>γ</mi><mo>̂</mo></mover><mo>∈</mo><mo stretchy="false">(</mo><mo>-</mo><mn mathvariant="normal">1,1</mn><mo stretchy="false">)</mo><mo>,</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn><mo>,</mo><msub><mrow><mover accent="true"><mi>α</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub><mo>+</mo><msub><mrow><mover accent="true"><mi>β</mi><mo>̂</mo></mover></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo><</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779861&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779841&type= 58.50466156 5.50333309 ，这说明模型基本是稳定的。 表1 表1 中 <math id="M163"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779865&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779862&type= 1.94733346 3.64066648 的均值为 <math id="M164"><mn mathvariant="normal">0.026</mn><mtext> </mtext><mn mathvariant="normal">127</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779869&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779852&type= 14.90133286 2.87866688 ， 表2 表2 中 <math id="M165"><mi>μ</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779873&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779856&type= 1.94733346 3.64066648 的均值为 <math id="M166"><mn mathvariant="normal">0.026</mn><mtext> </mtext><mn mathvariant="normal">237</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779893&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779875&type= 14.90133286 2.87866688 ，说明两组数据的收盘价均有一个缓慢上涨的趋势；二者数值差别不大，说明在这前后两个时间段该指数的收盘行情基本相差不大。 表1 表1 中 <math id="M167"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779882&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779896&type= 4.14866638 3.72533321 和 <math id="M168"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779902&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779885&type= 4.14866638 3.72533321 的均值相差仅为0.000 205，变化率仅为0.08%，说明这段时期该指数的杠杆效应并不明显，好消息和坏消息对该指数产生的影响可以大致抵消；但在 表2 表2 中，可以看到 <math id="M169"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">11</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779921&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779905&type= 4.14866638 3.72533321 的均值明显高于 <math id="M170"><msub><mrow><mi>α</mi></mrow><mrow><mn mathvariant="normal">21</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779909&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=29779924&type= 4.14866638 3.72533321 的均值，说明在这段时间好消息对该指数的影响超过了坏消息。结合该时期分析，这段时间我国爆发了新冠疫情，疫情的出现使得原本相对平稳的医药生物产业走进了人们的视野，一时间导致大量投资者涌入了该产业，并且随着疫情转为长期存在的情形，许多投资者开始长期在该板块中投入。 综上，我们的模型一定程度上反映了该指数在一段时期内的背景信息，利用该模型可以对市场信息作一定的解释。 4　 结 语 1 本文利用随机梯度哈密尔顿蒙特卡洛方法在后验密度解析形式未知的情况下得到了含混合高斯项的AR-GJR-GARCH模型各参数的贝叶斯估计。该方法相比于前人的Griddy-Gibbs方法优势在于无需在每次迭代前再对一个复杂积分项进行数值估计，降低了计算复杂度。实例分析给出了AR（1）-GJR-GARCH（1，1）模型在中证医药2019年3月13日—2020年11月3日指数收盘价数据变化率下的参数贝叶斯估计值，并根据得到的估计值推算市场背后信息，发现其较好地反映了该指数的市场波动情况与好坏消息对该指数的影响，验证了模型和方法的合理性。 参考文献 ENGLE R F . 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DOI： 10.14188/j.1671-8836.2021.0031（Ch http://dx.doi.org/10.14188/j.1671-8836.2021.0031(Ch ）. WANG Xiangsai.Bayesian Estimation of AR-GJR-GARCH Model with Mixed Gaussian Terms ［J］.J Wuhan Univ （Nat Sci Ed），2022，68（3）：317-323. DOI：10.14188/j.1671-8836.2021.0031（Ch）. 王祥赛.含混合高斯项的AR-GJR-GARCH模型的贝叶斯估计[J].武汉大学学报（理学版）,2022,68(3):317-323. DOI:10.14188/j.1671-8836.2021.0031 王祥赛.含混合高斯项的AR-GJR-GARCH模型的贝叶斯估计[J].武汉大学学报（理学版）,2022,68(3):317-323. DOI:10.14188/j.1671-8836.2021.0031 国家自然科学基金（11731012） 2021-02-07 2022-06-24 2022-06-21 武汉大学学报（理学版） 3 68