查看“︁偏差信息量准则”︁的源代码

{{multiple issues|
{{copyedit|time=2018-10-05T00:00:46+00:00}}
{{no footnotes|time=2018-10-05T00:00:46+00:00}}
{{roughtranslation|time=2018-10-05T00:00:46+00:00}}
}}
'''偏差信息量准则'''（{{lang-en| deviance information criterion}}，DIC）是[[等级线性模型|等级模型]]化的[[赤池信息量准则]]（AIC），被广泛应用于由[[马尔可夫链蒙特卡洛]]（MCMC）所模拟的[[后验概率|后验分布]]的[[贝叶斯推断|贝叶斯]][[统计模型|模型]]选择问题。然而，該項准則由於與[[赤池信息量准则]]同为僅随样本容量增加的[[渐近分析|渐近近似]]，因此只可应用于[[后验概率|后验分布]]呈[[多元正态分布]]的情况。


==定义==
定义偏差（{{lang|en|deviance}}）为 <math>D(\theta)=-2\log(p(y|\theta))+C</math>，其中 <math>y</math> 为数据，<math>\theta</math> 是模型中的未知参量，<math>p(y|\theta)</math> 是[[似然函数]]，<math>C</math> 是常量。

有两种计算模型参数的有效数量 <math>p_D</math>的方法。一种是 <math>p_D=\bar{D}-D(\bar{\theta})</math>，其中  <math>\bar{\theta}</math> 是 <math>\theta</math> 的[[期望值|期望]]{{harvcol|Spiegelhalter|Best|Carlin|van der Linde|2002|p=587}}。
第二种是 <math>p_D = p_V = \frac{1}{2}\widehat{\operatorname{var}}\left(D(\theta)\right)</math> {{harvcol|Gelman|Carlin|Stern|Rubin|2004|p=182}}。
有效数量 <math>p_D</math>越大，模型的参数就越多，模型就越容易拟合数据，但也需要更小的偏差。

偏差信息量准则 <math>\mathit{DIC}</math> 被定义为
:<math>\mathit{DIC} = p_D+\bar{D}</math>，
或等效于
:<math>\mathit{DIC} = D(\bar{\theta})+2 p_D</math>。
从第二种定义更能看出它和[[赤池信息量准则]]的联系。

==应用==
一般而言，偏差信息量准则 <math>\mathit{DIC}</math> 的值越小，模型越好。这一准则的优点是它很容易从[[马尔可夫链蒙特卡洛]]（MCMC）模拟产生的样本中计算出来。



==参见==
* [[赤池信息量准则]]
* [[相对熵]]

==参考文献==
{{refbegin}}

*{{cite journal

 | first = Tomohiro | last = Ando

 | year = 2007

 | title = Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models

 | url = https://archive.org/details/sim_biometrika_2007-06_94_2/page/443 | journal = Biometrika

 | volume = 94  | pages = 443–458

 | doi = 10.1093/biomet/asm017

 | issue = 2

 | ref = harv

}}

* Ando, T. (2010). ''Bayesian Model Selection and Statistical Modeling'', CRC Press. Chapter 7.

*{{cite journal

 | first = Tomohiro | last = Ando

 | year = 2011

 | title = Predictive Bayesian Model Selection

 | journal = American Journal of Mathematical and Management Sciences

 | volume = 31  | pages = 13–38

 | doi = 10.1080/01966324.2011.10737798

 | issue = 1-2

 | ref = harv

 }}

* Claeskens, G, and Hjort, N.L. (2008). ''Model Selection and Model Averaging'', Cambridge. Section 3.5.

*{{cite book

| last1 = Gelman  | first1 = Andrew 

| last2 = Carlin  | first2 = John B.

| last3 = Stern   | first3 = Hal S.

| last4 = Rubin   | first4 = Donald B.

| year = 2004 | title = Bayesian Data Analysis: Second Edition

| url = https://archive.org/details/bayesiandataanal0000unse_w6z6  | series = Texts in Statistical Science | publisher = CRC Press

| isbn = 1-58488-388-X | lccn = 2003051474 | mr = 2027492

| ref = harv

}}

* van der Linde, A. (2005). "DIC in variable selection", ''Statistica Neerlandica'', 59: 45-56. doi:[https://dx.doi.org/10.1111/j.1467-9574.2005.00278.x 10.1111/j.1467-9574.2005.00278.x]

*{{cite journal

 | first = David J. | last = Spiegelhalter 

 | first2 = Nicola G. |last2=Best

 | first3=Bradley P. |last3=Carlin

 | first4= Angelika |last4=van der Linde

 |date=2002

 | title = Bayesian measures of model complexity and fit (with discussion)

 | journal = Journal of the Royal Statistical Society, Series B

 | volume = 64  | issue = 4  | pages = 583–639

 | doi = 10.1111/1467-9868.00353 |mr=1979380 | jstor = 3088806

 | ref = harv

}}

*{{cite journal

 | first = David J. | last = Spiegelhalter 

 | first2 = Nicola G. |last2=Best

 | first3=Bradley P. |last3=Carlin

 | first4= Angelika |last4=van der Linde

 |date= 2014

 | title = The deviance information criterion: 12 years on (with discussion)

 | journal = Journal of the Royal Statistical Society, Series B

 | volume = 76  | issue = 3  | pages = 485–493

 | ref = harv

}}

{{refend}}

==外部链接==
* {{cite web |first=Richard |last=McElreath |date=2015-01-29 |title=Statistical Rethinking Lecture 8 (on DIC and other information criteria) |via=[[YouTube]] |url=https://www.youtube.com/watch?v=vSjL2Zc-gEQ&list=PLDcUM9US4XdMdZOhJWJJD4mDBMnbTWw_z&index=8#t=31m25s |access-date=2018-10-04 |archive-date=2019-03-13 |archive-url=https://web.archive.org/web/20190313210353/https://www.youtube.com/watch?v=vSjL2Zc-gEQ&list=PLDcUM9US4XdMdZOhJWJJD4mDBMnbTWw_z&index=8#t=31m25s }}

	
{{Authority control}}
[[Category:迴歸變數選擇]]
[[Category:贝叶斯统计]]