Eikosograms的一个应用

今天导师在组会上问了一个问题:

已知事件B和C在事件A的条件下独立,并且B和C在A-complement条件也也独立,问B和C是否独立?

类似涉及到条件概率的问题可以用Eikosograms直观的理解。关于Eikosograms可以参考http://sas.uwaterloo.ca/~rwoldfor/papers/venn/eikosograms/paperpdf.pdf。我们学概率的时候常常使用文氏图,但是文氏图有时候会比较misleading,比如 disjoint 的事件和 independent 的事件文氏图并不容易区分。

diagram

在一个Eikosogram中,region表示一个事件,region的面积表示此事件发生的概率。比如上面的图中有三个随机变量:A,B和C。这三个随机变量均包含0,1两个outcome (扩展到多个outcome也直接)。条件概率自然的可以用面积比表示(方框的总面积为1)。比如定义

C= 0: shaded region;

C=1:un-shaded region。

那么

Pr( C=1 | A = 0, B = 0 ) / Pr( C=0 | A = 0, B = 0 )  = Area of T1 / Area of (S1).

基于这个定义,independence有很直观的表示。比如 S1 和 S3 的“水位”相同则表示 B 和 C 在 A=0 的条件下独立。同样的,S2 和 S4 的“水位”相同表示B 和 C 在 A=1(即A-complement) 的条件下独立。那么B 和 C 是否是 unconditionally independent 呢?这个时候我们只需要把第1列和第2列合并,第3列和第4列合并,然后看等效的“水位”是否相同即可

Area(  S1 + S2 ) / Area( T1 + T2 )  ?= Area(  S3 + S4 ) / Area( T3 + T4 )

我们可以想象第1列和第2列中间有个隔板,抽掉隔板以后有个新的水位,如虚线所示;同样的第3和第4列之间也有一个隔板,抽掉之后也有新的水位。显然两个虚线一般不会处于同样的水位。

 

 

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这边的课题终于要接近尾声了

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英语差到绝望了。。。

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对小明的看法与第一次接触时发生了180度转变,这说明了第一印象不一定靠谱。

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“怕什么真理无穷 进一寸有一寸的欢喜”

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New Year Resolution,虽然迟了一个月。

1.要完成的文章

  • NOMA,OAMP coding,小梁,志鹏,许师兄,PT分析
  • 至少与Arian合作一篇文章,实在不行就PT分析这个工作

2.要学的东西

  • ESL podcast继续听,每天读新闻
  • Optimization,duality,KKT这些搞到烂熟,computational statistics课认真上
  • Follow一些优化领域的热点,比如Goldfarb在UCLA的几个slides,stochastic optimization,tensor等等;读一些arXiv上optimization的文章,找到一个自己以后做PI时可以从事的方向

3. 继续postdoc,如果晓东没有funding,备选??

4. 养成写东西的习惯,be a creator rather than a consumer

  • 每个月在这里记录一次,越多越好
  • 学的知识多做note;比如对optimization基础知识的总结
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