## Learned something new about Bayesian formula last night and learned the hard way

Bayesian formula,
[P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|\bar{A})P(\bar{A})}]
is conceptually straightforward, but amazingly useful in statistics. It turns calculation of (P(A|B)) into finding out (P(B|A)) by simply making use of the rule of total probability,
[P(A\cap B) = P(A|B)P(B) = P(B|A)P(A)]
and
[P(A) = P(A\cap B) + P(A\cap \bar{B}). ]

This seems rather trivial to me. Here comes the surprising part. Let us now add another set (C) in the following way,
[P\left(A|B\right) = P\left(\left(A|C\right)|B\right)P\left(C|B\right) + P\left(\left(A|\bar{C}\right)|B\right)P\left(\bar{C}|B\right), \hspace{2cm} (1)]
or in this way,
[P\left(A|B\right) = P\left(\left(A|B\right)|C\right)P\left(C\right) + P\left(\left(A|B\right)|\bar{C}\right)P\left(\bar{C}\right). \hspace{2cm} (2)]

Now let us ask which one of the above two formulae is the proper one, or both, or none?

It is easy to verify the first one: Assuming
[P\left(\left(A|C\right)|B\right) = P\left(A|\left(C,B\right)\right) = \frac{A\cap B \cap C}{B \cap C}, \hspace{2cm} (3)]
then right-hand side of Equ(1) becomes
[\frac{A\cap B \cap C}{B \cap C}\frac{B\cap C}{B} + \frac{A\cap B \cap \bar{C}}{B \cap \bar{C}}\frac{B\cap \bar{C}}{B} = \frac{A\cap B \cap C}{B} + \frac{A\cap B \cap \bar{C}}{B} = \frac{A\cap B}{B}, \hspace{1cm} (4)]
which is exactly the left-hand side of Equ(1).

Verifying Equ(2) is however not easy. If the assumption in Equ(3) is right, then Equ(2) becomes
[\frac{A\cap B \cap C}{B \cap C}\frac{C}{\Omega} + \frac{A\cap B \cap \bar{C}}{B \cap \bar{C}}\frac{\bar{C}}{\Omega}. \hspace{1cm} (4)]
I can see no clue that this expression should be (\frac{A\cap B}{B}).

However, if (P\left(A|B\right)) is the probability of a set of events, then the second one should be correct too. So what is the problem? It seems to me that when discussing (P\left(A|B\right)), we have implicitly limited the whole set, which originally is (\Omega), to be (B), therefore, all the expressions derived from there should have carried the condition (B) forever. So lesson one: Keeping the condition (B) as the condition for all other events. Therefore, Equ(1), not Equ(2), should be used in our case.

Another lesson learned is that conditional probability is a tricky concept and one has to deal it with extra attention.

## 神书推荐（Recommending The Princeton Companion to Mathematics）

Recently, I found a great book on mathematics, The Princeton Companion to Mathematics. It is like a guide or a big-picture introduction to almost every subfields of mathematics, without losing any accuracy and attractiveness.

All mathematicians, physicists, and students in math, physics, or even other fields related to appplied math, should read at least certain parts of this great book.

I think physicists should produce a similar book on physics too. Or maybe every discpline should have a simiar one.

## 2012年春季《临界现象与复杂性》课程

1、线性代数与数值线性代数：矩阵、本征值与本征向量、线性变化、矢量空间、Blas与Lapack
2、概率论：古典概型、简单事件、复合事件、频率与概率、有限事件空间上的概率论，概率三元体，概率论的矩阵表示，Monte Carlo方法简介
3、分析力学：状态与状态空间、动力学过程、Hamilton方程、Lagrange方程
4、量子力学：二维系统的量子力学，态矢量、算符、Schroedinger方程、测量，密度矩阵
5、统计力学：状态空间、系综理论、配分函数，相变，Metropolis方法，量子统计初步
**********一学期的量，这次就开这么多，下次把《系统理论基础》跟这门课合起来开，讲两学期*************************
6、非线性动力学：动力系统、简单稳定性分析、不同定性行为分类并举例，混沌举例
7、高等统计力学：相变、重整化群、有限大小标度、临界现象与临界指数，产生湮灭算符形式的量子统计入门
8、高等量子力学：Hibert空间的矢量和算符，测量问题与量子力学基础，产生湮灭算符形式的量子力学