金融學中的貝葉斯方法讀書筆記-(Bayesian Methods in Finance)

第一章   介紹

頻率學派 VS 貝葉斯學派 （Frequentist v.s Bayesian)

• 頻率學派認爲一個變量的概率分佈是確定的，然而貝葉斯學派認爲變量的概率分佈是不確定的，它會隨着新的消息，新的觀測而發生改變，先有根據已有的經驗假設一個先驗概率分佈，而後利用觀測值得出後驗分佈

Proponents of the frequentist approachconsider the s ource of
uncertainty to be the r andomness inherent in realizations of a r
andom variable. The probability distributions of variabl es are not
subject to uncertainty. In contrast, Bayesian statistics treats pr
obability distributions as uncertain and subject to modification as
new information becomes available. Uncertainty is implicitly
incorporated by probab ility updating. T he probability beliefs based
on the existing knowledge base take the form of the prior probability.
The posterior probability represents the updated beliefs.

第二章   貝葉斯框架 - 似然函數

泊松分佈

1. 泊松概率分佈公式
   $p(X=k)=\frac{\theta^{k}}{k !} e^{-\theta}, \quad k=0,1,2, \ldots$

2. 假設有20個觀測值 $x_{1}, x_{2}, \ldots, x_{20}$， 那麼聯合概率分佈爲：
   $\begin{aligned} L\left(\theta | x_{1}, x_{2}, \ldots, x_{20}\right) &=\prod_{i=1}^{20} p\left(X=x_{i} | \theta\right)=\prod_{i=1}^{20} \frac{\theta^{x_{i}}}{x_{i} !} e^{-\theta} \\ &=\frac{\theta^{\sum_{i=1}^{20} x_{i}}}{\prod_{i=1}^{20} x_{i} !} e^{-20 \theta} \end{aligned}$

3. 上式可以改爲：
   $L\left(\theta | x_{1}, x_{2}, \ldots, x_{20}\right) \propto \theta^{\Sigma_{i=1}^{20} x_{i}} e^{-20 \theta}$

4. 利用極大似然估計法（maximum likelihood）得出 最大似然估計:
   $\widehat{\theta}=\bar{x}=\frac{\sum_{i=1}^{20} x_{i}}{20}$
   即爲20個觀測值的平均值

5. 上述推斷的一個隱含假設是20個觀測值相互獨立，互不干擾

正態分佈

1. 概率分佈公式
   $f(y)=\frac{1}{\sqrt{2 \pi} \sigma} e^{-\frac{(-x)^{2}}{2 \sigma^{2}}}$

貝葉斯公式

1. $P(E | D)=\frac{P(D | E) \times P(E)}{P(D)}$

貝葉斯推斷與二項分佈

• Beta分佈是二項分佈的共軛先驗

The beta distribution is the conjugate prior distribution for the
binomial parameter θ . This means that the posterior distribution of θ
is also a beta distribution (of course, with updated parameters)

• 有時候由於樣本數據量太大，不同先驗分佈的選擇其實不會造成後驗分佈的較大差異，因爲這時候大樣本可以掩蓋掉先驗信息的作用。

The two posterior estimates and the m aximum-likelihood estimate are the same for all practical purposes. The reason is that the sample
size is so large that the information contained in the data sample
‘‘swamps out’’ the prior information. In Chapter 3, we further
illustrate and comment on the role sample size plays in posterior
inference

第三章   先驗信息與後驗信息、以及預測推斷

先驗：

• $p(\theta | \boldsymbol{y}) \propto L(\theta | \boldsymbol{y}) \pi(\theta)$
  where:
  • θ = unknown parameter whose inference we are interested in.
  • y = a vector (or a matrix) of recorded observations.
  • π (θ ) = prior distribution of θ depending on one or more parameters, called hyperparameters.
  • L (θ |y) = likelihood function for θ .
  • p(θ |y) = posterior (updated) distribution of θ .

從上述公式可與i看出：

• 影響後驗分佈主要有兩個因素，一是先驗信息，二是觀察到的數據
• 先驗信息的影響效力通常會隨着觀測數據量變大而減弱
• 如果觀測數據量很小，那麼先驗信息將會很大決定後驗分佈