Note01.md

Notes on Chapter 1

The Philosophy of Bayesian Inference

先验概率: 我们对于某个事物的belief，称为prior probability
后验概率: 在先验概率基础上，经过一些evidence之后，updated belief称为posterior probability

The Frequentist inference function would return a number, whereas the Bayesian function would return probabilities.

贝叶斯推断

bernoulli分布: 0-1分布
beta分布: 作为bernoulli分布的密度函数，先验共轭函数

概率分布

随机变量分为3种: 离散值、连续值、混合值

离散值的分布称为 probability mass function (PMF)
连续值的分布称为 probability density function (PDF)

常见的离散值分布:
Poisson分布:

常见的连续值分布
Exponential分布:

Inferring behavior from text-message data

# 假设短信数据分布符合泊松分布, 且E(C|λ) = λ
observation = mc.Poisson( "obs", lambda_, value = count_data, observed = True)

#泊松分布的参数λ，我们假设有一个switch point
@mc.deterministic
def lambda_( tau = tau, lambda_1 = lambda_1, lambda_2 = lambda_2 ):
    out = np.zeros( n_count_data ) 
    out[:tau] = lambda_1     #lambda before tau is lambda1
    out[tau:] = lambda_2     #lambda after tau is lambda2
    return out

# 假设λ符合指数分布
lambda_1 = mc.Exponential( "lambda_1", alpha )
lambda_2 = mc.Exponential( "lambda_2", alpha )
    
# E(λ|α) = 1/α; E(C|λ) = λ; E(C) = count_data.mean()    
alpha = 1.0/count_data.mean()

# P(τ=k)=1/n_count_data    
tau = mc.DiscreteUniform( "tau", lower = 0, upper = n_count_data )