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Central Limit Theorem
Random variable X1, X2, ... Xn, independent and identically distributed, and have limited mathematical expectation and variance:
,
, then for any real number x, the distribution function
satisfies
this theorem, when a large n, a random variable
approximately follow a standard normal distribution N (0,1). Thus, when a large n,
approximately normal distribution
. The theorem of the central limit theorem is the simplest and most common form, in practical work, as long as n is large enough, they can be independent and identically distributed random variables and as a normal variable. This method is very common in mathematical statistics used when dealing with large sample, it is an important tool.
Simple application of the Central Limit Theorem
References [1]
Gaussian distribution
Gaussian distribution Gaussian distribution, also known as being too distributions Normal distribution, is a very important in mathematics, physics and engineering areas such as probability distributions, has a major influence on many aspects of statistics.
One yuan Gaussian distribution
If the random variable conforms to a Gaussian distribution
,则有如下的概率密度函数
满足
而如果我们对随机变量
进行标准化
, 那么变量
服从0均值,1方程的一元标准高斯分布。
多元高斯分布
多维高斯分布的公式:
其中
为一个n维向量,
是均值向量,
是协方差矩阵。
多元高斯分布的的线性变换
两个高斯分布的KL散度
参考资料[5]
两个一元(一维)高斯分布的KL散度
:
KL divergence two multi-dimensional Gaussian distribution
:
This algorithm will be used in VAE, record it, if we look at VAE time can be found.
Reference material
[1] the central limit theorem, Baidu Encyclopedia
[2] https://zhuanlan.zhihu.com/p/38501770
[. 3] https://zhuanlan.zhihu.com/p/58987388
[. 4] HTTPS: // zhuanlan. zhihu.com/p/90272131
[. 5] of VAE (. 1) - Starting from KL