# Linear Algebra and Probability Theory

2022-08-06 18:09:55

Talk about artificial intelligence and machine learning,You must have some basic knowledge of mathematics,Only then can we better understand its nature.The most important of these basic mathematical knowledge contains two pieces of content：线性代数和概率论.

## 线性代数

The core meaning of linear algebra：万事万物都可以被抽象成某些特征的组合,并在由预置规则定义的框架之下以静态和动态的方式加以观察.

Vectors that describe mathematical objects require a specific mathematical language,范数内积就是代表.

The norm is a measure of the size of a single vector,描述的是向量自身的性质,其作用是将向量映射为一个非负的数值.通用的L(p)范数定义如下： L(1)范数计算的是向量所有元素绝对值的和,L(2)范数计算的是通常意义上的向量长度,L(+)范数计算的则是向量中最大元素的取值.

The norm computes the scale of a single vector,The inner product computes the relationship between two vectors. The inner product expression of two vectors of the same dimension is ： This expression can be understood as a vector x 经过矩阵 A 所描述的变换,变成了向量 y;也可以理解为一个对象在坐标系 A 的度量下得到的结果为向量 x,在标准坐标系 I（单位矩阵：主对角线元素为 1,其余元素为 0）的度量下得到的结果为向量 y.

Describe the matrix⼀The important parameters are特征值（eigenvalue特征向量（eigenvector）.对于给定的矩阵 A,Suppose its eigenvalue is λ,特征向量为 x,Then the relationship between them is as follows： Ax=λx

A matrix represents a transformation of a vector,其效果通常是对原始向量同时施加方向变化和尺度变化.Available for some special vectors,矩阵的作用只有尺度变化而没有方向变化,也就是只有伸缩的效果而没有旋转的效果.对于给定的矩阵来说,这类特殊的向量就是矩阵的特征向量,特征向量的尺度变化系数就是特征值.

The dynamic significance of matrix eigenvalues ​​and eigenvectors is to represent the speed and direction of change.

## 概率论

Same as linear algebra,Probability theory also represents a way of looking at the world,其关注的焦点是无处不在的可能性.The formal mathematical description of the probability of random events is the axiomatic process of probability theory.The axiomatic structure of probability reflects an understanding of the nature of probability.

The method of recognizing probability from the frequency of events is called频率学派（frequentist probability）,In the mouth of the frequentist school“概率”,In fact, it is the limit of the frequency of occurrence of a single result in an independently repeatable random experiment.Because the stable frequency is the embodiment of statistical regularity,The frequency is thus calculated from a large number of independent replicates,It is a reasonable idea to use it to characterize the possibility of an event happening.   Frequentists believe that assumptions exist objectively and do not change,即存在固定的先验分布,只是作为观察者的我们无从知晓.

The idea of ​​maximum likelihood estimation is to maximize the probability that the training data will appear,依此确定概率分布中的未知参数,估计出的概率分布也就最符合训练数据的分布.The idea of ​​the maximum posterior probability method is based on the training data and other known conditions,使未知参数出现的可能性最大化,并选取最可能的未知参数取值作为估计值.

## 总结

Whether machine learning or artificial intelligence,These high-level nouns can finally be connected with the mathematics that I have learned for many years,Really happy.Although I am not a math major,But I have always been confident in mathematics,大学时候的《线性代数》,in graduate school《概率论》,I still have the impression of these basic knowledge,But deep understanding still requires practice and consolidation.