起因

在看 椭圆检测 这篇论文的时候，我一直在思考里面定义的那两个高斯分布的内积是什么意思。
后来才知道，原来就是单纯的积分，只是正好凑成了这样的形式，开文记录一下

问题描述

首先是高斯分布的形式：
高斯分布的内积推导 - 图1%20%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft(%20x-%5Cmu%20_1%20%5Cright)%20%5E2%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%C2%A0%0A#card=math&code=%5Cmathcal%7BN%7D%5Cleft%28%20x%3B%5Cmu%201%2C%5Csigma%20%7B1%7D%5E%7B2%7D%20%5Cright%29%20%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft%28%20x-%5Cmu%20_1%20%5Cright%29%20%5E2%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%C2%A0%0A)
然后计算两个高斯分布相乘的变形：
高斯分布的内积推导 - 图2%20%5Cmathcal%7BN%7D%5Cleft(%20x%3B%5Cmu%202%2C%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%5C%5C%09%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft(%20x-%5Cmu%201%20%5Cright)%20%5E2%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%7D-%5Cfrac%7B%5Cleft(%20x-%5Cmu%202%20%5Cright)%20%5E2%7D%7B2%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Csigma%20%7B2%7D%5E%7B2%7D%5Cleft(%20x-%5Cmu%20_1%20%5Cright)%20%5E2%2B%5Csigma%20%7B1%7D%5E%7B2%7D%5Cleft(%20x-%5Cmu%202%20%5Cright)%20%5E2%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft(%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%20x%5E2-2%5Cleft(%20%5Cmu%20_1%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%20%5Cright)%20x%2B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7Bx%5E2-2%5Cfrac%7B%5Cmu%201%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7Dx%7D%7B2%5Cfrac%7B%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7D%20%5Cright%5C%7D%20%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft(%20x-%5Cmu%20%5Cright)%20%5E2%7D%7B2%5Csigma%20%5E2%7D%20%5Cright%5C%7D%20%5Cexp%20%5Cleft%5C%7B%20%5Cleft(%20%5Cfrac%7B%5Cmu%201%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright)%20%5E2-%5Cfrac%7B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft(%20%5Cmu%201-%5Cmu%20_2%20%5Cright)%20%5E2%7D%7B2%5Cleft(%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%7D%20%5Cright%5C%7D%7D%7B%5Csqrt%7B2%5Cpi%20%5Cleft(%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%7D%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Cfrac%7B%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft(%20x-%5Cmu%20%5Cright)%20%5E2%7D%7B2%5Csigma%20%5E2%7D%20%5Cright%5C%7D%5C%5C%09%3DA%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%5E2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft(%20x-%5Cmu%20%5Cright)%20%5E2%7D%7B2%5Csigma%20%5E2%7D%20%5Cright%5C%7D%5C%5C%5Cend%7Barray%7D%0A#card=math&code=%5Cbegin%7Barray%7D%7Bl%7D%09%5Cmathcal%7BN%7D%5Cleft%28%20x%3B%5Cmu%20_1%2C%5Csigma%20%7B1%7D%5E%7B2%7D%20%5Cright%29%20%5Cmathcal%7BN%7D%5Cleft%28%20x%3B%5Cmu%202%2C%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright%29%5C%5C%09%3D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft%28%20x-%5Cmu%201%20%5Cright%29%20%5E2%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%7D-%5Cfrac%7B%5Cleft%28%20x-%5Cmu%202%20%5Cright%29%20%5E2%7D%7B2%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Csigma%20%7B2%7D%5E%7B2%7D%5Cleft%28%20x-%5Cmu%20_1%20%5Cright%29%20%5E2%2B%5Csigma%20%7B1%7D%5E%7B2%7D%5Cleft%28%20x-%5Cmu%202%20%5Cright%29%20%5E2%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft%28%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright%29%20x%5E2-2%5Cleft%28%20%5Cmu%20_1%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%20%5Cright%29%20x%2B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7Bx%5E2-2%5Cfrac%7B%5Cmu%201%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7Dx%7D%7B2%5Cfrac%7B%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7D%20%5Cright%5C%7D%20%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft%28%20x-%5Cmu%20%5Cright%29%20%5E2%7D%7B2%5Csigma%20%5E2%7D%20%5Cright%5C%7D%20%5Cexp%20%5Cleft%5C%7B%20%5Cleft%28%20%5Cfrac%7B%5Cmu%201%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%29%20%5E2-%5Cfrac%7B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%5C%7D%5C%5C%09%3D%5Cfrac%7B%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft%28%20%5Cmu%201-%5Cmu%20_2%20%5Cright%29%20%5E2%7D%7B2%5Cleft%28%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright%29%7D%20%5Cright%5C%7D%7D%7B%5Csqrt%7B2%5Cpi%20%5Cleft%28%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright%29%7D%7D%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Cfrac%7B%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft%28%20x-%5Cmu%20%5Cright%29%20%5E2%7D%7B2%5Csigma%20%5E2%7D%20%5Cright%5C%7D%5C%5C%09%3DA%5Cfrac%7B1%7D%7B%5Csqrt%7B2%5Cpi%20%5Csigma%20%5E2%7D%7D%5Cexp%20%5Cleft%5C%7B%20-%5Cfrac%7B%5Cleft%28%20x-%5Cmu%20%5Cright%29%20%5E2%7D%7B2%5Csigma%20%5E2%7D%20%5Cright%5C%7D%5C%5C%5Cend%7Barray%7D%0A)
其中：
![](https://g.yuque.com/gr/latex?%5Cmu%20%3D%5Cfrac%7B%5Cmu%20_1%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%0A#card=math&code=%5Cmu%20%3D%5Cfrac%7B%5Cmu%201%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%0A)
高斯分布的内积推导 - 图3
高斯分布的内积推导 - 图4%20%5E2-%5Cfrac%7B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%3D-%5Cfrac%7B%5Cleft(%20%5Cmu%201-%5Cmu%20_2%20%5Cright)%20%5E2%7D%7B2%5Cleft(%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%7D%0A#card=math&code=%5Cleft%28%20%5Cfrac%7B%5Cmu%20_1%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%202%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%7D%20%5Cright%29%20%5E2-%5Cfrac%7B%5Cmu%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%2B%5Cmu%20%7B2%7D%5E%7B2%7D%5Csigma%20%7B1%7D%5E%7B2%7D%7D%7B2%5Csigma%20%7B1%7D%5E%7B2%7D%5Csigma%20%7B2%7D%5E%7B2%7D%7D%3D-%5Cfrac%7B%5Cleft%28%20%5Cmu%201-%5Cmu%20_2%20%5Cright%29%20%5E2%7D%7B2%5Cleft%28%20%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright%29%7D%0A)
![](https://g.yuque.com/gr/latex?A%3D%5Cmathcal%7BN%7D%5Cleft(%200%3B%5Cmu%20_1-%5Cmu%20_2%2C%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%20%0A#card=math&code=A%3D%5Cmathcal%7BN%7D%5Cleft%28%200%3B%5Cmu%20_1-%5Cmu%20_2%2C%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright%29%20%0A)
这时我们就能计算出其内积了：
![](https://g.yuque.com/gr/latex?%5Cint%7B%5Cmathcal%7BN%7D%5Cleft(%20x%3B%5Cmu%20_1%2C%5Csigma%20%7B1%7D%5E%7B2%7D%20%5Cright)%20%5Cmathcal%7BN%7D%5Cleft(%20x%3B%5Cmu%202%2C%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%7D%5Cmathrm%7Bd%7Dx%3DA%3D%5Cmathcal%7BN%7D%5Cleft(%200%3B%5Cmu%201-%5Cmu%20_2%2C%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright)%C2%A0%0A#card=math&code=%5Cint%7B%5Cmathcal%7BN%7D%5Cleft%28%20x%3B%5Cmu%20_1%2C%5Csigma%20%7B1%7D%5E%7B2%7D%20%5Cright%29%20%5Cmathcal%7BN%7D%5Cleft%28%20x%3B%5Cmu%202%2C%5Csigma%20%7B2%7D%5E%7B2%7D%20%5Cright%29%7D%5Cmathrm%7Bd%7Dx%3DA%3D%5Cmathcal%7BN%7D%5Cleft%28%200%3B%5Cmu%201-%5Cmu%20_2%2C%5Csigma%20%7B1%7D%5E%7B2%7D%2B%5Csigma%20_%7B2%7D%5E%7B2%7D%20%5Cright%29%C2%A0%0A)

图片版

Untitled.png
Untitled 1.png

参考链接