MILtracking目标跟踪解析
三种概率
在传统的机器学习中，样本只有一个包标签。但是在MIL中，有样本与样本集合的概念，样本集合有包标签(y_{bag})（若集合含有正样本，包标签为1，否则0），故样本除了具有样本标签(y_{sample})还有包标签(y_{bag})。通过Noisy-OR模型可以由集合各元素的样本标签计算该集合的包标签：
[pleft( y_{bag}=1left|
ight.X_{n}
ight)=1-prod_{n} left(1-pleft(y_{bag}=1left|
ight. x_{i}
ight )
ight )
]
其中(X_{n})是含有n个样本的集合(x_{i})是集合中的一个元素。
样本包标签后验概率推导
对于任意一个样本(x)的(pleft( y_{bag}=1left| ight.x ight))，我们可以通过naive bayes导出:
[egin{split}
pleft( y_{bag}=1left|
ight.x
ight) &=frac{pleft( xleft|
ight. y_{bag}=1
ight)pleft(y_{bag}=1
ight)}{pleft( xleft|
ight. y_{bag}=0
ight)pleft(y_{bag}=0
ight)+pleft( xleft|
ight. y_{bag}=1
ight)pleft(y_{bag}=1
ight)} \
&=frac{A}{B+A}=frac{1}{{frac{A}{B}}^{-1}+1}=frac{1}{e^{-lnfrac{A}{B}}+1}=sigmaleft( lnfrac{A}{B}
ight)\
&=sigma left( lnfrac{pleft( xleft|
ight. y_{bag}=1
ight)pleft(y_{bag}=1
ight)}{pleft( xleft|
ight. y_{bag}=0
ight)pleft(y_{bag}=0
ight)}
ight)\
\
&又因为pleft(y_{bag}=1
ight)=pleft(y_{bag}=0
ight)\
\
&=sigma left( lnfrac{pleft( xleft|
ight. y_{bag}=1
ight)}{pleft( xleft|
ight. y_{bag}=0
ight)}
ight)\
\
&又因为服从naive bayes假设（观测样本维度独立）
\
&=sigma left( lnfrac{prod pleft( x_{i}left|
ight. y_{bag}=1
ight)}{prod pleft( x_{i}left|
ight. y_{bag}=0
ight)}
ight)=sigma left( ln prod frac{ pleft( x_{i}left|
ight. y_{bag}=1
ight)}{ pleft( x_{i}left|
ight. y_{bag}=0
ight)}
ight)\
&=sigma left( sum lnfrac{ pleft( x_{i}left|
ight. y_{bag}=1
ight)}{ pleft( x_{i}left|
ight. y_{bag}=0
ight)}
ight)
end{split}
]
弱分类器与强分类器
设弱分类器
[h_{i}= lnfrac{ pleft( x_{i}left|
ight. y_{bag}=1
ight)}{ pleft( x_{i}left|
ight. y_{bag}=0
ight)}
]
则(h_{1},h_{2},...,h_{n})级联构成的强分类器为
[H_{n}=sum lnfrac{ pleft( x_{i}left|
ight. y_{bag}=1
ight)}{ pleft( x_{i}left|
ight. y_{bag}=0
ight)}=sum h_{i}
]
在上一次跟踪结果附近某范围采样一个集合(X_{near})，远处采样一个集合(X_{far}).假设跟踪结果有漂移，但真实位置仍然落在(X_{near}),则(X_{near})的包标签为1，(X_{far})的包标签为0，即已知了包标签。进而可求取(pleft( x_{i}left| ight. y_{bag}=1 ight))与(pleft( x_{i}left| ight. y_{bag}=0 ight))的分布
[pleft( x_{i}left|
ight. y_{bag}=1
ight) sim Nleft( mu_{i}^{near},sigma_{i}^{near}
ight)，
pleft( x_{i}left|
ight. y_{bag}=0
ight) sim Nleft( mu_{i}^{far},sigma_{i}^{far}
ight)]