统计学中,似然函数是给定数据的统计模型的参数的函数。
变量值集合:θ,已知结果x的似然函数和这些观察已知变量值的观察结果的概率相等:
![\mathcal{L}(\theta |x) = P(x | \theta)](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d3451302e5f0de41793e3863bfa260733e46879)
似然函数在离散概率分布和连续概率分布中不同:
离散概率分布:
假设X为一个随机变量,符合离散概率分布p,基于参数θ。则函数为:
![{\displaystyle {\mathcal {L}}(\theta |x)=p_{\theta }(x)=P_{\theta }(X=x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d040f393442cbda850d14d79af09f1206c8388c2)
被认为是θ的函数,称之为似然函数。
连续概率分布:
设X为符合基于变量θ,密度函数为f的绝对连续分布。则函数为:
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
Log-likelihood
对于很多应用,似然函数的自然对数,称为对数-似然(log-likelihood),很适合计算。因为对数是单调函数。
找到一个函数的最大值,通常涉及对函数的求导,并求解参数使函数最大化。并且通常对log-likelihood最大化简单于直接对原始函数求解。
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)
- L ( θ | x ) = f θ ( x ) , {\displaystyle {\mathcal {L}}(\theta |x)=f_{\theta }(x),\,}
![\mathcal{L}(\theta |x) = f_{\theta} (x), \,](https://wikimedia.org/api/rest_v1/media/math/render/svg/a164b9795cd3322bed7386369d8f8cc0a79a6092)