Likelihood_Function-似然函数

Last updated Feb 11, 2022 Edit Source

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Tags: #Math/Statistics #MachineLearning

对于某个(某组)随机变量 $X$, 我们通过采样获得了数据集 $x$ :

• 似然函数$\mathcal{L}(\theta \mid x)$就是在某个参数(parameter) $\theta$ 下, 现有数据 $x$ 出现的概率大小, 也就是说: $$\mathcal{L}(\theta \mid x) = P(X=x\mid\theta)$$ $P(X=x\mid\theta)$ 也常常写作 $p_{\theta}(x)=P_{\theta}(X=x)=P(X=x\space ;\theta)$

• 因为数据集有许多样本点, 所以似然函数是一个联合概率分布(Joint Probability)

• The likelihood function (often simply called the likelihood) describes the joint probability of the observed data as a function of the parameters of the chosen statistical model.

• For each specific parameter value $θ$ in the parameter space, the likelihood function $p ( X | θ )$ therefore assigns a probabilistic prediction to the observed data $X$.

• It is essentially the product of sampling densities.¹

• Maximum_Likelihood_Estimation-极大似然估计