参数估计

R 中的混合模型

Victor Medina

Researcher at The University of Edinburgh

head(data)

假设

哪个分布？→ 高斯分布 ✓
簇数？→ 2 个簇 ✓
需要哪些参数？
- 2 个均值
- 2 个比例
- 2 个标准差 → 均为 1 ✓

⇒ 需估计 4 个参数！（2 个均值和 2 个比例）

1 已知概率 →

估计均值与比例

2 已知均值与比例 →

估计概率

步骤 1：已知概率

head(data_with_probs)

         x prob_red prob_blue
1 3.294453     0.64      0.36
2 5.818586     0.01      0.99
3 2.380493     0.92      0.08
4 4.415913     0.16      0.84
5 5.048659     0.05      0.95
6 4.750195     0.09      0.91

用于均值

means_estimates <- data_with_probs %>%
   summarise(mean_red = sum(x * prob_red) / sum(prob_red),
             mean_blue = sum(x * prob_blue) / sum(prob_blue))
means_estimates

  mean_red mean_blue
1  2.86925  5.062976

用于比例

proportions_estimates <- data_with_probs %>% 
   summarise(proportion_red = mean(prob_red),
             proportion_blue = 1 - proportion_red)
proportions_estimates

  proportion_red proportion_blue
1          0.305           0.695

步骤 2：已知均值与比例

步骤 2：缩放后的概率

$\text{Probability}_{\text{ blue}}=\frac{0.065}{0.115 + 0.065}=0.36$

data %>% 
   mutate(prob_from_red = 0.3 * dnorm(x, mean = 3),
          prob_from_blue = 0.7 * dnorm(x,mean = 5),
          prob_red = prob_from_red/(prob_from_red + prob_from_blue),
          prob_blue = prob_from_blue/(prob_from_red + prob_from_blue)) %>% 
   select(x, prob_red, prob_blue) %>% head(4)

         x   prob_red  prob_blue
1 3.294453 0.63733037 0.36266963
2 5.818586 0.01115698 0.98884302
3 2.380493 0.91619343 0.08380657
4 4.415913 0.15721146 0.84278854

小结

知道概率时 → 估计均值与比例
知道均值与比例时 → 估计概率

Vamos praticar!

R 中的混合模型