Beta distribution

\(\displaystyle x\in [0,1] \)일 때, (support가 이렇게 주어지기 때문에, 확률분포로 쓸 수 있다.)

\(\Large \text{Beta}(\alpha, \beta) = f(x; \alpha, \beta) = \text{contant}\cdot x^{\alpha -1 }(1-x)^{\beta -1} \)

\(\Large = \frac{1}{\text{B}(\alpha, \beta)} x^{\alpha -1 }(1-x)^{\beta -1} \)

B is beta function. (여기서는 normalizer역할)

hyperparameter \(\alpha, \beta\)를 주었을 때 위의 pdf를 가지는 분포가 beta분포.

conjugate prior probability distribution for the

• Bernoulli,
• binomial,
• negative binomial,
• geometric distributions.

shapes

위키[1]에 그림이 많은데 정말 다 주옥같다. 구경하기도 좋고 ㅎㅎ

• \(\alpha = \beta\)
  • \(\alpha = \beta < 1 \)
    U-shaped. (\(\alpha \neq \beta\)일 때도 \(\alpha < 1, \beta < 1\)이면 U-shaped)
  • \(\alpha = \beta = 1 \)
    uniform [0,1] distribution
  • \(\alpha = \beta > 1 \)
    unimodal[2]
    • \(\alpha = \beta = 2 \)
      parabolic
    • \(\alpha = \beta > 2 \)
      bell shaped
• \(\alpha \neq \beta\)
  • \(\alpha = 1, \beta > 1\)
    positively skewed.(right tail is long), strictly decreasing.
    • \(1 < \beta < 2\)
      concave
    • \( \beta = 2 \)
      straight line with slope -2
    • \(2 < \beta \)
      convex

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