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:[1]

• $\alpha = \beta$
  • $\alpha = \beta < 1$
    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

upper document

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