Poisson

definition

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• 포아송 분포(Poisson Distribution): 단위 시간 혹은 영역 내에서 발생하는 사건 수를 나타내는 분포

  \[X\sim\mathrm{Poisson}(\lambda),\quad X\in\mathbb{Z}\setminus\mathbb{Z}^{-}\]
  • $\lambda>0$: 단위 시간 당 평균 발생 횟수로서 스케일 파라미터의 역수

• probability mass function:

  \[\begin{aligned} p(x\mid\lambda) &=\frac{\lambda^{x}\exp{-\lambda}}{x!} \end{aligned}\]

• moment generating function:

  \[\begin{aligned} M_{X}(t) &=\mathbb{E}_{p(x)}\left[\exp{tX}\right]\\ &=\sum_{x=0}^{\infty}{\exp{tx}\cdot\frac{\lambda^{x}\exp{-\lambda}}{x!}}\\ &=\exp{-\lambda}\sum_{x=0}^{\infty}{\frac{\left(\lambda\exp{t}\right)^{x}}{x!}}\\ \\ \because\exp{x} &=\sum_{k=0}^{\infty}{\frac{x^{k}}{k!}}\\ \sum_{x=0}^{\infty}{\frac{\left(\lambda\exp{t}\right)^{x}}{x!}} &=\exp{\left[\lambda\cdot\exp{t}\right]}\\ \\ \therefore M_{X}(t) &=\exp{-\lambda}\cdot\exp{\left[\lambda\cdot\exp{t}\right]}\\ &=\exp{\left[\lambda\left(\exp{t}-1\right)\right]} \end{aligned}\]
  • $\mathbb{E}\left[X\right]=(\mathrm{d}/\mathrm{d}t)M_{X}(t)\vert_{t=0}=\lambda$
  • $\mathbb{E}\left[X^{2}\right]=(\mathrm{d}^{2}/\mathrm{d}t^{2})M_{X}(t)\vert_{t=0}=\lambda+\lambda^{2}$
  • $\mathrm{Var}\left[X\right]=\mathbb{E}\left[X^{2}\right]-\mathbb{E}\left[X\right]^{2}=\lambda$

• canonical form:

  \[\begin{aligned} p(x) &=\frac{\lambda^{x}\exp{-\lambda}}{x!}\\ &=\frac{1}{x!}\exp{\left[x\log{\lambda}-\lambda\right]} \end{aligned}\]
  • $T(x)=x$
  • $\eta(\theta)=\log{\lambda}$
  • $A(\eta)=\lambda$
  • $h(x)=1/x!$

conjugate prior

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• poisson model:

  \[X\mid\lambda\sim\mathrm{Poisson}(\lambda)\]

• canonical form:

  \[\begin{aligned} p(x\mid\lambda) &=\frac{\lambda^{x}\exp{-\lambda}}{x!}\\ &=\frac{1}{x!}\exp{\left[x\log{\lambda}-\lambda\right]} \end{aligned}\]
  • $T(x)=x$
  • $\eta(\theta)=\log{\lambda}$
  • $A(\eta)=\lambda$
  • $h(x)=1/x!$

• prior of $\eta$:

  \[\begin{aligned} p(\eta) &\propto\exp{\left[\alpha\cdot\eta(\theta)-\beta\cdot A(\eta)\right]}\\ &=\exp{\left[\alpha\cdot\log{\lambda}-\beta\cdot \lambda\right]}\\ &=\exp{\left[\alpha\cdot\eta-\beta\cdot\exp{\eta}\right]} \end{aligned}\]

• change of variables $\eta\to\log{\lambda}$:

  \[\begin{aligned} p_{\lambda}(\lambda)\mathrm{d}\lambda &=p_{\eta}(\eta)\mathrm{d}\eta\\ \therefore p_{\lambda}(\lambda) &=p_{\eta}(\log{\lambda})\left\vert\frac{\mathrm{d}\eta}{\mathrm{d}\lambda}\right\vert\\ &=\exp{\left[\alpha\cdot\log{\lambda}-\beta\cdot\lambda\right]}\cdot\frac{1}{\lambda}\\ &=\lambda^{\alpha-1}\exp{-\beta\lambda}\\ &\approx\mathrm{Gamma}(\alpha,\beta) \end{aligned}\]

• Therefore, rate parameter of poisson $\lambda$, average occurrence rate per unit time, follow gamma distribution. accordingly, $\alpha$ represents the number of events occurring, and $\beta$ represents the waiting time per event.

  \[\lambda\sim\mathrm{Gamma}(\alpha,\beta)\]