Proof: Conjugate prior distribution for Poisson-distributed data
Theorem: Let there be a Poisson-distributed data set $y = \left\lbrace y_1, \ldots, y_n \right\rbrace$:
\[\label{eq:Poiss} y_i \sim \mathrm{Poiss}(\lambda), \quad i = 1, \ldots, n \; .\]Then, the conjugate prior for the model parameter $\lambda$ is a gamma distribution:
\[\label{eq:Poiss-prior} p(\lambda) = \mathrm{Gam}(\lambda; a_0, b_0) \; .\]Proof: With the probability mass function of the Poisson distribution, the likelihood function for each observation implied by \eqref{eq:Poiss} is given by
\[\label{eq:Poiss-LF-s1} p(y_i|\lambda) = \mathrm{Poiss}(y_i; \lambda) = \frac{\lambda^{y_i} \cdot \exp\left[-\lambda\right]}{y_i !}\]and because observations are independent, the likelihood function for all observations is the product of the individual ones:
\[\label{eq:Poiss-LF-s2} p(y|\lambda) = \prod_{i=1}^n p(y_i|\lambda) = \prod_{i=1}^n \frac{\lambda^{y_i} \cdot \exp\left[-\lambda\right]}{y_i !} \; .\]Resolving the product in the likelihood function, we have
\[\label{eq:Poiss-LF-s3} \begin{split} p(y|\lambda) &= \prod_{i=1}^n \frac{1}{y_i !} \cdot \prod_{i=1}^n \lambda^{y_i} \cdot \prod_{i=1}^n \exp\left[-\lambda\right] \\ &= \prod_{i=1}^n \left(\frac{1}{y_i !}\right) \cdot \lambda^{n \bar{y}} \cdot \exp\left[-n \lambda\right] \end{split}\]where $\bar{y}$ is the mean of $y$:
\[\label{eq:y-mean} \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i \; .\]In other words, the likelihood function is proportional to a power of $\lambda$ times an exponential of $\lambda$:
\[\label{eq:Poiss-LF-prop} p(y|\lambda) \propto \lambda^{n \bar{y}} \cdot \exp\left[-n \lambda\right] \; .\]The same is true for a gamma distribution over $\lambda$
\[\label{eq:Poiss-prior-s1} p(\lambda) = \mathrm{Gam}(\lambda; a_0, b_0)\]the probability density function of which
\[\label{eq:Poiss-prior-s2} p(\lambda) = \frac{ {b_0}^{a_0}}{\Gamma(a_0)} \lambda^{a_0-1} \exp[-b_0 \lambda]\]exhibits the same proportionality
\[\label{eq:Poiss-prior-s3} p(\lambda) \propto \lambda^{a_0-1} \cdot \exp[-b_0 \lambda]\]and is therefore conjugate relative to the likelihood.
- Gelman A, Carlin JB, Stern HS, Dunson DB, Vehtari A, Rubin DB (2014): "Other standard single-parameter models"; in: Bayesian Data Analysis, 3rd edition, ch. 2.6, p. 45, eq. 2.14ff.; URL: http://www.stat.columbia.edu/~gelman/book/.
Metadata: ID: P225 | shortcut: poiss-prior | author: JoramSoch | date: 2020-04-21, 08:31.