For a random variable \(X\) that follows a Poisson distribution \(\mathcal{P}(\lambda)\) with rate parameter \(\lambda\) (where \(\lambda>0\)), a suitable family of conjugate priors is the Gamma distribution \(\mathcal{G}(\alpha, \beta)\) with shape parameter \(\alpha > 0\) and scale parameter \(\beta > 0\). To determine the Bayes estimator of \(\lambda\) under squared error loss, we aim to minimize the Bayes risk, which involves finding the estimator that minimizes the expected squared deviation between \(\lambda\) and the estimator, expressed as:
\[\begin{equation} \underset{\delta}{\min} \mathbb{E}[(\lambda -\delta(x))^2] = \underset{\delta}{\min} \int (\lambda -\delta(x))^2 \pi(\lambda | x)d\lambda. \end{equation}\]The posterior distribution, \(\pi(\lambda | x)\), is found using Bayes formula:
\[\begin{equation} \pi(\lambda | x) = \frac{f(x | \lambda)\pi(\lambda)}{\int f(x | \lambda)\pi(\lambda)d\lambda} \end{equation}\]where \(f(x | \lambda)\) is the sampling distribution (likelihood) and \(\pi(\lambda)\) is the prior distribution. In our example, \(x_1, \dots, x_n \overset{\text{i.i.d.}}{\sim} \mathcal{P}(\lambda)\) where
\[\begin{equation} \begin{split} f(x | \lambda) &= \prod_{i=1}^{n} \frac{\lambda^{x_i}e^{-\lambda}}{x_i !} \\ &= \frac{\lambda^{\sum_i x_i}e^{-n\lambda}}{\prod_{i}x_i !} \end{split} \end{equation}\]and our prior distribution is \(\pi(\lambda | \alpha, \beta)\) with density function:
\[\begin{equation} \pi(\lambda | \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma( \alpha)}\lambda^{\alpha - 1}e^{-\beta \lambda}. \end{equation}\]A convenient way of representing the posterior distribution is to examine our density up to some proportional constant, that is,
\[\begin{equation} \pi(\lambda | x) \propto f(x | \lambda)\pi(\lambda) \end{equation}\]since \(\int f(x | \lambda)\pi(\lambda)d\lambda\) is the normalizing constant that guarantees us a valid probability density function. Therefore, our posterior distribution is
\[\begin{equation} \begin{split} \pi(\lambda | x) &\propto f(x | \lambda)\pi(\lambda) \\ &= \frac{\lambda^{\sum_i x_i}e^{-n\lambda}}{\prod_{i}x_i !}\frac{\beta^{\alpha}}{\Gamma( \alpha)}\lambda^{\alpha - 1}e^{-\beta \lambda} \\ &= c \lambda^{\sum_i x_i + \alpha - 1}e^{-(\beta + n)\lambda} \end{split} \end{equation}\]where \(c\) is a constant not interacting with our unknown parameter \(\lambda\). As such, we recognize \(\pi(\lambda | x)\) follows a Gamma distribution with \(\alpha' = \sum_i x_i + \alpha\) and \(\beta'= \beta + n\). Since our prior distribution as Gamma, and the posterior also follows a Gamma distribution, we call this a conjugate prior.
Now, we can find the Bayes estimator of \(\lambda\) under squared error loss, which in this case is the posterior mean.
\[\begin{equation} \begin{split} \delta(x) = \mathbb{E}[\lambda | x] &= \frac{\alpha'}{\beta'} \\ &= \frac{\sum_i x_i + \alpha}{\beta + n} \end{split} \end{equation}\]A safety study aims to estimate the average number of traffic accidents per year in a city based on recent data collected over a 10-year period. The number of accidents \(X\) in a fixed period (e.g., a year) is modeled as a Poisson random variable with parameter \(\lambda\) which represents the average rate of accidents per year.
Based on historical data, the prior mean number of traffic accidents per year is set to \(10\) using a Gamma distribution with shape \(\alpha = 5\) and \(\beta = 0.5\)
After counting the number of accidents over \(10\) years, the total observed accidents are \(\sum_i x_i = 50\).
Using Bayesian inference, the updated shape parameter is \(\alpha' = \sum_i x_i + \alpha = 50 + 5 = 55\) and rate parameter is \(\beta' = \beta + n = 0.5 + 10 = 10.5\). Thus the posterior distribution is \(\mathcal{G}(55, 10.5)\)
The posterior mean is:
\[\begin{equation} \mathbb{E}[\lambda | x] = \frac{55}{10.5} \approx 5.24 \text{ accidents per year} \end{equation}\]The posterior mean \((5.24)\) is lower than the prior mean \((10)\), reflecting the updated information from the new data.
This updated estimate provides a refined prediction of the average number of traffic accidents per year, aiding in the development of safety protocols.