Here’s a simple bit of statistics for a Friday lunchtime. You count the number of galaxies in a certain area on the sky (with the galaxies satisfying some specific properties, if you like). What is the true number density? Let the expected number be \(\lambda\) (the true number density multiplied by the area on the sky) and the measured number be \(k\). Then, in true Bayesian fashion, what we want is
\[P(\lambda|k) = \frac{P(k|\lambda)P(\lambda)}{P(k)}\]Now, for the prior, \(P(\lambda)\), we assume a prior which is flat on a logarithmic scale. That is, we guess (before making the observation) that the expected number is as likely to lie between 1 and 10 as it is to lie between 1000 and 10,000. (The alternative, a flat prior on a linear scale, would mean that we guess the true density is just as likely to lie between 10,001 and 10,010 as it is to lie between 1 and 10, which is ridiculous.) So \(P(\lambda) \propto 1/\lambda\). The likelihood, \(P(k\vert \lambda)\) is given by the Poisson distribution. So, ignoring the normalizing factor of \(P(k)\),
\[P(\lambda|k) \propto \frac{\lambda^k}{k!} e^{-\lambda} \frac{1}{\lambda}\] \[\propto \lambda^{k-1}e^{-\lambda}\]And this is the Gamma distribution. Easy peasy.