75 Exponential ISIs and Gamma priors
If I have ISIs that are Exponentially distributed, I need to infer the spiking rate \(\beta\). Prove that if I have a Gamma prior for \(\beta\) that that posterior is also Gamma. That is, prove that the Gamma distribution if conjugate to the Exponential. If the set of ISIs is \(\mathbf{y} = (y_1, y_2, \ldots, y_3)\), how are the prior parameters updated in the posterior?