Fitting a Gamma model to data using Metropolis Hastings

We will use a gamma likelihood for this dataset with both gamma parameters unknown. We’ll use exponential priors for both parameters and since we don’t have much intuition about these parameters, we’ll make them a bit vague, picking both to be exponential with mean 10.

Now, we want to fit this model by drawing a posterior sample using the Metropolis-Hastings algorithm. First, we will need to compute the acceptance probabilities. So, in our gamma model the download times $$y_i$$ are distributed like

$y_i \sim Gamma(\alpha,\beta)$

So, the probability of a download time, conditional on the gamma parameters, is given by

$P(y_i | \alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}y_i^{\alpha -1}e^{-\beta y_i}$

The joint likelihood of $$\alpha$$ and $$\beta$$ given the sample of $$n$$ download times is thus

$L(\alpha,\beta | \underset{\sim}y) = f(\underset{\sim}y | \alpha,\beta) = \left(\frac{\beta^\alpha}{\Gamma(\alpha)}\right)^n\prod_{i=1}^ny_i^{\alpha-1}\left(e^{-\beta\sum\limits_{i=1}^n y_i}\right)$

With priors for $$\alpha$$ and $$\beta$$: $$P(\alpha) = \frac{1}{10}e^{\frac{\alpha}{10}}$$ and $$P(\beta) = \frac{1}{10}e^{\frac{\beta}{10}}$$ we have a posterior probability:

$\pi(\alpha,\beta | \underset{\sim}y) = \left(\frac{\beta^\alpha}{\Gamma(\alpha)}\right)^n\prod_{i=1}^ny_i^{\alpha-1}\left(e^{-\beta\sum\limits_{i=1}^n y_i}\right) \frac{1}{10}e^{\frac{\alpha}{10}} \frac{1}{10}e^{\frac{\beta}{10}}$

Since $$P(\beta) = \frac{1}{10}e^{\frac{\beta}{10}}$$ is a constant with respect to $$\alpha$$, the acceptance probability for $$\alpha$$ is… (TO BE CONTINUED…)

$a_{\alpha} = \frac{\pi(\alpha^*,\beta|\underset{\sim}y)}{\pi(\alpha_i,\beta|\underset{\sim}y)}=$