Fitting a Gamma model to data using Metropolis Hastings

James Durbin bio photo By James Durbin

Keynote Systems measures download times for selected web sites. For example, consider this data: Keynote Systems download times

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…)

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