The survival times, in days, of patients diagnosed with a severe form of a terminal illness are thought to be well modelled by an exponential($\theta$) distribution. We observe the survival times of n such patients. Our prior distribution for $\theta$ is a gamma(a, b) distribution.
$$L(x|\theta) = \theta^ne^{-\theta\sum_\limits{i=1}^{n}x_i}, \quad 0 \le x < \infty$$
Let $T = \sum_\limits{i=1}^{n}x_i$
Using a Gamma distribution as the prior for the exponential distribution likelihood.
$$p(\theta) = \frac{\beta^{\alpha}\theta^{\alpha - 1}e^{-\theta\beta}}{\Gamma(\alpha)}, \quad 0\le \theta < \infty$$
$$p(\theta|x) \propto L(T|\theta)p(\theta) =e^{-T\theta}\theta^{n} \theta^{\alpha - 1}e^{-\theta\beta}$$ $$\propto \theta^{\alpha + n - 1}e^{-\theta\big(T+\beta\big)}$$
$$p(\theta|T)\propto \text{ Gamma}\bigg(\text{ Shape }= \alpha + n, \text{ Rate } = \beta + T\bigg)$$
Where i'm getting confused is between rate and scale and the how to choose the hyperparameters.
Say if I had data for a year and the mean survival time for 100 patients was 50 days, I know that my T would be 5000. Also I know that my n would be 50.
I would have to develop the prior from other knowledge say, some database from a different state with similar study. So say my prior mean was 80 days, with standard deviation say of 15 days, I cannot seem to get my alpha and beta right such that when I put this model in R I get a reasonable looking posterior.
From the examples I see on the internet, I see that the exponential models the time between events, but I want to model the time of the event possibly? The only way I can get a reasonable looking distribution is if I use an invgamma function. Is this what I need to do? The inverse of the time between events is the time of the event?
Do I use $E[X] = \alpha \beta$ or do I use $E[X] = \frac{\alpha}{\beta}$ for example? A similar question would be life time of light bulbs I suppose. Has anybody got anything to share on this kind of problem?
Thanks.
