# Survival time problem exponential with gamma prior

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.

• The source of your confusion might be that for if $Y\sim Exp(\theta)$ then $E[Y]=\frac1\theta$. So you want the prior for $\theta$ to be concentrated around $\frac{1}{80}$, not around 80. – Robin Ryder Oct 13 '18 at 16:22
• Thanks @RobinRyder I think you're right. However I have tried many things and maybe I'm thinking I have got it wrong when it's not. It was for an assignment that I have just handed in. I really couldn't get the question out because of this sticking point. Thank for your help. – Bucephalus Oct 14 '18 at 7:43

Based on your comment, it sounds like you are interested in $$\lambda=\frac1\theta$$, ie $$X\sim Exp(\frac1\lambda)$$. For $$\theta$$, the conjugate prior is Gamma, so you are right that for $$\lambda$$, the conjugate prior is Inverse-Gamma.