# Exponential Distribution - Rate - Bayesian Prior?

I have gone through WinBugs documentation (for example, http://www.mrc-bsu.cam.ac.uk/bugs/thebugsbook/examples/html/Chapter-11-Specialised/Example-11_7_2-leukaemia.html). And also through this book (http://www.amazon.ca/Bayesian-Survival-Analysis-Joseph-Ibrahim/dp/0387952772).

Both use a gamma distribution prior for the rate (lambda) for the exponential distribution. They alternate between Gamma(0.01,0.01) and Gamma(0.001,0.001).

I would like to use a non-informative prior.... but I don't think this is it?

Could someone explain whether this is a non-informative prior? If not, can anyone suggest one?

Edit: This answer seems to entail some confusion about different ways to parameterize the Gamma distribution. It's probably best to ignore it.

I think I know what's going on. It has to do with a decision about what you want your prior to be uninformative about: the rate parameter or the distribution of survival times.

Gamma(0.001, 0.001) has a lot of very small values (close to 0).

When the Exponential distribution's rate parameter is close to zero ($\epsilon$), then it has a very high expected value (1/$\epsilon$) and is very flat over a wide range of values.

In R, you can see this by plotting an exponential distribution with mean .0001 from 0 to 100:

curve(dexp(x, .0001), ylim = c(0, 1E-4), to = 100)

It's essentially uniform (i.e. uninformative) over this range. It's much less flat if you look all the way out to 10000, though, which is why you might prefer an even smaller rate parameter.

Hope this makes sense.

• Are you sure that you're interpreting the Gamma distribution to be parametrized by shape and rate rather than shape and scale? That is, having density $f(x) \propto 1/x$ – Neil G Feb 17 '13 at 21:32
• Also, I don't understand why being certain that the exponential is uniform should be uninformative. If we wanted that, we could choose a Gamma prior with shape 0 and rate 10000, which is much more concentrated on small values. This is more informative. – Neil G Feb 17 '13 at 21:38
• @NeilG I'm not sure. user13&c accepted my answer, which may indicate that my description matches their parameterization. Setting the shape to 0 results in an improper distribution, I think, which may be why BUGS doesn't like it. – David J. Harris Feb 17 '13 at 23:29
• @NeilG also, I'm not the first one to suggest that a prior with lots of values close to 0 is uninformative: the Jeffreys prior for lambda is also very concentrated: see Eqn 2.5 here: interstat.statjournals.net/YEAR/2008/articles/0804002.pdf – David J. Harris Feb 17 '13 at 23:31
• Oops, I think I linked to the wrong Jeffreys prior. Try page 64 here. The general point still stands. seor.gmu.edu/~klaskey/SYST664/Bayes_Unit3.pdf – David J. Harris Feb 17 '13 at 23:35

What does uninformative prior mean to you?

If you mean the Jeffreys prior, then it is $\beta \sim \textrm{Gamma}(0,0)$ as @Daniel points out.

If you mean a flat prior (which isn't uninformative, although it gives the illusion of being uninformative), then it is simply $\beta \sim \textrm{Gamma}(1,0)$, which you can verify by looking at the pdf to be an improper flat prior.

However, if there is a big difference between the priors, then you probably don't have enough data.

Using $\text{Gamma}(a,b), \; a\approx b \approx 0$ is uniform logarithmic-ally. This gives the non-informative prior, in terms of what your model want to do with the input data. As an example of this, you can see this: http://jmlr.csail.mit.edu/papers/volume1/tipping01a/tipping01a.pdf search for "gamma". Also check this out: http://www.stats.org.uk/priors/noninformative/YangBerger1998.pdf

• From my own experience, gamma distribution with parameters close to zero might be quite informative, i.e. even with sample of moderate size. In addition, if you are using WinBUGs, gamma distribution migh give a "headache" for the software. My suggestion would be to use completely flat prior distribution. – Tomas Feb 15 '13 at 10:20
• @Tomas: No, he's right. Gamma(0,0) is the Jeffreys prior, and thus truly uninformative. – Neil G Feb 17 '13 at 6:38