## The confusing gamma parameter

**B**oris from Ottawa sent me this email about *Introducing Monte Carlo Methods with R:*

As I went through the exercises and examples, I believe I found a typo in exercise 6.4 on page 176 that is not in the list of typos posted on your website. For simulation of Gamma(a,1) random variables with candidate distribution Gamma([a],b), the optimal choice of b seems to be a/[a] rather than [a]/a as suggested in the book. Since the ratio dgamma(x,a,1)/dgamma(x,a,[a]/a) is unbounded, simulations with candidate distribution Gamma([a],[a]/a) yields poor approximation to the target distribution.

**T**he problem with this exercise and the gamma distribution

in general is that it can be parameterised in terms of the scale or in terms of the rate, as recognised by the R [d/p/q/r]gamma functions:

GammaDist package:stats R Documentation The Gamma Distribution Description: Density, distribution function, quantile function and random generation for the Gamma distribution with parameters ‘shape’ and ‘scale’. Usage: dgamma(x, shape, rate = 1, scale = 1/rate, log = FALSE) pgamma(q, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) qgamma(p, shape, rate = 1, scale = 1/rate, lower.tail = TRUE, log.p = FALSE) rgamma(n, shape, rate = 1, scale = 1/rate) Arguments: x, q: vector of quantiles. p: vector of probabilities. n: number of observations. If ‘length(n) > 1’, the length is taken to be the number required. rate: an alternative way to specify the scale. shape, scale: shape and scale parameters. Must be positive, ‘scale’ strictly.

**T**hus, Boris understood *b* to be the scale parameter, while we meant *b* to be the rate parameter, meaning we are *in fine* in agreement about the solution! The deeper question is, why use a duplicated and hence confusing parameterisation?! The reason for doing so is that, while the scale is the natural parameter, the rate has the nicer (Bayesian) property of enjoying a gamma conjugate prior (rather than an inverse gamma conjugate prior). This is why the gamma distribution is implicitly calibrated by the rate, instead of the scale, in most of the Bayesian literature.

May 16, 2011 at 8:54 pm

But why didn’t we [statisticians] come up with a better convention, e.g. using $\gamma$ for the scale so that, for the uninitiated, the notation suggests something is different here. Or, better yet, utilize the notation we use for the normal distribution, i.e. $N(0,\tau^{-1})$?

May 16, 2011 at 9:34 pm

There is actually the same difficulty with the normal, could mean the standard deviation, the variance or even the older “probable error” or “modulus” … Because standard distributions have been introduced and used in many fields, it is somehow hopeless to try to impose a unique notation ex-post…