# Why is Inverted Gamma (3,0.0005) a diffuse prior when variance of this random variable is small?

A few times I came across the statement that an Inverted Gamma (3, 0.0005) prior for the variance is quite diffuse but proper. Hence, my understanding is that the prior variance of this random variable is large. However, Wikipedia says that the variance with these parameters equals 0.2500e-04, which is tiny. Indeed, when I sample 1000 draws from this distribution, the standard deviation across draws equals 0.00028099. Similarly, evaluating the cdf at, say, 0.5 yields 1, which means that values larger than 0.5 are very unlikely.

Why is this prior considered diffuse nevertheless?

Many thanks

• Are you sure about the parameterization? Please take a look at one of your sources and double-check that its notation agrees with yours. – whuber Nov 4 '16 at 15:40
• The prior comes from this paper: jstor.org/stable/2527349?seq=1#page_scan_tab_contents They write that the prior density for the variance is given by IG(nu/2,delta/2) (p. 1001) and further below that "the parameters for the inverted gamma distribution were nu= 6, delta = 0.001. This (proper) prior is quite diffuse; for example, third and higher moments do not exist." (p. 1006). So I would say that my notation agrees with theirs. – LuckyLuke Nov 4 '16 at 15:58
• Because they do not define their notation, you have to assume whatever makes sense. In particular, if assuming the second parameter is a rate gives obviously bad results, then they must intend it to be a scale. – whuber Nov 4 '16 at 16:02
• Flipping the parameters I get infinitely large values, which doesn't make any sense either. – LuckyLuke Nov 4 '16 at 16:46
• That makes loads of sense for a diffuse prior! You likely have overflowed your floating point arithmetic, that's all. – whuber Nov 4 '16 at 16:49

## 1 Answer

At the beginning, I misunderstood the concept "diffuse prior", too. Actually, "diffuse prior" not necessarily deduces the variance of parameters is high. The only way to determine whether the prior is diffuse is to check the extent to which the prior imposes impacts on posterior.

For example, if $x_i|\mu, \sigma^2 \sim N(\mu, \sigma^2) i.i.d.$ and $\sigma^2 \sim IG(\alpha, \beta).$ Then: $\sigma^2|x_1,...,x_n \sim IG(\alpha + n/2, \beta + 1/2\sum(x_i-\mu)^2)$. We can find the smaller $\alpha, \beta$, the smaller the impacts prior exerts on posterior.