# Finite integral vs Finite probability Density

I was reading the Bayesian Data Analysis book from Gelman et al. I was going through Appendix A which describes probability distributions, and realize the book talks about something I don't really understand.

For example for the inverse wishart distribution, the book says: The Wishart ... The integral is finite if the degrees of freedom parameter,$$\nu$$, is greater than or equal to the dimension, $$k$$. The density is finite if $$\nu \geq k +1$$. A noninformative distribution is obtained as $$\nu \rightarrow 0$$

I dont really get what is the difference between the integral being finite and the density being finite. From my understanding the density is finite (hence proper prior) if the integral is 1. So being the density finite implies that the integral is also finite. Hence I dont get what they are really referring to, i.e. what is the difference between the density being finite and the integral being finite; or what do they refer by the integral in this context. Any thoughts?. This appears in page 582 from the book. A similar discussion about the integral and the density is done for other distributions in the same appendix.

Thank you

• The density is the integrand, not the integral. Do not confuse the two. For instance, the integrand $$f(x) = \frac{\exp(-|x|)}{2\sqrt{\pi |x|}}$$ diverges at $0$ but its integral is $1.$ On the other hand, the integrand $f(x)=1$ is finite but its integral diverges.
– whuber
Commented Nov 19, 2020 at 15:37
• aaaa yes I see. Now I understand. I was understanding the density as its integral. Thanks!!. An analogy to your's would be the Dirac measure right? Commented Nov 19, 2020 at 15:41
• The Dirac measure is a different animal altogether. As a measure, it is finite: its value on any measurable set is either $0$ or $1.$ It is true that it is a weak limit of a sequence of absolutely continuous measures whose densities, although they may all be bounded, have upper bounds that grow arbitrarily large. But one should not impute any properties of elements of a sequence to its limit!
– whuber
Commented Nov 19, 2020 at 15:54