Improper Prior Distribution

Question

What is the clear mathematics definition about improper prior distributions? Can you give me some book or article links about it?

Answer 1

Degroot & Schervish Probability and Statistics (4th ed.) Def. 7.3.2

Improper Prior. Let ξ be a non-negative function whose domain includes the parameter space of a statistical model. Suppose that $\intξ(θ)dθ = ∞$. If we pretend as if ξ(θ) is the prior p.d.f. of θ, then we are using an improper prior for θ.

Other:

• The entire section of 7.3 on improper priors in Degroot & Schervish (see above)
• Jeffreys Prior: Section 5 here & "The Jeffreys Prior" mid-section here (contain good information about improper priors in general as well)
• Hoff's A First Course in Bayesian Statistical Methods on pp. 78-79 (1st ed.) chapter 5

There are other books & articles out there but I'm not familiar enough with them to hand out a recommendation