What is the clear mathematics definition about improper prior distributions? Can you give me some book or article links about it?
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http://en.wikipedia.org/wiki/Prior_probability#Improper_priors – Did May 15 '14 at 18:52
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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
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An improper prior density is a nonintegrable prior density. The nonintegrable density can be an infinite measure that is countably additive (as stated). Another possibility is that it is a finitely additive probability measure assigning zero to all subsets of the parameter space. – Anne van Rossum May 08 '16 at 12:52