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Alright, I am reading this book called "Bayesian Data Analysis" and it states this idea of being uniform in log of something which I don't have any clue what that might mean??

Example 1: Introduction of Dirichlet distribution as a conjugate for Multinomial distribution:

$$p(\theta|\alpha) \propto \prod\limits_{j=1}^{k}{\theta_{j}^{\alpha_{j}-1}}$$ where the distribution is restricted to nonnegative $\theta_{j}$'s with $\sum\limits_{i=1}^{k}{\theta_j}=1$. ... Setting $\alpha_j=0$ for all $j$ results in an improper prior distribution that is uniform in the $log(\theta_j)$.

Example 2: Or somewhere else in the book about normal distributions it mentions that considering $p(\sigma^2)\propto \frac{1}{\sigma^2}$ is makes a uniform distribution with regard to $\sigma$ on log scale.

Could somebody please explain the situation?

Cupitor
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  • Re "don't have a clue": are you asking for an explanation of what a uniform distribution is or what a logarithm is (or both)? – whuber Oct 05 '13 at 15:11
  • @whuber I am asking how can this distribution be uniform regarding the log of a parameter. The definition that I know of being uniform is $p(\theta)=c$. – Cupitor Oct 05 '13 at 15:25
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    It sounds like you are wondering how to find the distribution of $\log(\theta)$ given the distribution of $\theta$. This is a special case of the situation where $\log$ is replaced by any measurable transformation $f$, which is discussed in general at http://stats.stackexchange.com/q/14483 using--for illustration--the specific example of $f(\theta)=\theta^2$, but everything there applies to the logarithm and an easy calculation shows that $p(\log(\theta)|\alpha=-1) = 1$ for $0 \lt \theta \lt 1$. – whuber Oct 05 '13 at 16:03

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