The background of the problem is this: 

Assume that we have a parameter vector $\Theta$ which satisfies $\Theta^\prime\Theta=1$. If we let this vector have the uniform prior, the density of the prior is
$$
\pi(\Theta)=\frac{\Gamma(q/2)}{(2\pi)^{q/2}}
$$
where $q$ is the dimension of $\Theta$. Now, we have a set of possible models in the set $\mathcal{M}=\{2, \dots, k\}$, where each (integer) $i\in\mathcal{M}$ corresponds to the dimension of $\Theta\_i$.

What I am supposed to answer is how this prior behaves when comparing dimensions $q+1$ to $q$. A hint I got was to look at what happens to the ratio of a priori probabilities when $q$ increases from for instance 2 to 10 (actually, he said when $k$ increases, but I assume he means $q$). Doing this yields:
$$
\frac{\pi(\Theta\_{10})}{\pi(\Theta\_{2})}=\frac{\frac{\Gamma(5)}{(2\pi)^5}}{\frac{1}{2\pi}}=\frac{\Gamma(5)}{(2\pi)^4}
$$

but what on earth does this tell me? 

Also, I think I found something interesting [here on page 3.](http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture6.pdf) More specifically, this:

> 
> This becomes more problematic in higher dimensions: the uniform prior in large dimension does not integrate anymore. [...] it tells that most of the probability mass lies at $+\infty$.
> 
> 
> 

This sounds very interesting, but is not expanded on as far as I can tell. How do I show this? I'm guessing that is what I need to show (even though my vector of uniform distributions is restricted to length 1). Furthermore, the next sentence is very interesting too as I in another exercise am to do the same thing but with standard normal priors;

> 
> If instead one considers a high-dimensional Gaussian distribution $X\sim N (0,1)$, most of the mass is concentrated in a (high dimensional) unit sphere centered at the origin.
> 
> 
> 

So, can anyone shed some light on this issue? I'm a bit lost, and would appreciate help!