Probability of heads in a biased coin Given $N$ flips of the same coin resulting in $k$ occurrences of 'heads', what is the probability density function of heads-probability of the coin?
 A: 
Reading about priors, the article on wikipedia (en.wikipedia.org/wiki/Prior_probability) seems to recommend Jeffreys' prior (en.wikipedia.org/wiki/Jeffreys_prior#Bernoulli_trial) which is 1/sqrt[p(1-p)], although I didnt understand the explanation of why.

You're not clear as to whether you're confused with how they arrived at that particular prior, or the purpose of the Jeffreys prior. 
The Wikipedia article has a pretty good summary of some of the advantages and disadvantages of Jeffreys priors. You can google around if you're still confused or just say so :) . 
The way you find the Jeffreys prior is you need to first find the Fisher information of the parameter. Here is a paper that derives the binomial Fisher information. After we do that, we take the square root of this, and then use this as the prior. The reason why '$ \propto $' is used is because when you're finding the posterior distribution, it's easier to find with up to proportion to the parameter and then solve for the normalizing constant for the posterior. 
I have to go to class now, but I'll try to elaborate more later if no one else does. 
A: Because the $N$ (independent) coin flips occur with probability proportional to $p^k(1-p)^{N-k}$, the likelihood induced on the coin's bias is $\textrm{Beta}(k + 1, N-k + 1)$.
You could have picked any parametrization of the bias.  You chose to represent it as a probability $0 \le p \le 1$, but it could have been an "odds" $0\le o$, or a log-odds $\ell$.  Since this choice is arbitrary, your prior should be independent of this choice. Jeffreys found the only prior that satisfies this "indifference" to the choice of parametrization: the Jeffreys prior, $\textrm{Beta}(\frac12, \frac12)$. 
Pointwise product of densities gives the posterior $\textrm{Beta}(k+\frac12, N-k+\frac12)$.
