How to use the measurement in quantum phase estimation?

Reading the quantum phase estimation algorithm on Wikipedia, I am wondering how exactly the measurements are used to obtain the phase $$\delta$$. I understand that the value of phase is encoded into the binary string that represents a computational state. My question is more about what extra information can be extracted from the probabilities. If I perform the measurement in the computational basis, i.e., $$|a\rangle$$ in the notation used on Wikipedia, what I obtain is a list of probabilities for each state. (Imagine the case $$\delta\neq 0$$). How exactly can I get this $$\delta$$ from these probabilities? Since the probability $$\text{Pr}(a)$$ is what I get and is expressed in terms of $$\delta$$, do I have to inverse the following function

$$\text{Pr}(a) = \frac{1}{2^{2^{n}}}\frac{|\sin(\pi 2^{n}\delta)|^{2}}{|\sin(\pi\delta)|^{2}}$$

in order to get $$\delta$$?

Meanwhile, how does the algorithm recognise the nearest integer to $$2^{n}\theta$$? The only way I can imagine is that the probability that corresponds to a certain state is greater than other. Imagine I do not approximate, $$\textit{a priori}$$, $$2^{n}\theta = a + 2^{n}\delta$$, then what I obtain is this list of probabilities for each $$a$$. It appears I can use any of them to estimate the phase. In this case, which probability shall I use?