Say you have a coin. You don't know the bias of this coin. That it, it flips heads with probability $p$, and tails with $1-p$ for $p \in [0, 1]$.
Somebody flips the coin 1000 times. It comes up heads on every time and then asks you: "What is the bias of this coin"?
You cannot guess it with certainty, as any non-zero $p$ is possible. However, what is the most likely value of $p$?
I'm not sure how to approach this. It seems actually that all values of p have 0 chance of being correct, as there's an infinite number of values $p$ can be. So I thought about $P(p \in [0, 0.1])$, $P(p \in (0.1, 0.2])$ etc. Then finding the group with the highest probability, in this case would be the last one. Then disecting that group into equal parts and starting again, eventually converging on the actual value.
I'm not sure how to formalize this process, or how to actually find $P(p \in [lower, upper])$