If you have no information, you choose a prior that reflects the fact that you have no information. One way of doing this is the principle of maximum entropy, which tells you in particular that if you’re trying to choose a prior over a parameter that lies in the interval [0, 1] (the bias of the coin), the maximum entropy prior is the uniform prior. If you have other information, you incorporate that into your choice of prior.
If you have no information, you choose a prior that reflects the fact that you have no information. One way of doing this is the principle of maximum entropy, which tells you in particular that if you’re trying to choose a prior over a parameter that lies in the interval [0, 1] (the bias of the coin), the maximum entropy prior is the uniform prior. If you have other information, you incorporate that into your choice of prior.