"The prior probability is properly so-called, not because it is a priori (it is not), but because it is in
place prior to one's taking the new evidence into account."
Different priors are sometimes used as a form of sensitivity analysis, often to show that the choice of prior is not overly influential.
Choosing the prior is the eternal problem faced by Bayesians.
Depending on the problem, there are three ways of choosing a prior:
subjective (your personal degree of belief);
objective and informative (use historical data, or data from previous experiments); or
objective and uninformative (e.g. uniform prior, reference priors, maximum entropy, Jeffreys prior).