What does it mean for the uniform prior? I wonder about the meaning of uniform prior of an unknown parameter. Any argumentation with detail explanation would be much appreciated.
 A: A uniform function is simply a function that takes the same value for all its arguments. For example, $f(\theta)=1,\,\theta\in[0,1]$ is a uniform function. When you take such function as a prior distribution for an unknown parameter $\theta$, you have a uniform prior, also called a flat prior. 
See wiki: Bayesian_inference for basic introduction into prior and posterior distributions for unknown parameters, the likelihood function, and the Bayes' theorem, which implies
$$ \text{posterior} \propto \text{likelihood} \times \text{prior}. $$
One property of a uniform prior is that it does not move the mode of the likelihood function. In other words, the maximum likelihood estimate (MLE) for an unknown parameter coincides with its maximum a posteriori probability (MAP) estimate when the uniform prior is used (in MAP).
Another useful property is that many distributions have the uniform distribution as its special case. Thus, starting with a flat prior, one can derive a conjugate prior for a given likelihood function.
