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I've heard that the $\alpha$ and $\beta$ parameters of the Beta distribution intuitively represent the number of successes and failures, respectively.
1) If so, what's the purpose of subtracting $1$ from them in the exponent?
2) If so, why would then $\alpha=1$ and $\beta=1$ represent a uniform distribution? Do we assume that there is one success and failure initially? Does this have anything to do with Laplace's Rule of Succession?
3) How can we intuitively understand positive real-valued shape parameters for the distribution?