Properly interpret the alpha / beta parameters in the Beta Distribution

For quite a while I believed that the proper interpretation of a Beta distribution with $\alpha$ and $\beta$ is: "what is the most likely $P$ given $\alpha -1$ success (heads), and $\beta -1$ of failures (tails)", which also made sense when $\alpha -1$ and $\beta$ are equal to 1, which means that you don't have any heads / tails, i.e. a uniform distribution.

Yesterday, after asking a different question Is the beta distribution really better than the normal distribution for testing the difference of two proportions? , this understanding was undermined.

Can anyone help me to better understand the interpretation of the $\alpha$ and $\beta$ parameters?

I tried to get help from this post: here but that didn't help either.

That is one useful interpretation of the Beta distribution when it is used as a conjugate prior distribution to the binomial distribution. It breaks down a bit when you consider the possibility that it is perfectly legitimate for $\alpha + \beta < 1$ or even for $\alpha + \beta < 1/2$, meaning that $\alpha + \beta$ being the prior sample size is also only one possible interpretation of the parameters.
More generally, these are concentration parameters, which are a class of parameters that govern probability distributions over probability distributions. Concentration parameters have an interesting property. The smaller they are, the more sparse the distribution is. In the case of the Beta distribution, as $\alpha,\beta \rightarrow 0$, more and more of the probability is concentrated on the probability parameter $p$ being 0 or 1. Another interesting property of a concentration parameter is that when they all equal one, all possibilities are equally likely. Yet another property is that, as the concentration parameters get larger, the distributions tighten about the expectations.
• Wow, this is a very impressive and usfull explanation, i have to say that i cant think of any of my real world data including $alpha$ + $beta$ $<1$ so ill stick to the explanation im used to – Yehoshaphat Schellekens Jul 7 '14 at 6:45