Intuition of Beta distribution with less-than-one parameters According to Wikipedia, Beta distribution models the distribution of parameter $p$ of Binomial distribution. Concretely, consider a Binomial distribution
$$f(k;n,p)=\binom{n}{k} p^k (1 - p)^{n-k}$$
Then the Beta distribution corresponding to $k$ occurrences of positive outcome is given as
$$g(p;\alpha,\beta) = \frac{p^{\alpha-1}(1-p)^{\beta-1}}{B(\alpha,\beta)}$$
where $\alpha=k+1$ and $\beta=n-k+1$. The intuition of Beta distribution is then obvious when $\alpha,\beta$ are integers not less than $1$. However, I cannot find any intuition for the following case:

*

*When $\alpha,\beta$ are real numbers and not less than $1$, rather than integers

*When $\alpha,\beta$ are real numbers within the interval $(0,1)$, i.e. the number of positive occurrences $k$ is less than $0$???

I have searched through many articles. This one turns out to give me some hint, but only for the case $\alpha=\beta$ and $\alpha\in(0,1)$. Can anybody help me with the intuition for the cases above?
 A: Non-Integer Values Generally
To your first question, in cases where outcomes are neither entirely positive nor entirely negative, it makes sense to allow $\alpha$ and $\beta$ to reflect those mixed, fractional results.  A very simple example would be a game that can end in a tie (chess, football, etc.).  In some cases, when using a Beta distribution to model wins and losses, one might want to treat a tie as incrementing both $\alpha$ and $\beta$ by $1/2$.  Or perhaps one wants to aggregate Likert-style responses (strongly agree, agree, neutral, disagree, strongly disagree) or star ratings (1-5 stars).  One could translate such responses into fractional updates to a Beta distribution reflecting the aggregate:
strongly agree: increment $\alpha +1,$ leave $\beta$ unchanged
agree: increment $\alpha +3/4, \beta +1/4$
neutral: increment $\alpha +1/2, \beta +1/2$
disagree: increment $\alpha +1/4, \beta +3/4$
strongly disagree: leave $\alpha$ unchanged, increment $\beta$ by 1
When aggregated, the following combinations would thus all yield equivalent effects on the total $\alpha$ and total $\beta$ counts--a result that seems reasonable:
1 strongly agree + 1 strongly disagree = 2 neutrals = 1 agree + 1 disagree
Of course, one will often end up with total $\alpha$ and $\beta$ counts that are not integers, yet nevertheless model meaningful fractional outcomes.
$\alpha, \beta < 1$
Your second question concerns cases when $\alpha$ and $\beta$ are both less than 1, and one gets a bimodal, U-shaped distribution where most of the mass is located at the extremes (near 0 and 1).  George Casella discusses one nice example at the end of his paper Introduction to Empirical Bayes: if you ask consumers how likely they are, percentage-wise, to make a large purchase in the next year (say, a house or a car), the distribution of their answers will likely follow a U-shaped distribution.
Imagine surveying consumers about their likelihood of purchasing a car within the next year.  Most answers will be single-digit probabilities since most people will only buy a car if they lose use of their old one (through theft or accident)--a small chance.  But some percentage of people will be more than 90% confident that they will be purchasing a car in the next year.  Relatively few, however, will be more than 20% confident but less than 80% confident of purchasing a car.  As result, the distribution will be bimodal: it you were to fit a Beta distribution to your survey data, you would get $\alpha, \beta < 1$.  The survey data discussed by Casella resulted in a Beta(.25, .43):
