I was looking at the Bernoulli Distribution and its relation to the Prior Belief Distribution. The equation is written as $$ \frac{x^{\alpha - 1}(1-x)^{\beta -1}}{B(\alpha, \beta)}. $$
I've also studied the properties of the of the Beta function which is define as $$ B(x,y) = \int_0^1 t^{x-1}(1-t)^{y-1} \, \mathrm dt. $$
The Beta function is basically the integral of the Bernoulli Distribution from what I can see.
I graphed the Prior Belief Distribution for a couple of variations for $\alpha$ and $\beta$ and noticed that dividing by the Beta function squashes the height of the distribution by some amount.
My question is what is the Beta function doing exactly. Is it squashing the distribution to within some range? If so, can someone show me a proof as to why this works so that I can internalize it better?
So the main job of the $B(\alpha, \beta)$ function here is to normalise the distribution, i.e. ensure that it has integral $1$ when integrated over the whole range. Can you see why that is?