if a fraction of user 1 is p1 (with parameters a1 and b1), and a fraction of user 2 is p2=1-p1 with parameters a2 and b2, then the overall distribution density will be

f(x, p1,p2, a1, b1, a2, b2)= p1* 1/B(a1,b1)*x^(a1-1)*(1-x)^(b1-1) +

p2* 1/B(a2,b2)*x^(a2-1)*(1-x)^(b2-1)

An important question, I am waiting for the answer, too...

Hi Akhilkumar,

Just to elaborate a bit on Alexander Kolker's already helpful answer: The sum of two beta distributions will not result in a beta distribution so it can not be described with alpha & beta parameters alone.

The outcome will be a mixture distribution, and can even be bimodal. There are tools to fit these distributions though. See the following example in CrossValidated and the R package betareg might be useful:

CrossValidated

https://stats.stackexchange.com/questions/160838/calculating-parameters-for-a-mixture-of-beta-distributions

R package betareg

https://www.jstatsoft.org/article/view/v048i11

Hope that helps

Alexander Kolker, Gi-Mick Wu Thanks a lot for your input.

It seems not to be helpful to start by thinking of merging only two curves ... better to think immediately of combining many curves. You also need to decide whether your combined-curve result needs to be within the same family as the "observed" ones

Combining the curves might be done by averaging some characteristic of the curves:

(a) average percentage points, either leaving results raw for many such percentage points, or fitting a distribution to a pair of percentage points

(b) average moments of the "distribution" curves, and fit a distribution to the average moments;

(c) average the parameter values of the "beta distributions" or, for example, logarithms of the parameter values, to get parameters of a combined curve.

In the above you might consider a "robust"version of the average.

It would be good to try something out with some fictitious starting curves and examine the results produced graphically to see what you think most suitable to your problem. Looking at things graphically for many curves may lead you to think about dealing with understanding the differences between opinions, rather than just finding an a average opinion.