# Beta distribution vs beta binomial distribution: alpha and beta

I have been attempting to estimate alpha and beta from a beta binomial distribution given my data. There are R packages like VGAM to do this. I am wondering if there is a difference between estimating these parameters for a beta binomial distribution and if I estimated them for a beta distribution. Is there a way to fit my data to a beta distribution and get the same alpha and beta as when I fit it to a beta binomial distribution?

• The beta-binomial distribution is a distribution over non-negative integers less than the number of trials, while the beta distribution is a distribution on reals in the unit interval. Given data that is either integer- or real-valued, one will be an obviously incorrect substitute for the other. What kind of data do you have? – Sycorax says Reinstate Monica Feb 9 '15 at 17:41

## 1 Answer

Beta and Beta-binomial are two different distributions. Beta is a continuous distribution (values in the $[0, 1]$ range) while Beta-binomial is discrete (integers from $0$ to $n$). So it is not a good idea to use both distributions to approximate your data since it can be discrete or continuous, but not both.

• Yes, but parameters alpha and beta are the shape parameters for the underlying beta distribution. So given a list of integers x and another list of integers n-x. Is there a way to get alpha and beta by fitting to a beta distribution. Possibly by fitting x/n to a beta distribution? – user3266890 Feb 9 '15 at 17:57
• You just need estimate $\alpha$ and $\beta$ for the beta-binomial distribution. That gives you the parameters for the underlying beta distribution. – Sycorax says Reinstate Monica Feb 9 '15 at 18:01
• You could even try to fit normal distribution to a vector of zeros and ones - it will just not make much sens. Distributions are simple functions, you can fit them to everything, but they are designed for certain cases and make sens when used properly. – Tim Feb 9 '15 at 18:01
• @user3266890 and dividing x/n does not make a variable continuous. Discrete variable does not have to be made of integers, just from values that could be possibly counted. See: en.wikipedia.org/wiki/… – Tim Feb 9 '15 at 18:06
• My question originally stems from the need to estimate alpha and beta of a beta binomial distribution given my data. However, in python there are only ways to fit data to a beta distribution and not a beta binomial. So I wanted to know if there was a way to get from one to the other. It seems that there is not. – user3266890 Feb 9 '15 at 18:10