What is the equation for beta1 prime, beta2 prime and the normalizing constant in the following beta distribution?

Question

Can someone tell me what the separate equations for beta1 prime (B_1) and beta2 prime (B_2) and the normalizing constant are in this beta distribution? How does one go about computing them?

θ ^(k+β_1 -1) (1 − θ)(n−k+β_2 −1)/B(k+β_1, n-k + k+β_2)

If you could help me, I'd be very thankful. Thanks!

Answer 1

Some preliminaries:

The Beta distribution pdf is:

[ (\theta)^(\alpha - 1)*(1- \ theta)^(\beta - 1) ] / B(\alpha, \beta)

Where:

• \theta is the random variable between 0 and 1 that we usually try to solve for. For example, using maximum likelihood estimation (MLE) or MAP estimation.
• \alpha and \beta are parameters for the Beta distribution, called the shape and rate
• B(\alpha, \beta) is the Beta function - NOT to be confused with the Beta probability distribution. The Beta function is:

B(a,b) = [ Gamma(a)*Gamma(b) ] / Gamma(a+b)

Where Gamma is the gamma function, given by:

Gamma(a) = (a-1)!

For positive integers a, b. There is a more complicated form when a,b are not integers. So you can calculate the Beta function using whatever built in factorial function your software program uses.

So in your case, \alpha = k + Beta_1, and \beta = n - k + Beta_2. This looks like a posterior distribution for a Beta prior with Binomial Likelihood.

I assume you're performing a Bayesian inference. If that's the case, then usually we set:

• \alpha = "Number of successes"
• \beta = "Number of failures" = "Number of total observations - number of successes"

when performing Bernoulli experiments, i.e. like coin flipping or users subscribing to a website.

If you provide more information on what you're trying to solve, perhaps we can be of more help.