Beta Distribution

#1
When we are using Beta distribution, does the low values of alpha and beta have any effect on posterior and meant that the data weighed much more heavily in forming the posterior than did the prior?

I have a test with these data:
experiment 1: 30 success in 75 trials
experiment 1: 60 success in 75 trials
for the first time I chose alpha and beta=1
and then change the prior to alpha=30 and beta=10 and calculate the posterior. but couldn't get how to interpret the posterior result.

Could some one help on that?
Thanks,
-Shani
 

Dason

Ambassador to the humans
#2
You should be a little more clear with your notation and the models you're fitting. My guess is that you're assuming a binomial response and you put a beta prior on the probability of success for the binomial. If that's the case then yes lower values of alpha and beta result in the prior having less weight in the posterior. You can see this fairly easily by recognizing that the beta is the conjugate prior for the binomial success probability parameter and that the posterior also has a beta distribution \(Beta(\alpha + n_{success}, \beta + n_{failures})\).

So if you use use large values for alpha and beta in the prior then the posterior parameters seem fairly close to the prior parameters.
 

Dason

Ambassador to the humans
#3
Please only post a topic once. If you want the thread moved to a different subform just let us know. Thank you.
 
#4
Yes. I am using Beta distribution as a prior for binomial and here I am trying to compare the effect of two different alpha and beta prior values on posteriors based on number of success in each experiment:

I run Binom.test in R got these p-values for experiment 1:
alpha and beta equal to 1:
sum(dbinom(30:75,75, prob=0.5))=[1] 0.968025
pbeta(0.5,31,45)=[1] 0.9473289
for alpha=30 and beta=10:
pbeta(0.5,60,54)=[1] 0.2863282

for experiment 2:

sum(dbinom(60:75,75, prob=0.5))=[1] 7.938995e-08


alpha and beta equal to 1:
pbeta(0.5,61,16)=[1] 4.921424e-08
for alpha=30 and beta=10:
pbeta(0.5,90,25)=[1] 1.793108e-10

How can I interpret the result?
 
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