We know that Beta distribution is conjugate prior for binomial likelihood. In a course, I am doing on coursera, instructor used beta distribution with alpha as k + 1 and beta as n-k+1 as likelihood instead of using binomial distribution, I am not able to understand the reason behind it.
Also, when the shape of binomial and beta likelihood distribution are very different (see attached chart) - binomial likelihood is given by orange solid line and beta likelihood is given by green dashed line. The two are very different in shape and value.
So, it will be helpful if anyone can explain the reason for using beta likelihood instead of binomial and why do we get different distribution with binomial and beta likelihood?
Adding more details - Course is https://www.coursera.org/learn/statistical-inferences/home/welcome And question under discussion can be found in second assignment of week 2.
In the code shared for the assignment - instructor uses prior, likelihood and posterior as given below by the snippet of R code
theta<-seq(0,1,0.001) n <- 20 k <- 10 prior <- dbeta(theta, 1, 1) # as alpha -1 = k = 10 and beta - 1 = n - k = 10 likelihood <- dbeta(theta,10 , 10 ) # beta likelihood, green dashed line in chart posterior <- dbeta(theta, 11, 11)
While, I am fine with prior and posterior, but as my per understanding likelihood should be
likelihood <- dbinom(10,20,theta) #binomial likelihood, solid orange line in chart
However, plot of binomial likelihood (solid orange line) is very different from beta likelihood (dashed green line).
Now, I am not sure why instructor used beta likelihood as given in the code snippet above and not binomial likelihood in the manner defined by me in the second snippet.
I understand that by modifying the constant of binomial distribution, we can convert it to beta after we have replaced alpha with k + 1 and beta with n - k + 1. So, is the difference I am observing in the chart below is due to constant or something else? Also, how can I get same result by using binomial likelihood
I hope, this clarifies!