Binomial Likelihood for bayesian statistics

Question

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!

Answer 1

From the picture, it looks like the prior is $p\sim\textsf{Beta}(1,1)$ for the unknown proportion $p$, which is the same as $p\sim\textsf{Unif}(0,1)$, and thus using a binomial likelihood with $k$ successes in $n$ trials, $k|n,p\sim\textsf{Bin}(n,p)$, the posterior is $p|n,k\sim\textsf{Beta}(k+1,(n-k)+1)$.

I don't know what they mean when they say beta likelihood. A likelihood is a pmf or pdf expressed as a function of it's usual parameters, keeping the usual input fixed (https://en.wikipedia.org/wiki/Likelihood_function), e.g. the binomial likelihood above is a binomial pmf where the $k$ successes and $n$ trials are fixed, and it's viewed as a function of the usual parameter $p$. The posterior isn't a likelihood, it's a probability distribution. Calling something a beta likelihood would mean you're fixing the usual input $p$ and viewing the beta pdf as a function of its usual parameters $\alpha$ and $\beta$ (or just one of them if you keep one fixed). This obviously isn't what they're plotting, so I think they just misused terminology.