# Derivation of the posterior given a “non-standard” likelihood and prior

I have an iid random sample $Y=(y_1,...y_N)$ with $p(y_i|\theta) = \theta^{y_i}(1-\theta)^{y_i}$ if $0\leq y_i \leq1$ and $0$ otherwise. The prior is $$\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}\theta^{\alpha-1}(1-\theta)^{\beta-1} \quad \text{if} \quad 0 < \theta <1$$

and $0$ otherwise.

Now my thinking was the following:

$p(y|\theta) = \Pi_{i=1}^N \theta^{y_i}(1-\theta)^{y_i} = \theta^{\Sigma y_i}(1-\theta)^{\Sigma y_i}$

The posterior is Likelihood times Prior thus:

$p(\theta | y) \propto \theta^{\Sigma y_i}(1-\theta)^{\Sigma y_i} \theta^{\alpha-1}(1-\theta)^{\beta-1} = \theta^{\Sigma y_i +\alpha-1}(1-\theta)^{\Sigma y_i + \beta-1}$

This looks like a (kernel of a?) Beta distribution.

So I did the following in R:

a <- 2
b <- 4

scale <- gamma(a+b) / (gamma(a) * gamma(b))
p_grid<-seq(0,1,length.out = 100)

prior_theta <- function(a,b){
length_x <- 100
theta <- seq(0,1,length.out = length_x)
p_theta <- scale * (theta^(a-1)) * ((1-theta)^(b-1))
return(p_theta/100)
}

lik_y <- function(ptheta){
length_y <- 100
y <- seq(0,1, length.out = length_y)
p_y <- (ptheta^y)*((1-ptheta)^y)
return(p_y)
}

theta <- prior_theta(a,b)

prob_y <- lik_y(theta)
prob_y

unstandardized_posterior <- prob_y*theta

standardized_posterior <- unstandardized_posterior/sum(unstandardized_posterior)

post_check <- dbeta(seq(0,1,length.out = 100), 2, 7)/100
plot(standardized_posterior);lines(post_check, col = "red")


And found that choosing $\alpha = 2,\beta=4$ results approximately in a beta distribution with $\alpha = 2, \beta = 7$. However, according to the calculation above the same number should be added to both parameters. Thus my twofold question: How would I determine $\Sigma_i y_i$ and how would I apply that to the example below? Help is much appreciated.

• @Daniel Winkler - are you sure you've correctly written the likelihood? It's not a valid density function for most values of $\theta$. – Taimur May 6 '17 at 13:11
• Yes that is the given likelihood. I was wondering about that too but figured it is proportional to a beta distribution with alpha=beta=sum(y)+1. – Daniel Winkler May 6 '17 at 14:56
• Also theta has only positive probabilities between 0 and 1 – Daniel Winkler May 6 '17 at 14:59