# Posterior Distribution for parameter with beta distribution given binomial data

I am currently working on a problem but I'm having issues due to not having much experience with Bayesian statistics at the moment

I have a sequence of observations $X_1, X_2, \dots X_N$ the are distributed Binomially iid with parameter $\theta$. $\theta$ is distributed by a beta distribution with parameters $a,b$. I'm asked to find the likelihood function $L(\vec{x};\theta)$ and the posterior distribution for $\theta$ given my $N$ observations.

Here is my work \begin{align*} L( \vec{x} ; \theta) &= Pr(\vec{x}; \theta) \\ &= \prod_{i=1}^N \theta^{x_i}(1-\theta)^{1-x_i} \\ &= \theta^{\Sigma x_i} (1-\theta)^{N - \Sigma x_i} \end{align*} This one I think is correct

Now for the posterior distribution of $\theta$ I am having some issues: \begin{align*} Pr(\theta \vert \vec{x}) &= \frac{Pr(\vec{x} \mid \theta) Pr(\theta)}{Pr(\vec{x})} \\ &= \frac { \theta^{\Sigma x_i} (1-\theta)^{N - \Sigma x_i} \cdot Beta(a,b) } { \sum_{i=1}^N \theta_i^{\Sigma x_i} (1-\theta_i)^{N - \Sigma x_i} \cdot Beta(a,b) } \end{align*}

Mainly with the denominator, $Pr(\vec{x})$. Is it true that I have to calculate the total probability for all possible values of each $\theta_i$ to get $Pr(\vec{x})$ since the observations depend on $\theta?$ The question says that I must express the posterior distribution in terms of $a,b,$ and the data, but Im not sure how to get rid of the $\theta$.

• You need a self-study tag. See discussion of homework-related questions in the help center – Glen_b Nov 4 '17 at 3:46

You're not supposed to "get rid of" the $\theta$. Since it's a posterior distribution for $\theta$, you're seeking an expression in $\theta$. Once you have it, you will be able to recognize the density -- it is the parameters of that density will be in terms of $a,b$ and the data.
Firstly, your prior is a prior on $\theta$ so explicitly write down that density you've got as "Beta(a,b)" as a density function that would be a prior on $\theta$ ($p(\theta)$)
(i) you already used $i$ up in the powers of $\theta$ and $1-\theta$, so you couldn't index your sum by $i$
(ii) it shouldn't be a sum at all, $\theta$ is continuous.