Expectance of predictive posterior on binomial-beta modeling

Question

I'm struggle with figuring out how to prove:

$E(\tilde{x}|x)=n\frac{a1}{a1+b1}$

where $x|\theta \sim Bin(n,\theta)$, $\theta \sim Beta(\alpha,\beta)$ and so $\theta|x\sim Beta(a_1=\sum{x_i}+\alpha,a_2=n.N-\sum_{}x_i+\beta)$ ($N$ as sample size)

I tried:

$p(\tilde{x}|x)=\displaystyle\int_{\theta}p(\tilde{x}|\theta)f(\theta|x)d\theta$

$p(\tilde{x}|\theta)=\displaystyle\frac{n!}{\tilde{x}!(n-\tilde{x})!}\theta^{\tilde{x}}(1-\theta)^{n-\tilde{x}}$

$f(\theta|x)=\displaystyle\frac{1}{B(\sum x_i+\alpha, n.N-\sum x_i + \beta)}\theta^{\sum x_i+\alpha-1 }(1-\theta)^{n.N-\sum x_i +\beta -1}$

so

$p(\tilde{x}|x)\propto \displaystyle\frac{n!}{\tilde{x}!(n-\tilde{x})!}\displaystyle\int_{\theta} \theta^{\tilde{x}+\sum x_i +\alpha -1}(1-\theta)^{n-\tilde{x}+n.N-\sum x_i +\beta -1}d\theta $

$p(\tilde{x}|x)\propto \displaystyle\frac{n!}{\tilde{x}!(n-\tilde{x})!}B(\tilde{x}+\sum x_i +\alpha, n-\tilde{x}+n.N-\sum x_i +\beta)$

I don't know what that distribution is so I've explicited the formula:

$p(\tilde{x}|x)=\displaystyle\frac{n!}{\tilde{x}!(n-\tilde{x})!}\frac{B(\tilde{x}+\sum x_i +\alpha, n-\tilde{x}+n.N-\sum x_i +\beta)}{B(\sum x_i +\alpha, n.N-\sum x_i +\beta)}$

so:

$E(\tilde{x}|x)=\displaystyle\sum_{\tilde{x}=1}^{n}\tilde{x}\displaystyle\frac{n!}{\tilde{x}!(n-\tilde{x})!}\frac{B(\tilde{x}+\sum x_i +\alpha, n-\tilde{x}+n.N-\sum x_i +\beta)}{B(\sum x_i +\alpha, n.N-\sum x_i +\beta)}$

Making some math I got:

$E(\tilde{x}|x)=\displaystyle\sum_{\tilde{x}=1}^{n}\frac{1}{B(\tilde{x},n-\tilde{x}+1)}\frac{B(\tilde{x}+\sum x_i +\alpha, n-\tilde{x}+n.N-\sum x_i +\beta)}{B(\sum x_i +\alpha, n.N-\sum x_i +\beta)}$

So here is the question:

1- Is the $\tilde{x}$ support the same as $x|\theta$ support?

2- if so then how to solve this summation?

Answer 1

There's a simpler way to tackle it, using the Law of Iterated Expectations. For notational clarity, I will define $y$ as "future $x$", so your first line would be $\mathbb{E}(y|x) = \dots$. Writing out the Law as applied to this case gives us:

$$\mathbb{E}[y|x] = \mathbb{E}_{\theta | x} \mathbb{E}[y|\theta]$$

We have $\mathbb{E}[y|\theta] = n\theta$. Taking the expectation of this with respect to $\theta | x$ gives us:

$$\mathbb{E}[y|x] = n\mathbb{E}[\theta | x] = n {a_1 \over a_1 + b_1}$$

The derivation of $a_1$ and $b_1$ as parameters of the posterior $p(\theta | x)$ I believe you have done above, although the Latex formatting seems not to be 100% correct in the relevant section at the time of this writing.