Show that $\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$

Question

Suppose you assign a $Beta(\alpha,\beta)$ prior distribution for $\theta$, and the you observed $y$ heads out of $n$ spins. Show algebraically that your posterior mean of $\theta$ always lies between your prior mean $\frac{\alpha}{\alpha+\beta}$ and the observed relative frequency of heads if $\frac{y}{n}$.

I have that $X\sim Bin(n,\theta)$ and $\theta\sim Beta(\alpha,\beta)$ so the posterior is $$\pi(\theta|x)\propto \pi(\theta)f(x|\theta)= \theta^{\alpha-1}(1-\theta)^{\beta-1}\theta^y(1-\theta)^{n-y}$$ $$=\theta^{a+y-1}(1-\theta)^{\beta+n-y-1}\sim Beta(a+y,n+\beta-y)$$ and the posterior mean is $$E[\pi(\theta|x)]=\frac{a+y}{a+n+\beta}$$

Now how I show that $$\frac{\alpha+y}{\alpha+n+\beta}\in (\frac{\alpha}{\alpha+\beta};\frac{y}{n})$$ ?

Answer 1

$\dfrac{\alpha+y}{\alpha+n+\beta}=\dfrac{\alpha+\beta}{\alpha+n+\beta}\cdot\dfrac{\alpha}{\alpha+\beta}+\dfrac{n}{\alpha+n+\beta}\cdot\dfrac{y}{n}$

Take $\gamma=\dfrac{n}{\alpha+n+\beta}\in(0,1)$ then $\dfrac{\alpha+y}{\alpha+n+\beta}=(1-\gamma)\dfrac{\alpha}{\alpha+\beta}+\gamma\dfrac{y}{n}$.

It is a convex combination of $\dfrac{\alpha}{\alpha+\beta}$ and $\dfrac{y}{n}$ so lies between the two.

Answer 2

Proposition: Let $a,b,c,d$ be positive real numbers such that $ad-bc<0$. Then $$\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$$

First, notice that the requirement is equivalent to $a/b<c/d$, so we can use either as the supposition in the proposition.

$$ad-bc=ad+ac-(ab+bc)<0\Rightarrow\frac{a}{b}<\frac{a+c}{b+d}$$ by dividing both sides by $b(b+d)$. Likewise

$$ac-bd=cd+ac-(cd+bd)<0\Rightarrow\frac{a+c}{b+d}<\frac{b}{d}$$ by dividing both sides by $d(b+d)$.

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This proposition is actually hugely useful in combinatorics, as you can use it to prove that the probability that an event occurs is strictly between $0$ and $1$ if and only if it can both occur and not occur. We use this principle in combinatorics to probabilistically prove that graphs have certain properties by showing the probability a randomly created large graph has the property is non-zero