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If we have a Beta likelihood and a binomial prior, we get a beta posterior. Can someone please explain why this approaches a binomial as $n\rightarrow\infty$. I plotted it and this appears to be the case...

$$f(X|P)=\begin{pmatrix} n\\ x \end{pmatrix}\cdot p^x \cdot (1-p)^{n-x}$$

$$\xi(P) \propto \frac{p^{\alpha-1} \cdot (1-p)^{\beta-1}\cdot\Gamma{(\alpha)}\cdot\Gamma{(\beta)}}{\Gamma(\alpha+\beta)} $$

$$\xi(P) \propto p^{\alpha-1} \cdot (1-p)^{\beta-1} $$

$$\xi(P|X=x)\propto f(P| X=x) \cdot \xi(P)$$

$$\Rightarrow Beta Posterior \propto p^{x+\alpha-1}(1-p)^{n-x+\beta-1}$$

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    $\begingroup$ It can't approach a binomial (a discrete distribution) as a posterior because $p|X\sim\text{Beta}(x+\alpha,n-x+\beta),$ which is a continuous distribution. Could you clarify what you mean? $\endgroup$ – Sycorax Apr 19 '16 at 2:24
  • $\begingroup$ @C11H17N2O2SNa I had the same thought, but I think his wording is wrong. See my answer? $\endgroup$ – Neil G Apr 19 '16 at 2:25
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Both the prior and the likelihood are Beta-distributed. You have a Beta-binomial model:

$$p \sim \rm{Beta}(a, b)$$ $$x \sim \rm{Binomial}(n, p)$$

Observations from that model induce a Beta likelihood on the parameter $p$. That's your $f$. You have a Beta prior on the parameter $p$. That's your $\xi$.

The Beta distribution is an exponential family with zero carrier measure, so it is closed under pointwise product of densities; therefore, the posterior is also Beta-distributed.

In the limit, as more and more samples are collected, the posterior on $p$ converges to a delta distribution. As that happens, your Beta-binomial model collapses to a simple Binomial distribution.

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