prior and posterior predictive distributions, Bayes Theory

Question

Consider the binomial sampling model with a Beta prior on $\theta$ and the prior predictive distribution. Let $n$ be the binomial sample size. \begin{align} p(y^{new}) &= \int_{\theta}f(y^{new}|\theta)\pi(\theta)d\theta \\ &= \binom{n}{y^{new}} \frac{\Gamma(\alpha+y^{new}) \Gamma(n - y^{new} + \beta)}{\Gamma(n + \alpha + \beta)} \cdot \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) \Gamma(\beta)} \end{align}
This is a Beta-Binomial distribution, which we denote Beta-Binomial($n$, $\alpha$, $\beta$).

For the posterior predictive distribution, given that the posterior distribution for $\theta$ is still Beta, one can substitute $α^{post}= α + y^{old}$ and $β^{post}= β + n − y^{old}$ in the above results. Letting $n^{new}$ be the new sample size, the posterior predictive distribution is then: \begin{align}p(y^{new}|y^{old}) = \binom{n^{new}}{y^{new}} \frac{\Gamma(\alpha^{post}+y^{new}) \Gamma(n^{new} - y^{new} + \beta^{post})}{\Gamma(n^{new} + \alpha^{post} + \beta^{post})} \cdot \frac{\Gamma(\alpha^{post} + \beta^{post})}{\Gamma(\alpha^{post}) \Gamma(\beta^{post})}\end{align} I wonder how the formula $p(y^{new}|y^{old})$ is derived?

Answer 1

As Demetri says, the answer is right there - maybe an example helps?

Consider $f(y^{new}=1|\theta ,y)=P(y_{f}=1|\theta)$, i.e., the probability that the next attempt will be a success, assuming random sampling. Note $P(y_{f}=1|\theta)=E(y|\theta)=\theta$.

Hence, with $m$ and $k$ the number of attempts and successes in the first sample, \begin{eqnarray} p(y^{new}=1|y)&=&\int f(y^{new}=1|\theta)\pi(\theta|y)d\theta\notag\\ &=&\frac{\Gamma\left(\alpha _{0}+\beta_{0}+n\right)}{\Gamma\left(\alpha_{0}+k\right)\Gamma\left(\beta_{0}+n-k\right)}\int \theta\theta ^{\alpha _{0}+k-1}\left( 1-\theta \right) ^{\beta _{0}+n-k-1}d\theta\notag\\ &=&\frac{\Gamma\left(\alpha_{0}+\beta_{0}+n\right)}{\Gamma\left(\alpha_{0}+k\right)\Gamma\left(\beta_{0}+n-k\right)}\frac{\Gamma\left(\alpha_{0}+k+1\right)\Gamma\left(\beta_{0}+n-k\right)}{\Gamma\left(\alpha_{0}+\beta_{0}+n+1\right)}\notag\\ &=&\frac{\alpha_{0}+k}{\alpha _{0}+\beta_{0}+n}, \end{eqnarray} using $\Gamma(\alpha)=(\alpha-1)\Gamma(\alpha-1)$.

Answer 2

You have the answer in your first line. The posterior predictive is

$$ p(y^{new} \mid y^{old} ) = \int_\theta p(y^{new} \mid \theta) p(\theta \mid y^{old}) \, d\theta $$

The functions in the integrand are the likelihood and posterior respectively. The likelihood is binomial, the posterior is beta, and so as in your first line, the posterior predictive is beta-binomial.

Are you asking for the derivation of the beta binomial density?