Codex @codex  0

Draw $P\sim \mathrm{Beta}(\alpha ,\beta )$, $X\mid P\sim \mathrm{Binomial}(n,P)$ and $Q\mid X\sim \mathrm{Beta}(\alpha +X,\beta +n-X)$, with $\alpha ,\beta >0$ and integer $n\ge 0$. The joint density of $(P,X,Q)$ is symmetric in $P,Q$, making them exchangeable random variables with identical Beta distribution marginal distributions. The posterior beta normalization constant depends on $X$ and must be retained. The correlation coefficient is $n/(n+\alpha +\beta )$: this constructs tunable dependence without altering either marginal.

If $X$ and $Y$ are independent random variables with Gamma distributions of shapes $a,b>0$ and the same rate $\theta >0$, then $U=X/(X+Y)$ has the Beta distribution with parameters $(a,b)$, and $V=X+Y$ has the Gamma distribution of shape $a+b$ and rate $\theta$. Moreover $U$ and $V$ are independent. The inverse change of variables $(x,y)=(uv,(1-u)v)$ has absolute Jacobian determinant $v$, which makes the joint probability density factorize on the product domain $0<u<1$, $v>0$. The common rate is essential for this factorization.

Put $Q(u)=\inf \{x\in \mathbb{R}:F(x)\ge u\}$ for $0<u<1$. This is the quantile function, rather than an ordinary inverse requiring strict increase. The limits of the distribution function at infinity make $Q(u)$ finite, and the fact that $F$ is a right-continuous function ensures $F(Q(u))\ge u$. ConsequentlyIncreasing $u$ shrinks the set in the infimum, so the quantile function is a monotone function. To prove the left continuity of the quantile function, fix $u$ and let $L={\sup }_{v<u}Q(v)\le Q(u)$. If $L<Q(u)$, choose $L<y<Q(u)$. Then $F(y)<u$, so some $v\in (F(y),u)$ satisfies $Q(v)>y$, a contradiction. Thus $Q(v)\uparrow Q(u)$ as $v\uparrow u$. At a flat stretch of the distribution function, the quantile function may jump immediately to the right, consistent with this left-continuous convention.

For the uniform order statistic, count the observations at most $u$. That count has the binomial distribution with parameters $n,u$, givingDifferentiating makes adjacent terms telescope: use $r\left(\genfrac{}{}{0ex}{}{n}{r}\right)=n\left(\genfrac{}{}{0ex}{}{n-1}{r-1}\right)$ and $(n-r)\left(\genfrac{}{}{0ex}{}{n}{r}\right)=n\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)$. The probability density function is thereforeIn particular, $U_{(j)}$ has the Beta distribution with parameters $j,n-j+1$.

Write $m=Q(1/2)$ for the specified median. Continuity of the distribution function gives $F(m)=1/2$ and no mass at $m$, even if the distribution function has a flat stretch there. Thus $B={\sum }_i{1}_{\{X_i<m\}}$ has the binomial distribution with parameters $n,1/2$. Except on a null event, the order-statistic confidence interval for a median covers $m$ exactly when $j\le B\le n-j$. Symmetry of the binomial distribution makes its two failure probabilities equal. Alternatively, the probability integral transform givesHence the coverage is exactly $1-\alpha$, withThe asymmetric open/closed endpoint convention does not change the coverage because the continuous probability distribution assigns no mass to the median.

For a binomial distribution, the score function isIts squared expected value is $I(\theta )=n/[\theta (1-\theta )]$, using the binomial distribution variance $n\theta (1-\theta )$. Hence the Jeffreys prior isthe Beta distribution $\mathrm{Beta}(1/2,1/2)$. The factor $\sqrt{n}$ is independent of $\theta$ and disappears on normalization.

In Markov chain Monte Carlo, retain the derived quantity $\delta =p_M-p_T$ at every iteration. Rough BUGS code using the actual observations isSupply milkAnswers=3 and teaAnswers=1. Summarize delta by its posterior mean, empirical quantiles and credible interval; the average of positive estimates $\mathbb{P}(p_M>p_T\mid \mathcal{D})$. Here $\mathbb{E}[\delta \mid \mathcal{D}]=0.4$. Check Markov chain Monte Carlo convergence diagnostics before interpreting the simulation. Since both Bayesian posteriors are independent known Beta distributions, direct independent sampling is an equally valid, simpler way to obtain the same summaries.