Past exam of the statistics course of the University of Oxford / 2021 / 3

Set $S={\sum }_i{x}_i$ and $N=nm$. Conditional independence of the binomial distribution observations gives the likelihood functionThe prior distribution is the beta distribution with parameters $(1,\beta )$. By Bayes' theorem, multiplying the prior density by the likelihood function gives a posterior density proportional toRecognizing and normalizing this beta distribution givesfor $0<\theta <1$. This is Beta-binomial conjugacy; the original binomial coefficients cancel in the normalization because they are independent of $\theta$.

The prior mean, posterior mean, and maximum-likelihood estimate areFor $0<S<N$, the last follows by maximizing $S\log \theta +(N-S)\log (1-\theta )$. At $S=0$ or $S=N$, use the usual binomial likelihood convention that admits the endpoints $0$ and $1$ as maximum-likelihood estimates; if the parameter domain is strictly $(0,1)$, these are boundary suprema rather than attained maxima. The posterior mean remains interior in all cases.

TakinggivesSince $N=nm>0$ and $\beta +1>0$, $0<\lambda <1$. Thus the posterior mean is a convex combination of the prior mean and observed binomial proportion, with larger prior weight when $\beta +1$ is large relative to $N$.

The prior mean $1/(\beta +1)$ decreases as $\beta$ increases. More strongly, its cumulative distribution function and upper-tail probability areFor every fixed $t$, that upper-tail probability decreases with $\beta$. The statistician increasingly favors smaller values of $\theta$; as $\beta \to \infty$, the prior distribution concentrates near zero. For $\beta >1$, its variance $\beta /[(\beta +1{)}^2(\beta +2)]$ also decreases. This is concentration of a beta prior near zero.

With the data fixed, the derivatives areThus the prior weight increases and the posterior mean decreases. Their limits are $\lambda \to 1$ and ${\tilde{\theta }}_1\to 0$. Both the changing prior mean and the increasing weight contribute to the change in the posterior mean.

Let $S_{xx}={\sum }_i{x}_i^2$, $S_{xy}={\sum }_i{x}_i{y}_i$, and $\bar{y}=n^{-1}{\sum }_i{y}_i$. The independent observations in the normal linear model giveThe centered design condition ${\sum }_i{x}_i=0$ removes the $\alpha \beta$ term:The stated prior density factors into independent normal distributions for the regression intercept $\alpha$ and regression coefficient $\beta$. Multiplication by the likelihood function therefore leaves separate quadratic expressions in the two coefficients. Define the positive posterior variancesand the posterior meansBy completing the square, the joint posterior density isEquivalently,This is Gaussian conjugacy for a normal linear model. Each posterior precision parameter is the sum of its data and prior precision parameters. The formulas remain valid when $S_{xx}=0$: all $x_i$ are then zero, so the posterior distribution of $\beta$ equals its prior distribution.

Put $M_0=\alpha +x_0\beta$. As a linear combination of independent normal random variables under the posterior distribution,The shortest 95% credible interval isHere $\Phi$ is the standard normal distribution cumulative distribution function. To justify shortest length, consider all intervals of a fixed length $d$. If $f$ is this normal distribution posterior density, the interval probability ${\int }_a^{a+d}f(t)dt$ has derivative $f(a+d)-f(a)$ with respect to $a$. The normal distribution is symmetric and its density strictly decreases with distance from its mean, so this probability is uniquely maximized when the midpoint is $m_0$. The centered interval can therefore attain 95% posterior probability with the least length. It is both an equal-tailed credible interval and a highest posterior density interval.

The target is the experiment's mean at $x_0$. A posterior predictive distribution for a new noisy observation would instead have variance $v_0+{\sigma }^2$; that extra observation variance does not belong in the requested interval.

An unknown variance requires a joint prior distribution for $(\alpha ,\beta ,{\sigma }^2)$. The analysis is not determined until a prior for the unknown variance, and its relation to the coefficients, is specified. A maximum-likelihood estimate substituted for ${\sigma }^2$ does not by itself account for its posterior uncertainty.

For example, retain the independent coefficient prior distributions from part (b) and independently assign $s={\sigma }^2$ an inverse-gamma distribution with shape $a_0>0$ and scale $b_0>0$, whose probability density function is proportional to $s^{-a_0-1}{e}^{-b_0/s}$. Then the joint posterior density is proportional toConditional on $s$, the two coefficient posterior distributions are still the independent normal distributions in part (b), with $s$ substituted for ${\sigma }^2$. The other full conditional distribution isThe marginal coefficient posterior distribution requires marginalization over $s$; it is generally nonnormal and the two coefficients need not remain independent. In particular, the posterior cumulative distribution function of the requested mean becomesIts shortest 95% credible interval must be found from this marginal posterior distribution, for example by numerical integration or posterior simulation. This is normal regression with independent coefficient and variance priors.

A different, convenient choice is a normal-inverse-gamma prior, in which the conditional coefficient prior covariance matrix is proportional to ${\sigma }^2$. That choice gives a multivariate Student's t-distribution after marginalization, so the interval uses its univariate Student's t-distribution quantiles. It changes the coefficient prior specification; a Student interval does not follow automatically from retaining the fixed-${\tau }_0,{\tau }_1$ priors in part (b).