Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 (Astrostatistics)

The expected number in a spherical shell is $4\pi r^2{\rho }_0{e}^{-r/r_0}dr$. Sincethe normalized radial density is the Gamma distribution

In the stated units, $\alpha =1/r$. The change of variables formula givesThus, with ${\alpha }_0=1/r_0$,

Writing $r_s=1/{\alpha }_s$, the log likelihood isHenceThe score variance and negative expected Hessian give Fisher informationSince $r_s$ is Gamma with mean $3r_0$ and variance $3r_0^2$,The estimator is unbiased and attains the Cramér-Rao lower bound.

The inverse-square law gives $f=L_0/(4\pi r^2)$, so observation is equivalent toFor $u=R/r_0$, integrating the Gamma density givesTherefore

At distance $r$, inclusion requires $L\ge 4\pi r^2{f}_{\min }$. ThusIt equals $1/2$ when

The squared chord distance isHence the restricted Gaussian process has covarianceIt depends only on $t-t^{\prime }$, so it is stationary. It is periodic in either argument with period $T=2\pi /\omega$, and Gaussian-process realizations inherit that period almost surely because $f(t+T)-f(t)$ has zero variance.

Let $K_{\theta }$ have entries $k_{\theta }(t_i,t_j)$ and letIndependent Gaussian measurement errors give the multivariate normal density

The posterior isA random-walk Metropolis–Hastings algorithm proposes ${\vartheta }^{\prime }$ from a symmetric density about the current $\vartheta =(\mu ,\theta )$ and accepts with probabilityFor distinct states, multiplying the transition density by the target density giveswhich is symmetric and proves detailed balance. Run multiple dispersed chains, tune proposals during warm-up, inspect traces, effective sample sizes and convergence diagnostics, then estimate the period mean by averaging $2\pi /{\omega }_m$ over retained draws.

For draw $m$, let $C_m$ be the observed covariance matrix, $k_{\ast m}=(k_{{\theta }_m}(t_{\ast },t_i){)}_i$, andThe posterior predictive distribution is a mixture of these conditional Gaussians. Its Monte Carlo mean and variance are

The scale separation makes correlations between distinct observation times negligible, so $K_{\theta }\simeq AI$. Put $v_i=A+{\sigma }_i^2$. With a flat prior,The next latent value is likewise approximately independent of the past conditional on $\mu$, with $f_{\ast }\mid \mu \sim N(\mu ,A)$. Marginalizing $\mu$ givesWhen every ${\sigma }_i=0$, $\bar{\mu }=\bar{y}$, $V_{\mu }=A/N$, and the predictive variance is $A(1+1/N)$.

Normal-normal conjugacy givesWith $b={\tau }^2/({\tau }^2+{\sigma }^2)$,

Write $D_s=M_s+{\epsilon }_s$. ThenThe terms are independent and centered, soThe unpooled estimate $D_s$ has mean squared error ${\sigma }^2$, so population shrinkage improves it.

Convolving the population and measurement distributions givesPut $V={\tau }^2+{\sigma }^2$. With a flat prior on $M_0$,

Let $S={\sum }_s(D_s-\bar{D}{)}^2$. Integrating out $M_0$ givesNear zero, integrability requires $k>-1$; at infinity it requiresFor integer $k$ the posterior is proper exactly whenThe choice $k=0$ is allowed for $N>3$ and yields a proper posterior, but it is an improper flat prior on a scale parameter and is not invariant under reparameterization, so sensitivity to more principled scale priors should be checked.

Up to normalization, the joint posterior isA Gibbs sampler cycle consists of:

Each proposal is the exact full conditional and is therefore accepted. If the cycle begins with density $p({\theta }^t\mid D)$, integrating the product of the current posterior and successive conditional kernels over all overwritten coordinates leavesso a full Gibbs sweep preserves the joint posterior.

Completing the square gives

The evidence is the convolution of the likelihood with the prior:

Sincewe obtain

The Kullback-Leibler divergence between the posterior normal distribution and $N(0,{\tau }^2)$ is

Substituting $\tilde{\theta }={\tau }^{2}y/({\sigma }^2+{\tau }^2)$ and ${\sigma }_{\theta }^2={\sigma }^2{\tau }^2/({\sigma }^2+{\tau }^2)$ into parts c and d and simplifying givesThus the equality holds.

It holds generally whenever the posterior is well defined and the expectations exist. Bayes' theorem givesTaking posterior expectations yieldswhich rearranges to the claimed evidence decomposition.