Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 219 / 2

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 $A(1+1/N)$.