Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 43 / 3

Write $v_i={\sigma }_i^2$ for each component variance and ${\tau }^2$ for the fixed prior variance denoted ${\sigma }^2$ in the source; these must not be confused. Let $\phi (x;\mu ,v)$ be the normal distribution probability density function and $D_j={\sum }_{i=1}^k{\omega }_i\phi (x_j;{\mu }_i,v_i)$. Independent observations and independent parameter priors give the posterior distributionIts support has $v_i>0$, ${\omega }_i>0$ and ${\sum }_i{\omega }_i=1$, with $\alpha ,\beta ,{\tau }^2,{ϵ}_i>0$. The component means have independent fixed-variance priors with a normal distribution, not means with variances proportional to $v_i$. The factor $e^{-\beta /v_i}$ follows from the stated inverse-gamma distribution. The later printed augmented expression instead has $e^{-\beta v_i}$ and an undefined prior mean; those are inconsistent with this specification. The intended model uses $e^{-\beta /v_i}$ and prior mean zero throughout.

Hold all parameters except ${\mu }_i$ fixed and propose ${\mu }_i^{\prime }={\mu }_i+\eta$ with $\eta \sim N(0,s_{\mu i}^2)$. This random-walk Metropolis algorithm has a symmetric proposal distribution. DefineCancellation of unchanged likelihood and prior factors gives the Metropolis–Hastings acceptance probabilityDraw an independent uniform and accept if it is at most $a_i$; otherwise retain the old value. Update each component in turn, recomputing the needed mixture contributions. On a logarithmic scale, compare $\log U$ with $\min (0,{\sum }_j\log (D_j^{\prime }/D_j)-[({\mu }_i^{\prime }{)}^2-{\mu }_i^2]/(2{\tau }^2))$ to avoid numerical underflow.

A valid symmetric proposal distribution is $v_i^{\prime }=v_i+\eta$ with $\eta \sim N(0,s_{vi}^2)$ on the entire real line. Reject immediately if $v_i^{\prime }\le 0$. Otherwise use the positive-parameter random walk with boundary rejection. Withthe Metropolis–Hastings acceptance probability isNo Jacobian determinant is needed for this additive proposal in $v_i$. Do not repeatedly redraw until $v_i^{\prime }>0$: that would create a state-dependent truncated proposal and require an additional Hastings correction. Alternatively a symmetric walk in $u_i=\log v_i$ respects positivity, but then the transformed target has a factor $v_i$; equivalently the acceptance ratio in variance coordinates has the extra factor $v_i^{\prime }/v_i$.

Choose a fixed Dirichlet distribution proposal $q(\mathbf{\omega })\propto {\prod }_i{\omega }_i^{{\eta }_i-1}$ with every ${\eta }_i>0$, independent of the current state, and propose the entire weight vector. This automatically respects the simplex constraint. With $D_j^{\prime }={\sum }_i{\omega }_i^{\prime }\phi (x_j;{\mu }_i,v_i)$, the Independence Metropolis–Hastings algorithm accepts withThis includes $q(\mathbf{\omega })/q({\mathbf{\omega }}^{\prime })$, not merely the posterior ratio. A particularly simple choice is ${\eta }_i={ϵ}_i$, the weight prior: its factors cancel and the ratio is just the mixture likelihood ratio. A proposal concentrated around a pilot estimate can mix better, provided it retains positive density throughout the relevant simplex. Both target and proposal densities use the same $k-1$ independent simplex coordinates, so this direct weight proposal has no extra transformation Jacobian determinant.

Use the data augmentation labels to separate the mixture into groups. Define $n_i={\sum }_j{1}_{\{z_j=i\}}$, $s_i={\sum }_{z_j=i}{x}_j$ and $Q_i({\mu }_i)={\sum }_{z_j=i}(x_j-{\mu }_i{)}^2$. The intended joint posterior distribution is proportional toSumming over all allocation labels recovers the marginal mixture posterior. The Gaussian mixture Gibbs updates with independent priors follow by collecting the terms in each parameter.

For ${\mu }_i$, the conditional exponent isCompleting the square givesFor $v_i$, the conditional density is proportional to $v_i^{-(\alpha +n_i/2)-1}\exp [-(\beta +Q_i({\mu }_i)/2)/v_i]$, henceThe independent prior on ${\mu }_i$ adds neither a $v_i^{-1/2}$ factor nor a ${\mu }_i^2/v_i$ penalty. Finally, combining the allocation counts with the Dirichlet distribution prior givesA Gibbs sampler successively draws from these conditionals, using the most recently updated values. An empty component has $n_i=s_i=Q_i=0$, so its mean and variance conditionals reduce to their proper priors without dividing by an undefined group sample mean.

Conditioned on the parameters, the data augmentation labels are independent. By Bayes' theorem, the conditional probability of component $i$ for observation $j$ isThis is the mixture responsibility. Draw an independent uniform random variable and select $i$ using the cumulative sums of $(r_{j1},\dots ,r_{jk})$. Refresh every label before recomputing the component sufficient statistics for the next Gibbs sampler sweep. Numerically, form the log scores $\log {\omega }_i-\frac{1}{2}\log v_i-(x_j-{\mu }_i{)}^2/(2v_i)$, subtract their maximum and exponentiate before normalization.

The marginal Metropolis–Hastings algorithm avoids storing allocation labels, but each component update changes every mixture likelihood term. Cache component contributions, calculate log-likelihoods stably and tune the proposal scale and random-walk Metropolis efficiency during a pilot run; then fix proposal scales or use a justified adaptive scheme. Means and variances may be strongly dependent, making separate random walks inefficient. Weight proposals must respect the simplex, variance proposals must respect positivity, and the displayed acceptance ratios must retain all relevant priors and proposal corrections.

The augmented Gibbs sampler uses elementary conditionals and typically costs order $nk$ for a label sweep, followed by sufficient-statistic updates. It avoids Metropolis rejections in these conditional draws, but does not remove slow mixing: ambiguous allocations can lock means, variances and labels into strongly coupled configurations. Empty components must be updated from their priors, not deleted automatically. The proper inverse-gamma scale $\beta >0$ suppresses arbitrarily small variances even though an unregularized Gaussian-mixture likelihood can be singular.

With exchangeable component priors, label switching creates $k!$ equivalent posterior modes. Ordering component means is an identification constraint, not an innocuous change unless the intended inferential model is made explicit. Predictive density and permutation-invariant summaries avoid arbitrary labelling; component-specific summaries require an appropriate relabelling convention. If the Dirichlet parameters differ between components, exact label symmetry may be broken, but substantial multimodality can remain.

Use dispersed initial states, multiple chains and diagnostics on several scientifically relevant functions. Report effective sample size of a Markov chain and Monte Carlo error estimates, not just a large iteration count. Burn-in can reduce initialization bias but does not prove convergence; thinning cannot by itself cure poor exploration. Closed-form Gibbs updates simplify computation, while reliable inference still requires adequate exploration of the posterior.