Mixtures versus K-means
Two analysts cluster the same two-dimensional data. One fits K-means. The other fits a three-component Gaussian mixture with unrestricted component covariance matrices.
Explain two important differences between the fitted models. For each difference, explain its practical consequence.
Write the posterior probability that an observation belongs to mixture component k, defining every quantity.
If the fitted mixture has three components, must there be three biological populations? Explain.
One distinction is hard nearest-centre assignment against probabilistic membership from a density model. The mixture reports a responsibility for an observation that sits between components, so an overlap is visible as a probability rather than as a forced label. Another distinction is Euclidean nearest-centre geometry against component-specific covariances and mixing weights. The mixture can give the components different orientations, spreads, and prevalences, and the maximum-a-posteriori boundary can be curved.
With component probability \pi_k, mean \boldsymbol\mu_k, and covariance \Sigma_k,
\tau_k(\mathbf{x}) =\frac{\pi_k\,\varphi_p(\mathbf{x};\boldsymbol\mu_k,\Sigma_k)} {\sum_h\pi_h\,\varphi_p(\mathbf{x};\boldsymbol\mu_h,\Sigma_h)}, \qquad \pi_k\ge 0, \qquad \sum_k\pi_k=1.
The denominator is the mixture density. The probabilities sum to one.
- Components are terms in an approximation to the observed density. Several components can represent one non-normal population, and overlap or misspecification can obscure distinct populations. A biological reading needs evidence beyond the component count.