A tied observation keeps fractional Gaussian responsibilities

Assume AC. Take one observation $x=0\in \mathbb{R}$, two Gaussian components with means ${\mu }_1=-1$ and ${\mu }_2=1$, common variance ${\sigma }^2>0$, and mixture weights ${\pi }_1=1\mathrm{/}3$ and ${\pi }_2=2\mathrm{/}3$. Their component density is ${\varphi }_1(x;\mu ,{\sigma }^2)=(2\pi {\sigma }^2{)}^{-1\mathrm{/}2}\exp (-(x-\mu {)}^2\mathrm{/}(2{\sigma }^2))$, and the responsibility of component $k$ is ${\gamma }_k={\pi }_k{\varphi }_1(0;{\mu }_k,{\sigma }^2)\mathrm{/}{\sum }_{j=1}^2{\pi }_j{\varphi }_1(0;{\mu }_j,{\sigma }^2)$. The two squared distances from the observation to the component means are both $1$. For every $\sigma >0$, the responsibility vector is exactly $(1\mathrm{/}3,2\mathrm{/}3)$; therefore as $\sigma \downarrow 0$ it remains fractional rather than converging to $(1,0)$ or $(0,1)$.

Finite Gaussian mixture with positive-definite component covariances — the positive-weight Gaussian component formula and responsibility definition.

Equal-spherical mixture assignments and objective converge to k-means — tied nearest centers receive limiting responsibilities proportional to their fixed mixture weights.

Positive-definite multivariate Gaussian density is normalized — each component formula is a normalized Gaussian density.