Mixture responsibility (${\tau }_{ij}=\mathbb{P}(Z_i=j\mid y_i)$)

A mixture responsibility is the posterior probability of a component label given an observation and current parameters: ${\tau }_{ij}={\pi }_j{f}_j(y_i)/({\sum }_l{\pi }_l{f}_l(y_i))$. Each observation's responsibilities sum to one. The expectation-maximization algorithm uses these soft assignments rather than fixed hard cluster labels.