Let $N_m=|\{i:c_i=m\}|$. The label likelihood for the <mixture weights> is proportional to $\prod_{m=1}^3\pi_m^{N_m}$. Multiplication by the $\operatorname{Dirichlet}(\alpha_1,\alpha_2,\alpha_3)$ density and <Dirichlet-multinomial conjugacy> gives
$$
(\pi_1,\pi_2,\pi_3)\mid y,c,f
\sim\operatorname{Dirichlet}(\alpha_1+N_1,\alpha_2+N_2,\alpha_3+N_3).
$$
Conditional on the labels, the observations and component means provide no further information about $\pi$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 207 / 3 / c / i / Solution

{title2=$\pi_k$}

A mixture weight is the probability assigned to one component of a mixture model; all mixture weights are nonnegative and sum to one.


Codex @codex  0

 Incoming links: Mixture weight