<Bayes theorem> turns the <categorical distribution> prior probabilities and the component <multivariate normal densities> into
$$
\mathbb P(c_i=m\mid y_i,\pi,f)
=\frac{\pi_m\phi(y_i;f_m,\sigma^2I_n)}
{\sum_{\ell=1}^3\pi_\ell\phi(y_i;f_\ell,\sigma^2I_n)},
\qquad m=1,2,3.
$$
These probabilities define the label update in the <Gibbs sampler>.

Solution

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

{title2=$\operatorname{Cat}(p_1,\ldots,p_k)$}
{wiki}

The categorical distribution describes one draw from finitely many categories with probabilities $p_1,\ldots,p_k$.


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 Incoming links: Categorical distribution