Mathematical Statistics II · Bayesian inference and decision rules
A categorical probability vector (P₁,P₂,P₃) has a Dirichlet prior with parameters (1,2,3)
Problem
A categorical probability vector \((P_1,P_2,P_3)\) has a Dirichlet prior with parameters \((1,2,3)\). Conditional on this vector, observations are independent categorical draws. The observed counts are \((3,0,2)\). For positive parameters \((a_1,a_2,a_3)\), the Dirichlet density on the simplex is proportional to \[ p_1^{a_1-1}p_2^{a_2-1}p_3^{a_3-1}. \] Find the posterior parameters. Now define the grouped probability and within-group fraction \[ R=P_1+P_2,\qquad W=\frac{P_1}{P_1+P_2}. \] Find the posterior distributions of \(R\) and \(W\), and determine whether they are posterior-independent. Calculate the posterior predictive probability that the next observation belongs to category 1 or 2. Show that updating only the grouped counts gives the same posterior for \(R\). Does that mean the detailed counts provide no information about \(W\)?
Hint
Add the observed category counts to the prior exponents.
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