For a decision $x$ in the <unit ball>, <conditional expectation> under the posterior gives
$$
\mathbb E[(x^T\beta)^2\mid Y]
=x^T\mathbb E[\beta\beta^T\mid Y]x
=x^T(C+mm^T)x.
$$
The symmetric <positive semidefinite matrix> $C+mm^T$ has a <Rayleigh quotient> maximized over $\lVert x\rVert\leq1$ by any unit <eigenvector> corresponding to its largest <eigenvalue>. <Bayes decision rule> therefore chooses any such eigenvector for the stated utility.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 1 / c / Solution

{c}

A Bayes decision rule maximizes posterior expected utility, equivalently minimizes posterior expected loss, separately at each observed data value.


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