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Combining the Gaussian likelihood with the prior $\theta \sim N_p(0,c^2{I}_p)$ gives the posterior precision and meanThus the posterior in the Gaussian normal-mean shrinkage Bayes estimator isThe Bayes estimator under squared error loss is the posterior mean, soIts posterior expected loss is the trace of the posterior covariance and does not depend on $X$. Therefore its Bayes risk is

For $0<\pi <1$, let ${\delta }_{\pi }$ be the Bayes classifier with class-zero prior $\pi$, and define its two class-conditional error probabilities byWriting the likelihood ratio as $L=f_1/f_0$, the rule chooses class one whenThe Gaussian likelihood-ratio level sets have probability zero, so dominated convergence shows that $e_0$ and $e_1$ depend continuously on $\pi$. As $\pi \downarrow 0$, the threshold tends to zero and the classifier chooses class one almost surely, givingAs $\pi \uparrow 1$, it chooses class zero almost surely, giving the opposite limit $(0,1)$. The intermediate value theorem therefore provides ${\pi }^{\ast }\in (0,1)$ such that

The rule ${\delta }_{{\pi }^{\ast }}$ is an equalizer rule with worst-case risk $r$. For any classifier $\delta$,because ${\delta }_{{\pi }^{\ast }}$ minimizes the integrated risk for its prior. Hence ${\delta }_{{\pi }^{\ast }}$ is a minimax decision rule, as summarized by the minimax Gaussian Bayes classifier from equal class errors.

The prior is indeed least favorable. Its Bayes risk is $r$, while for any other prior $q$ the Bayes risk is at most the integrated risk of the same equalizer rule ${\delta }_{{\pi }^{\ast }}$, namelyThus no prior has larger Bayes risk.

The conditional class probability is $p_k(x)=\mathbb{P}(Y=k\mid X=x)$. A Bayes classifier choosesand its Bayes risk isA sequence of classifiers is consistent when $R(h_n)\to R_{\mathrm{Bayes}}$ as $n\to \infty$, with convergence interpreted in probability or in expectation according to whether the training sample is conditioned upon.

By the Cauchy-Schwarz inequality, part (b), and part (e),This bound tends uniformly to zero ifThe plug-in classifier excess-risk bound then shows that the misclassification risk of the nearest-neighbour classifier converges to the Bayes risk.

A sequence of classifiers $h_n$ is risk-consistent when its classification risk converges to the Bayes risk as the training sample size tends to infinity.

For a known mean-zero noise law with finite variance ${\sigma }^2$, observing only $Y=a+ϵ$ with unrestricted $a\in \mathbb{R}$ has minimax risk ${\sigma }^2$ under squared-error loss. The observation itself attains this risk. For the lower bound, use a flat prior distribution on $[-A,A]$ and condition an easier experiment on $|ϵ|\le M$. Away from the prior endpoints, the Bayesian posterior noise has its original truncated law. Let $A\to \infty$ and then $M\to \infty$ in the resulting Bayes risk lower bound. No normal distribution or density for the noise is required.