The Prior Overrules the Peak

A one-dimensional Gaussian generative classifier with a shared variance:

class $k$	prior $\pi_k$	mean $\mu_k$	variance
0	0.9	0	1
1	0.1	2	1

Classify $x = 2$ with $\hat y = \arg\max_k ; \pi_k , N(x;, \mu_k,, 1)$ .

Undecidable. Without the normalizing constant $1/\sqrt{2\pi\sigma^2}$ these are not probabilities and cannot be compared.

Class 0. The prior's 9-to-1 edge beats the likelihood's $e^{2} \approx 7.4$ : once the shared constant cancels, $0.9 \times 0.135 = 0.122$ against $0.1 \times 1 = 0.100$ .

Class 1. $x = 2$ sits exactly on $\mu_1$ and two standard deviations from $\mu_0$ , so it belongs to the nearer class.

Class 1. The likelihood ratio is 7.4 to 1, and the priors only break ties when the two densities are equal.