Likelihood Ratio Approach to Classification

Let's continue from the previous post.

For the two-class case, the decision rule for $\bf x$ in class $\omega_1$ is

\[P({\bf x}|\omega_1)P(\omega_1) > P({\bf x}|\omega_2)P(\omega_2)\]
\[L_R({\bf x})=\frac{P({\bf x}|\omega_1)}{P({\bf x}|\omega_2)}>\frac{P(\omega_2)}{P(\omega_1)}=\tau_c\]
That is, the likelihood ratio of $\bf x$ belonging to class $\omega_1$ to $\omega_2$ is equal to the ratio of probability of class $\omega_1$ to $\omega_2$, called a threshold value $\tau_c$.

For the same, two-class case, let x a value from -8 to 8 in 0.1 stepping. The values of x that can be classified as class $\omega_1$  (blue) are

xl=-8:0.1:8;
w1=find(normpdf(xl,-1,1)./normpdf(xl,1,1) > 0.4/0.6);
h=bar(xl(w1),0.6*normpdf(xl(w1),-1,1),1); hold on;
set(h,'EdgeColor',get(h,'FaceColor'));
w2=setxor(1:length(xl),w1);
h=bar(xl(w2),0.4*normpdf(xl(w2),1,1),1,'r'); hold off;
set(h,'EdgeColor',get(h,'FaceColor'));
xlabel('x');

The last value of x that belongs to class $\omega_1$ is

>> xl(w1(end))

ans =

    0.2000

which is also at the decision boundary.