Maximum Likelihood vs Maximum A Posteriori in action

Specify for the following figure, how the Maximum likelihood decision rule applied to to the red class and to the blue class, respectively, will decide

• ML Classifier chooses the class with the the highest Likelihood $p(x|s)$
• From Graph it is visible that
  • $0<X<11$, $p(x|s=2)$ has the highest value. ⇒ Classification is RED
  • $11<X<17$, $p(x|s=2)$ has the highest value. ⇒ Classification is BLUE

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Now let the red class be only $1/3$ as frequent as the blue class

$$\mathrm{P}(s=2)=\frac{1}{3}\cdot \mathrm{P}(s=1)$$

Specify depending on the feature $x$, to which class the ML classifier, and to which class the MAP classifier decides.

MAP uses the Prior which can be calculated as follows

Given: $P(s=2)=\frac{1}{3}\cdot P(s=1)$

From the Prior the decision boundary can be expected to shift with the new decision boundary $x\ast$ The Decision Boundary in MAP is where the Posterior Probabilities are equal.

• $x<x\ast$ i.e. $3\cdot p(x|s=1)<p(x|s=2)$, MAP chooses red class
• $x>x\ast$ i.e. $3\cdot p(x|s=1)>p(x|s=2)$, MAP chooses blue class

The exact value of $x\ast$ is where $3p(x|s=1)=p(x|s=2)$, which appears to be around $x=11.9$ in the graph.