Why is the entire area of a join probability distribution considered when it comes to calculating misclassification?

In the image given below, I do not understand a few things

1) Why is an entire area colored to signify misclassification? For the given decision boundary, only the points between $$x_0$$ and the decision boundary signify misclassification right? It's supposed to be only a set of points on the x-axis, not an area.

2) Why is the green area with $$x < x_0$$ a misclassification? It's classified as $$C_1$$ and it is supposed to be $$C_1$$ right?

3) Similarly, why is the blue area a misclassification? Any $$x >$$ the decision boundary belongs to $$C_2$$ and is also classified as such...

• It would be interesting to know your views on the answer and help improve it, if needed. Aug 18, 2019 at 15:29

The misclassifications that could arise if $$\hat{x}$$ is used as decision boundary are:

a) Classifying a point as $$C_2$$ when actually it was $$C_1$$ -- which will only happen when $$x > \hat{x}$$ as only the points greater than $$\hat{x}$$ are being classified as $$C_2$$.

b) Classifying a point as $$C_1$$ when actually it was $$C_2$$ -- which will only happen when $$x < \hat{x}$$ as only the points less than $$\hat{x}$$ are being classified to be from class $$C_1$$.

2) Why is the green area with x < x0 a misclassification?

If we label a point -- the $$x$$ which is from the interval on the horizontal axis which corresponds to the green area -- which is a probability as being from either class then there is a good chance that it could be misclassified. This is because both the joint distributions' curves are above the axis for that interval on the axis and have some area (probability) for that interval on the horizontal axis.

There is a positive probability that a point that was drawn from the interval (a subset of the sample space, an event), which corresponds to the green area, was drawn from either of the two joint distributions or belongs to either of the two classes $$C_1$$ and $$C_2$$. The probability is the area under the curve.

It's classified as C1 and it is supposed to be C1 right?

The same data point could well be generated from multiple different probability distributions. Here, you have an illustration of two probability distributions for which the author is trying to find the optimal decision boundary that will minimize the misclassification "region".

The point is not supposed to be from $$C_1$$, even if the density of the joint prob. distribution of $$(x, C_1)$$ has higher values. It could well be from $$C_2$$. And, when one classifies the same data point as being from class $$C_1$$ and make an error. This is why the whole area under the curve $$p(x, C_2)$$ has been painted green -- which means that there is some probability that a point could be from the distribution $$p(x, C_2)$$ and if we blindly label all the points to be from $$C_1$$ will certainly lead to some misclassifications.