Classification
Classification is the problem of guessing to which class it belongs(often represented by the symbol $\omega$) given a data point. For this problem, it is quite common to calculate the posterior probability of a data point for each class and classifying that point as the group with the largest probability.
Given this definition, a classifier can be defined in the following ways:
- $p(\omega_1 | x) > p(\omega_2 | x)$
- $p(\omega_1 | x) - p(\omega_2 | x) > 0$
- $\frac{p(\omega_1 | x)}{p(\omega_2 | x)} > 1$
- $ln(p(\omega_1 | x)) - ln(p(\omega_2 | x)) > 0$
However, since the posterior is hard to estimate in most cases, it is quite common to use the Bayes’ theorem. It maps to classification like so:
$$ p(\omega | x) = \frac{p(x|\omega)p(\omega)}{p(x)} $$
In this formula the terms are:
- $p(x | \omega)$ : The distribution of the class $\omega$
- $p(\omega|x)$ : Class conditional
- $p(\omega)$ : Class prior
- $p(x)$ : Data distribution
How is class distribution calculated
It is guessed using contextual information on the data. More often than not, it is a gaussian distribution.