
## And Denote It S.

Recall, that in Bayesian decision theory [42] the classification decision for instance X at equal costs of mistake is made based on maximal a posteriori probability, which for a class ciis:

$$p(c_{i}|\mathbf{X})={\frac{P(c_{i})p(\mathbf{X}|c_{i})}{p(\mathbf{X})}},$$
$$\left(2\right)$$

$\overset{\star}{\underset{\star}{\mathrm{d}}}$ . 
p(X), (2)
where p(X) is an evidence of X, which is constant for all the classes ci.

As first presented by Kelly et al [76], concept drift may occur in thee ways.

1. Class priors P(c) might change over time.

2. The distributions of one or several classes p(X|c) might change. 3. The posterior distributions of the class memberships p(c|X) might change.

Note, that the distributions p(X|c) might change in such a way that the class membership is not affected (e.g. symmetric movement to opposite directions).

Sometimes change in p(X|c) (independently whether it affects p(c|X) or not) is referred as virtual drift and change in p(c|X) is referred as real drift [153]. We argue, that from practical point of view it is not essential whether the drift is real or virtual, since p(c|X) depends on p(X|c) as in Equation
(2). In this thesis now on we do not make a distinction between the real and virtual drifts.

## 2 How Do Concept Drift Learners Work?

Following the framework, which was set-up in the previous section, the learner should provide the most accurate generalization for the data at time t + 1. In order to build such a learner, four main design sub-problems need to be solved.

A.1 Future assumption: a designer needs to make an assumption about the future data source St+1.

A.2 Change type: a designer needs to identify possible change patterns. A.3 Learner adaptivity: based on the change type and the future assumption a designer chooses the mechanisms which make the learner adaptive.

A.4 Model selection: a designer needs a criterion to choose a particular parametrization of the selected learner at every time step (e.g. the weights for an ensemble members, the window size for variable window method).

All these sub-problems are the choices to be made when designing a learner. In Figure 3 we depict a positioning of each design sub-problem within the established learning framework. In the next subsections we discuss each of the design sub-problems individually.