4. Gibbs estimators
4.1. General framework 
 
 
4.2. Dichotomic histograms 
 
 
4.3. Mathematical framework for density estimation 
 
 
4.4. Main oracle inequality 
 
 
4.5. Checking the accuracy of the bounds on the Gaussian shift model 
 
 
4.6. Application to adaptive classification 
 
 
4.6.1. Randomized classification rule, general case 
 
 
4.6.2. Randomized classification rule for “non-ambiguous” classification problems 
 
 
4.6.3. Deterministic classification rule 
 
 
4.6.4. Counter example 
 
 
 
 
 
 
4.7. Two stage adaptive least square regression 
 
 
4.8. One stage piecewise constant regression 
 
 
4.9. Some abstract inference problem 
 
 
4.10. Another type of bound