You have classification data with classes Y ∈ {+1, −1} and features Fi ∈ {+1, −1} for i ∈ {1, . . . , K}. Say you duplicate each feature, so now each example has 2K features, with F_K+i = F_i for i ∈ {1, . . . , K}. Compare the original feature set with the doubled one and indicate whether the below statements are true or false for Na¨ıve Bayes. You may assume that in the case of ties, class +1 is always chosen. Assume that there are equal numbers of training examples in each class.

(i) The test accuracy could be higher with the original features. 

(ii) The test accuracy could be higher with the doubled features. 

(iii) The test accuracy will be the same with either feature set. 

(iv) On a given training instance, the conditional probability P(Y |F₁, . . .) on a training instance could be more extreme (i.e. closer to 0 or 1) with the original features. 

(v) On a given training instance, the conditional probability P(Y |F₁, . . .) on a training instance could be more extreme (i.e. closer to 0 or 1) with the doubled features. 

(vi) On a given training instance, the conditional probability P(Y |F₁, . . .) on a training instance will be the same with either feature set.