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This is a question from a data mining exam:

You are working with a dataset that contains descriptions of toxic and non-toxic substances.

The dataset, which consists of 1000 examples from each of the two classes, is described in terms of a class label and a number of attributes. The dataset is sorted so that the 1000 toxic examples come first, followed by the 1000 non-toxic examples. Someone tells you that they have confirmed that, for this data set, the conditional probability that is gained from knowledge about a specific attribute X is not different from the prior class probability. Assume that they are correct.

The question is to state whether the following statements are correct or incorrect:

g) For each example, the value of attribute X is sampled from {“true”, “false”} with a uniform distribution. h) For each example, the value of attribute X is sampled from {“true”, “false”} such that the probability of selecting “true” is 0.75, while the probability of selecting “false” is 0.25.

both g and h were solved as correct independently but I find this hard to understand. Can anyone explain?

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    $\begingroup$ The information seems pretty garbled to me. E.g., "The conditional probability" that what? And toxic or nontoxic substances are going to have their own attributes: the attributes would not be assigned by some sampling process. $\endgroup$
    – rolando2
    Commented Sep 29, 2012 at 20:53

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Since the sizes of classes are equal, prior class probability is {0.5,0.5}. If the conditional probability that is gained from knowledge about a specific attribute X is not different from the prior class probability - that means that this attribute values have the same mean (or probabilities for nominal attribute) for each class. Uniform discrete distribution with two events has equal "True" and "False" probabilities: {0.5,0.5}. The g) and h) cannot be satisfied simultaneously. Thus I think the question was whether the following statements can be correct or incorrect. The both can be correct: g) is correct when there are 1000 True and 1000 False (uniform distribution), such that class 1 has 500 True and 500 False, and class 2 is the same; h) is correct when there are 1500 True and 500 False (prespecified distribution) such that class 1 has 750 True and 250 False and class 2 is the same.

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  • $\begingroup$ Yes the question was whether the following statements can be correct or incorrect. But the next statement is: i) For each example, the value of attribute X is sampled from {“true”, “false”} such that if the example is toxic, then the probability of selecting “true” is 0.75, while the probability of selecting “false” is 0.25, and if the example is non-toxic, the probability of selecting “true” is 0.25, while the probability of selecting “false” is 0.75. False .. Why is it not ok to change probabilities in this case? $\endgroup$
    – Mona Rifaat
    Commented Sep 30, 2012 at 13:22
  • $\begingroup$ Becouse in this case the conditional probability will be different from prior probability: P(toxic|X=True)=0.75; P(toxic|X=False)=0.25 against prior P(toxic)=0.5 $\endgroup$
    – O_Devinyak
    Commented Sep 30, 2012 at 20:18
  • $\begingroup$ The conditional and prior probabilities equivalence is observed when both classes have the same number of __True__s and __False__s. Statements g) and h) agree with this demand, but i) does not. In i) class 1 (toxic) has 1000*0.75=750 True and 250 False, but class 2 (non-toxic) is not the same. Class 2 has vice versa: 250 True and 750 False. $\endgroup$
    – O_Devinyak
    Commented Sep 30, 2012 at 20:25

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