Let's say we have a dataset that consists of two classes (i.e. "y+" &"y-") in the target column. For a particular feature the probability of both the
classes (i.e. p(y+) & p(y-)) are 0.5 respectively. What will be the entropy of the feature?

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Accepted Answer

⚹ To calculate the entropy of a feature in a dataset for a binary classification problem where the probabilities of each class are equal (p(y+) = p(y-) = 0.5), we use the formula for entropy in the context of information theory ⚹ ⚹ The entropy $ H $ of a random variable is a measure of the amount of uncertainty or impurity in the dataset and is given by the formula: ⚹ ⚹ $ H(Y) = -\sum_{i=1}^{n} p(y_i) \log_2 p(y_i) $ ⚹ ⚹ In this case, since we have two classes with equal probability, the formula simplifies to: ⚹ ⚹ $ H(Y) = -[p(y+) \log_2 p(y+) + p(y-) \log_2 p(y-)] $ ⚹ ⚹ Substituting the given probabilities into the formula, we get: ⚹ ⚹ $ H(Y) = -[0.5 \log_2 0.5 + 0.5 \log_2 0.5] $ ⚹ ⚹ $ H(Y) = -[0.5 \times (-1) + 0.5 \times (-1)] $ ⚹ ⚹ $ H(Y) = 1 $ ⚹ ⚹ Therefore, the entropy of the feature with equal class probabilities of 0.5 is 1 bit. This indicates maximum uncertainty or impurity, as the classes are equally likely. ⚹

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