Probabilistic Modelling I was reading Hal Daume's ML notes and it decomposes a training example into its features and uses a mean and variance for each feature in an example. 


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*It asks , for a given class 'k' when can we assume all features have equal variance ? 


It seems that if the class 'k' is independent of those features, all of them should have equal variance. Am I correct or is there a better answer


*When can we assume a feature i has equal variance for each class ?


When the class label is independent of feature i, then that is true.
Am I correct ?
 A: 
It seems that if the class 'k' is independent of those features, all of them should have equal variance. Am I correct or is there a better answer

Here's a counterexample showing why this can't be true. Imagine some set of features with unequal variances. Now, generate class labels randomly from some distribution that doesn't depend on the features (e.g. flip a coin). Class is therefore independent of the features. However, the variance of the features hasn't changed--it's still unequal. Furthermore, under these conditions, the variance given any particular class is equal to the overall variance (see answer to the second question, below). So, the class-conditional variances are also unequal.

When can we assume a feature i has equal variance for each class? When the class label is independent of feature i, then that is true. Am I correct?

That's correct. Say we have feature $X_i$ and class $C$. If they're independent then $p(X_i \mid C) = p(X_i)$. That is, the distribution of the feature given the class is equal to the marginal distribution of the feature. Therefore, the variance is always the same, no matter the class.
