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Difference between training and test data distribution


I think you're confusing the underlying distribution from which both training and test distributions are drawn, with the distributions of the specific train and test draws. Unless the underlying distribution is eg time-sensitive, changed during the time between eg drawing the training and the testing samples, the underlying distribution is identical each time. The goal in learning a machine learning model is typically not to learn the training distribution, but to learn the latent underlying distribution, of which the training distribution is only a sample. Of course, you cannot actually see the underlying distribution, but eg, if you only really cared about learning the training samples, you could simply memorize the training samples in a lookup table, end of story. In reality, you are using the training sample as a proxy into the underlying distribution.

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