How to test two data sets from the same distribution?

Suppose I have two data sets, how can I tell that they are from two distributions? what's the difference of these two data sets?

This is actually very generic question emerged in machine learning.

Some intuitive ideas:
1. estimate the density of data samples for each data set. This method might be very weak. In reality, density estimation is a very difficult task compared with "simple classification". This approach is generally not application in reality.

2. Estimate the sufficient statistics of each data set. Like the mean, variance of each feature, the class conditional distribution. This can be interpreted as analogy in cognition. An analogy can be derived if the relationship between multiple symbols can be maintained. The problem is when can we conclude that the difference is large enough? It seems some hypothesis testing is required.

3. Transformation. A data set can be transformed into another data set. But how do you know the feature mapping? A more reasonable way is to enforce equivalence of sufficient statistics in a newly generated space.

4. Dimensionality reduction. Assume that two data set shared the same distribution on a fixed number of dimensions. By projecting the two data set into those dimensions, probably we can find some interesting patterns.

5. Learn the classification function. Use the decision function to measure the difference. this is quite related to transfer learning.

6. Any other ways?

OK. To endeavor this direction, where can we find the data set?

The approach of Kernel Maximum Mean discrepancy might be related.