Total Variation Distance of Probability Measures

The total variation distance is a way of measuring how different two probability distributions are from one another, also known as the statistical distance, statistical difference, or variational distance. It is defined as the largest possible gap between the probabilities that the two distributions assign to the same event, taken over every event that could be considered. Geometrically, when the two distributions are drawn as curves over the same space, the total variation distance equals half of the area lying between the two curves.