Editing Total variation (section)

====Total variation of probability measures====
{{unreferenced section|date=May 2012}}
{{main|Total variation distance of probability measures}}
The total variation of any [[probability measure]] is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are [[probability measure]]s, the '''[[total variation distance of probability measures]]''' can be defined as <math>\| \mu - \nu \|</math> where the norm is the total variation norm of signed measures.  Using the property that <math>(\mu-\nu)(X)=0</math>, we eventually arrive at the equivalent definition

:<math>\|\mu-\nu\| = |\mu-\nu|(X)=2 \sup\left\{\,\left|\mu(A)-\nu(A)\right| : A\in \Sigma\,\right\}</math>

and its values are non-trivial. The factor <math>2</math> above is usually dropped (as is the convention in the article [[total variation distance of probability measures]]). Informally, this is the largest possible difference between the probabilities that the two [[probability distribution]]s can assign to the same event. For a [[categorical distribution]] it is possible to write the total variation distance as follows

:<math>\delta(\mu,\nu) = \sum_x \left| \mu(x) - \nu(x) \right|\;.</math>

It may also be normalized to values in  <math>[0, 1]</math> by halving the previous definition as follows

:<math>\delta(\mu,\nu) = \frac{1}{2}\sum_x \left| \mu(x) - \nu(x) \right|</math><ref>{{cite web|last1=Gibbs|first1=Alison|author2=Francis Edward Su|title=On Choosing and Bounding Probability Metrics|url=https://www.math.hmc.edu/~su/papers.dir/metrics.pdf|access-date=8 April 2017|pages=7|date=2002}}</ref>