where is the Kullback–Leibler divergence from q to p. Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection is the "closest" distribution to q of all the distributions in P.
The I-projection is useful in setting up information geometry, notably because of the following inequality, valid when P is convex:
This inequality can be interpreted as an information-geometric version of Pythagoras' triangle inequality theorem, where KL divergence is viewed as squared distance in a Euclidean space.
It is worthwhile to note that since and continuous in p, if P is closed and non-empty, then there exists at least one minimizer to the optimization problem framed above. Furthermore if P is convex, then the optimum distribution is unique.
The reverse I-projection also known as moment projection or M-projection is
Since the KL divergence is not symmetric in its arguments, the I-projection and the M-projection will exhibit different behavior. For I-projection, will typically under-estimate the support of and will lock onto one of its modes. This is due to , whenever to make sure KL divergence stays finite. For M-projection, will typically over-estimate the support of . This is due to whenever to make sure KL divergence stays finite.
The concept of information projection can be extended to arbitrary statistical f-divergences and other divergences.
- Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory (2 ed.). Hoboken, New Jersey: Wiley Interscience. pp. 367(theorem 11.6.1).
- Nielsen, Frank (2018). "What is... an information projection?" (PDF). 65 (3). AMS: 321–324.
- K. Murphy, "Machine Learning: a Probabilistic Perspective", The MIT Press, 2012.
- F. Nielsen, "What is... an information projection?", AMS Notices, (65) 3, pp. 321-324, 2018
|This probability-related article is a stub. You can help Wikipedia by expanding it.|