In information theory, the information projection or I-projection of a probability distribution q onto a set of distributions P is
p ∗ = arg min p ∈ P D K L ( p | | q ) {\displaystyle p^{*}={\underset {p\in P}{\arg \min }}\operatorname {D} _{\mathrm {KL} }(p||q)} . where D K L {\displaystyle D_{\mathrm {KL} }} is the Kullback–Leibler divergence from q to p. Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection p ∗ {\displaystyle p^{*}} 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:
D K L ( p | | q ) ≥ D K L ( p | | p ∗ ) + D K L ( p ∗ | | q ) {\displaystyle \operatorname {D} _{\mathrm {KL} }(p||q)\geq \operatorname {D} _{\mathrm {KL} }(p||p^{*})+\operatorname {D} _{\mathrm {KL} }(p^{*}||q)} . 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 D K L ( p | | q ) ≥ 0 {\displaystyle \operatorname {D} _{\mathrm {KL} }(p||q)\geq 0} 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
p ∗ = arg min p ∈ P D K L ( q | | p ) {\displaystyle p^{*}={\underset {p\in P}{\arg \min }}\operatorname {D} _{\mathrm {KL} }(q||p)} . Since the KL divergence is not symmetric in its arguments, the I-projection and the M-projection will exhibit different behavior. For I-projection, p ( x ) {\displaystyle p(x)} will typically under-estimate the support of q ( x ) {\displaystyle q(x)} and will lock onto one of its modes. This is due to p ( x ) = 0 {\displaystyle p(x)=0} , whenever q ( x ) = 0 {\displaystyle q(x)=0} to make sure KL divergence stays finite. For M-projection, p ( x ) {\displaystyle p(x)} will typically over-estimate the support of q ( x ) {\displaystyle q(x)} . This is due to p ( x ) > 0 {\displaystyle p(x)>0} whenever q ( x ) > 0 {\displaystyle q(x)>0} to make sure KL divergence stays finite. The reverse I-projection plays a fundamental role in the construction of optimal e-variables.
The concept of information projection can be extended to arbitrary f-divergences and other divergences.
See also Sanov's theorem
References
K. Murphy, "Machine Learning: a Probabilistic Perspective", The MIT Press, 2012.