Information projection

In information theory, the information projection or I-projection of a probability distribution q onto a set of distributions P is

${\displaystyle p^{*}={\underset {p\in P}{\arg \min }}\operatorname {D} _{\mathrm {KL} }(p||q)}$.

where ${\displaystyle D_{\mathrm {KL} }}$ is the Kullback–Leibler divergence from q to p. Viewing the Kullback–Leibler divergence as a measure of distance, the I-projection ${\displaystyle p^{*}}$ 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: ^[1]

${\displaystyle \operatorname {D} _{\mathrm {KL} }(p||q)\geq \operatorname {D} _{\mathrm {KL} }(p||p^{*})+\operatorname {D} _{\mathrm {KL} }(p^{*}||q)}$.

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 ${\displaystyle \operatorname {D} _{\mathrm {KL} }(p||q)\geq 0}$ 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

${\displaystyle p^{*}={\underset {p\in P}{\arg \min }}\operatorname {D} _{\mathrm {KL} }(q||p)}$.

Since the KL divergence is not symmetric in its arguments, the I-projection and the M-projection will exhibit different behavior. For I-projection, ${\displaystyle p(x)}$ will typically under-estimate the support of ${\displaystyle q(x)}$ and will lock onto one of its modes. This is due to ${\displaystyle p(x)=0}$, whenever ${\displaystyle q(x)=0}$ to make sure KL divergence stays finite. For M-projection, ${\displaystyle p(x)}$ will typically over-estimate the support of ${\displaystyle q(x)}$. This is due to ${\displaystyle p(x)>0}$ whenever ${\displaystyle q(x)>0}$ 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. ^[2]

See also

• Sanov's theorem

References

1. Cover, Thomas M.; Thomas, Joy A. (2006). Elements of Information Theory (2 ed.). Hoboken, New Jersey: Wiley Interscience. p. 367 (Theorem 11.6.1).
2. Nielsen, Frank (2018). "What is... an information projection?" (PDF). 65 (3). AMS: 321–324.{{cite journal}}: Cite journal requires |journal= (help)

• K. Murphy, "Machine Learning: a Probabilistic Perspective", The MIT Press, 2012.