Information projection

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


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

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

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

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


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

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