Information projection

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

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

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

[math]\displaystyle{ \operatorname{D}_{\mathrm{KL}}(p||q) \geq \operatorname{D}_{\mathrm{KL}}(p||p^*) + \operatorname{D}_{\mathrm{KL}}(p^*||q) }[/math].

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 [math]\displaystyle{ \operatorname{D}_{\mathrm{KL}}(p||q) \geq 0 }[/math] 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

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

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