After Boaz posted on the mother of all inequalities, it seemed about the right time to get around to the next series of posts on entropy optimality. The approach is the same as before, but now we consider entropy optimality on a path space. After finding an appropriate entropy-maximizer, the Brascamp-Lieb inequality will admit a gorgeous one-line proof. Our argument is taken from the beautiful paper of Lehec.
For simplicity, we start first with an entropy optimization on a discrete path space. Then we move on to Brownian motion.
1.1 Entropy optimality on discrete path spaces
Consider a finite state space and a transition kernel . Also fix some time .
Let denote the space of all paths . There is a natural measure on coming from the transition kernel:
Now suppose we are given a starting point , and a target distribution specified by a function scaled so that . If we let denote the law of , then this simply says that is a density with respect to . One should think about as the natural law at time (given ), and describes a perturbation of this law.
Let us finally define the set of all measures on that start at and end at , i.e. those measures satisfying
and for every ,
One should verify that, like many times before, we are minimizing the relative entropy over a polytope.
One can think of the optimization as simply computing the most likely way for a mass of particles sitting at to end up in the distribution at time .
where is the heat semigroup of our chain , i.e.
Let denote the time-inhomogeneous chain with transition kernels and and let denote the law of the random path . We will now verify that is the optimal solution to (1).
We first need to confirm that , i.e. that has law . To this end, we will verify inductively that has law . For , this follows by definition. For the inductive step:
We have confirmed that . Let us now verify its optimality by writing
where the final equality uses the fact we just proved: has law . Continuing, we have
where the final inequality uses the definition of in (2). The latter quantity is precisely by the chain rule for relative entropy.
We conclude that
and from this one immediately concludes that . Indeed, for any measure , we must have . This follows because is the law of the endpoint of a path drawn from and is the law of the endpoint of a path drawn from . The relative entropy between the endpoints is certainly less than along the entire path. (This intuitive fact can again be proved via the chain rule for relative entropy by conditioning on the endpoint of the path.)
1.2. The Brownian version
Let us now do the same thing for processes driven by Brownian motion in . Let be a Brownian motion with . Let be the standard Gaussian measure and recall that has law .
We recall that if we have two measures and on such that is absolutely continuous with respect to , we define the relative entropy
where denotes the drift. We require that is progressively measurable, i.e. that the law of is determined by the past up to time , and that . Note that we can write such a process in differential notation as
Fix a smooth density with . In analogy with the discrete setting, let us use to denote the set of processes that can be realized in the form (4) and such that and has law .
As in the discrete setting, this problem has a unique optimal solution (in the sense of stochastic processes). Here is the main result.
Theorem 1 (Föllmer) If is the optimal solution to (5), then
Just as for the discrete case, one should think of this as asserting that the optimal process only uses as much entropy as is needed for the difference in laws at the endpoint. The RHS should be thought of as an integral over the expected relative entropy generated at time (just as in the chain rule expression (3)).
The reason for the quadratic term is the usual relative entropy approximation for infinitesimal perturbations. For instance, consider the relative entropy between a binary random variable with expected value and a binary random variable with expected value :
I am going to delay the proof of Theorem 1 to the next post because doing it in an elementary way will require some discussion of Ito calculus. For now, let us prove the following.
Proof: The proof will be somewhat informal. It can be done easily using Girsanov’s theorem, but we try to keep the presentation here elementary and in correspondence with the discrete version above.
Note that has the law of a standard -dimensional of covariance .
If is an -dimensional Gaussian with covariance and , then
where the latter expectation is understood to be conditioned on the past up to time .
In particular, plugging this into (6), we have
The proof is taken directly from Lehec. We will use the entropic formulation of Brascamp-Lieb due to Carlen and Cordero-Erausquin.
By (8), we have for all :
The latter equality uses the fact that each is an orthogonal projection.
Let denote a standard Gaussian on , and let denote a standard Gaussian on for each .
Theorem 3 (Carlen & Cordero-Erausquin version of Brascamp-Lieb) For any random vector , it holds that
Proof: Let with denote the entropy-optimal drift process such that has the law of . Then by Theorem 1,
where the latter inequality uses Lemma 2 and the fact that has law .