My first post is about a beautiful hypothesis testing inequality due to Le Cam [1, Chapter 16, Section 4]. Here is the setting. There is an unknown distribution on which we have two hypotheses. The first hypothesis is and the second is where and denote classes of probability measures. Given i.i.d data from , the goal is to figure out which of the two hypothesis is the right one.
As one gets more and more data points i.e., as increases, this problem intuitively should get easier. Le Cam’s inequality quantifies this by asserting that the error in this hypothesis testing problem decreases exponentially in . The precise statement and a proof sketch of this inequality is given below. This inequality has many applications; a particularly interesting one is its role in establishing frequentist convergence rates of posterior distributions; this will be outlined in a future post.
We first need to specify the notion of error in hypothesis testing. Given a test (which is simply a -valued function of the data), its type I error is while its type II error is . For ease of notation, let me write for where is the -fold product of with itself. We define the error of a test to be the sum of its type I and type II errors. The smallest error in testing against is therefore
Le Cam’s inequality states that decreases exponentially in . The rate of this decrease depends on how close the classes and are. This closeness is measured by the so-called Hellinger affinity between and defined as
where and denote the densities of and with respect to a common dominating measure . measures closeness between and with large values indicating closeness. Hence the name affinity.
The precise statement of Le Cam’s inequality is
where denotes the convex hull of i.e., the class of all finite linear combinations of probability measures in . As a consequence, we have that decreases exponentially in as long as is bounded from above by a constant strictly smaller than one. It must be noted that (1) is a finite sample result i.e., it is allowed that and depend on . It might seem strange that (1) involves the convex hulls of and instead of and themselves. This can be explained perhaps by the simple observation: .
Let me now provide a sketch of the proof of (1). Let and similarly define . Rewrite as
The first idea is to interchange the inf and sup above. This is formally done via a minimax theorem. See this note by David Pollard for a clear exposition of a minimax theorem and its application to this particular example. This gives
Minimax theorems require the domains (on which the inf and sup are taken) to be convex which is why one gets convex hulls above. The advantage of using the minimax theorem is that the infimum above can be easily evaluated explicitly. This is because
We thus get
The quantity also measures closeness between and . It is actually called the total variation affinity between and . The trouble with it is that it is difficult to see how the right hand side above changes with . This is the reason why one switches to Hellinger affinity. The elementary inequality gives
The proof is now completed by showing that
This inequality is a crucial part of the proof. This is the reason why one works with Hellinger affinity as opposed to the total variation affinity although the latter is more directly related to . Let us see why (2) holds when . The general case can be tackled similarly.
Any probability measure in is of the form for some . Similarly, any probability measure in is of the form for some . The Hellinger affinity between and is therefore
Now if and , we can rewrite the above as
Fixing and looking at the inner integral above, the densities (as functions of ) involved are all convex combinations of densities in and respectively. Therefore the inner integral above is at most which means that the above quantity is atmost
By the same logic, . This proves (2) for . The general case is not much more difficult. This completes the sketch of the proof of (1).
 L. Le Cam. Asymptotic Methods in Statistical Decision Theory. Springer-Verlag, New York, 1986.