Remark on the asymptotics of the likelihood ratio and the K.-L. divergence

The problem.

Let f, g, h be three densities and suppose that, x_i \sim h, i \in \mathbb{N}, independently. What happens to the likelihood ratio

\prod_{i=1}^n \frac{f(x_i)}{g(x_i)}

as n\rightarrow \infty?

Clearly, it depends. If h = g \not = f, then

\prod_{i=1}^n \frac{f(x_i)}{g(x_i)} \rightarrow 0

almost surely at an exponential rate. More generally, if h is closer to g than to f, in some sense, we’d expect that \prod_{i=1}^n \frac{f(x_i)}{g(x_i)} \rightarrow 0. Such a measure of “closeness” of “divergence” between probability distributions is given by the Kullback-Leibler divergence

D_{KL}(f, g) = \int f \log\frac{f}{g}.

It can be verified that D_{KL}(f,g) \geq 0 with equality if and only if f=g, and that

D_{KL}(h,g) < D_{KL}(h,f) \Longrightarrow \prod_{i=1}^n \frac{f(x_i)}{g(x_i)} \rightarrow 0 \qquad (1)

almost surely at an exponential rate. Thus the K.L.-divergence can be used to solve our problem.

Better measures of divergence?

There are other measures of divergence that can determine the asymptotic behavior of the likelihood ratio as in (1) (e.g. the discrete distance). However, in this note, I give conditions under which the K.-L. divergence is, up to topological equivalence, the “best” measure of divergence.Read More »

Sudden jumps and almost surely slow convergence

Estimating a mean can be an arduous task. Observe, for instance, the “sudden jumps” in trajectories of the Monte Carlo estimate \overline X_n of \mathbb{E}X, where \mathbb{E}X^2 = \infty.

sudden jumpsProposition.
Let X_1, X_2, X_3, \dots be a sequence of i.i.d. and integrable random variables, and let \overline X_k = \frac{1}{k}\sum_{i=1}^k X_i. For all M > 0 we have that

P\left( \left| \overline X_{n} - \overline X_{n-1}\right| > Mn^{-1/2} \;\; \text{ i.o.} \right)

is 1 or 0 according to \text{Var}(X_n) =\infty or \text{Var}(X_n) < \infty.

Corollary.
If \text{Var}(X_k) = \infty, then \overline X_n converges towards \mathbb{E}X_k at a rate almost surely slower than \frac{1}{\sqrt{n}} :

\left|\overline X_n - \mathbb{E}X_n \right| \not = \mathcal{O}\left(n^{-1/2}\right) \quad \text{a.s.}

PDF proof.

 

Many thanks to Xi’an who wondered about the occurence of these jumps.

Fair unfavorable games

Fair gambling games should be… fair, right?!

Félix Locas, here at UQAM, introduced me to “fair but unfavorable” gambling games of the following type:

  • you pay \mu to draw a random positive gain X such that \mathbb{E}X = \mu;
  • you can play for as long as you want.

Precisely, we have a sequence X_1, X_2, X_3, \dots of independent and identically distributed random variables representing your gain (minus \mu) at the kth game. Thus X_k \in [-\mu, \infty) and \mathbb{E}X_k = 0.

The surprising fact is that there exists such “fair” gambling games for which the total gain S_n = \sum_{i=1}^n X_i tends to -\infty in probability. More precisely (and even worse), there are games for which

P\left( S_n < -\frac{(1-\varepsilon)n}{(\log n)^{a} }\right) \rightarrow 1

for all small a > 0. (See W. Feller, Note on the law of large number and “fair” games, 1945.)Read More »