When is the mode of a stochastic process a better statistic than the mean?

Question

This is a soft question.

I've been interested in Onsager-Machlup theory recently. Essentially, the Onsager-Machlup function serves the role of a density but it can exist on non locally compact spaces.

Given a measure $\mu$ on metric space $(X,d)$, if there is a function $F$ on $X$ for which

$$\lim_{\varepsilon\to 0}\frac{\mu(B_\varepsilon(z_2))}{\mu(B_\varepsilon(z_1))}=\exp\left(F(z_2)-F(z_1)\right)$$

(and the limit always exists) then $F$ is called the Onsager-Machlup function for $\mu$. The minimizers of $F$ are often called the modes of $\mu$.

When considering a probability measure of some space of paths, if the minimizer exists then it is referred to as the "mode" of the stochastic process whose law is that measure.

I have a bit of a soft question -

When does the mode of a stochastic process capture its behavior better than its mean?

Answer 1

A "mode" of the Onsager-Machlup action functional identifies a locally most probable transition pathway between metastable states. If there is a single minimizer then mode and mean will be equally informative, but there may well be multiple local minima of the action functional, and then the mean does not tell you which are the relevant transition pathways.

Answer 2

Old question, but there is something more to say here I think. I will stick to the finite dimensional case, where the probability density function takes the place of the Onsanger Machlup functional.

The first observation is that the mode is more relevant than the mean when you’re interested in $L^p$ norms of the probability density function for high $p$. In fact it gets more relevant as $p$ increases, in a sense I will elaborate on below.

To make things more precise, I am referring to the fact that for $X$ a random variable with pdf $f$,

$$\lim_{p \to \infty} \|f\|_{L^p} = \text{esssup} \, f = \text{Mode}(X)$$

while

$$\|f\|_{L^1} = \mathbb E[X].$$

So going from $L^1$ to $L^\infty$ “interpolates” between the mean being most relevant to the mode being most relevant. There may be implications for the infinite dimensional case if one can find a suitable proxy for $L^p$ norms.

Perhaps somewhat less obvious is the relevance to exponential integrals via the so called Laplace principle - in expressions like

$$\int e^{\theta f(x)} \, dx,$$

the mode dominates for large values of $\theta > 0$, in the sense that

$$\lim_{\theta \to \infty} \frac{1}{\theta} \log \int e^{\theta f(x)} = \text{esssup} \, f.$$

Such integrals appear in change of measure/variational formulae in large deviations theory. This second observation generalises naturally to the infinite dimesional case as well.