Rediscovering a function from samples

A friend shared this nice problem with me. Suppose you have a fixed function, \(f\), and a family of probability distributions, defined by, say, prob. density functions, \(P_t\), \(t \in A\). If we know \(E_t=E_{P_t}[f]\) for every \(t\in A\), can we recover \(f\)?

Clearly the answer depends on both how rigid \(f\) is and the family \(P_t\). We can cast this problem as a functional analysis problem by defining \(k(x,t)=P_t(x)\) to be a kernel and the expectation to be an integral transform. Then the question becomes: is there an inverse kernel, say, \(k^{-1}(x,t)\), such that \(\int k^{-1}(x,t)E(t)dt=f(x)?\) When does that exist and when is it unique? Hints can be taken from the Laplace transform, i.e., \(P_t(x)\propto e^{-tx}\) - up to a normalizing constant this is the just the exponential distribution. In general, this can be a hard problem though.

Fredholm equations

If we know \(E(t)=E_{P_t}[f]\) for every \(t\in A\), can we recover \(f\)? Formally, we have the equation: \(E(t)=\int k(x,t)f(x)dx,\) with appropriate limits for the integral. This equation is called a “Fredholm Equation of the first kind” and is closely studied in functional analysis and signal processing.

Practical stuff

If we assume the existence of an inverse kernel, how can we approximate it? One idea — which is also kind of a standard approach — is to fix a set of orthonormal basis functions, describe everything in terms of them, and then resolve them to arrive to a linear algebra problem.