Appropriate prior for Bayesian inference of classical fields given data

Suppose I want to infer the configuration of a function $q(t)$ with $t in Omega$, where $Omega subset mathbb{R}$, given a dataset $d=[{t_1, q_1 }, …, { t_N,q_N }]$, where $q_i equiv q(t_i) $. Suppose further I know the dynamics of the field, for instance $q(t)$ is assumed to obey the equation of the anharmonic oscillator

$$left( frac{d^2}{dt^2} + omega^2right)q(t)=g q^3(t),$$

deriving from a Hamiltonian

$$H(p,q) = frac{p^2}{2}+omega^2frac{q^2}{2}+frac{g}{4}q^4.$$

First, note that in this case I can just use two data points and solve the equation, but that’s just peculiar to the dimensionality of the problem (if this was a field theory and I just had scattered data, that wouldn’t work).

The natural way to do this statistically would be to use Bayes’ theorem

$$mathcal{P}[q(t) | d] = frac{mathcal{P}[d | q(t)] mathcal{P}[q(t)]}{Z_d},$$

where

$$Z_d=int mathcal{D}q(t)mathcal{P}[d | q(t)].$$

Here we need to make certain assumptions about the prior probability $mathcal{P}[q(t)]$, and this is where the knowledge about the field dynamics fits in. The question is: what is the appropriate prior that captures our knowledge of the field dynamics?

One idea might come from the following: rewrite the numerator of the left hand side as

$$mathcal{P}[d | q(t)] mathcal{P}[q(t)]=e^{-mathcal{H}[q(t),d]}$$

where the Hamiltonian $mathcal{H}[q(t),d]$ is defined as $mathcal{H}[q(t),d]= – log (mathcal{P}[d | q(t)])$ and is hence broken into two additive factors, $mathcal{H}[d|q(t)]$ and $mathcal{H}[q(t)]$.