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Background: Recall that a Langevin motion on a Riemannian manifold $(M, g)$ in $\mathbb{R}^D$ can be written down as the solution to the SDE in a local chart $U\subset \mathbb{R}^d$ (open) $$dY_t = [\nabla_g \cdot g^{-1}(Y_t) -\nabla_g V(Y_t)]dt+\sqrt{2g^{-1}(Y_t)}dB_t.$$ Here $V:U\to \mathbb{R}$ is the potential function, a scalar, $g$ is the metric tensor, $\nabla_g f$ for scalar $f$ is the manifold gradient, and $\nabla_g \cdot\vec{f}$ for vector fields $\vec{f}$ is the manifold-divergence. Precisely, $\nabla_g f := g^{-1} \nabla f$ and $\nabla_g \cdot \vec{f} = (\det g)^{-1/2} \nabla \cdot (\sqrt{\det g} \vec{f})$. We extend the manifold divergence to matrix fields by applying it row-wise.

It is well known that $Y$ has steady-state density $$p_\infty(y) \propto e^{-V(y)}\sqrt{\det g(y)},$$ i.e. its steady-state measure has Radon-Nikodym derivative $e^{-U(y)}$ with respect to the volume measure $\nu(dy) = \sqrt{\det g(y)}dy$ on $M$.

Now consider the scenario where either the potential or the geometry, or both, introduces some parameters $\theta$ (could be a vector or scalar). In that case, we might clarify our notation so that $$p_\infty(y;\theta) \propto e^{-V(y;\theta)} \sqrt{\det g(y;\theta)}$$ For each $\theta$ we must assume $V(\cdot ;\theta)$ is a potential, and $g(\cdot;\theta)$ is a metric tensor, etc. Now, we have a parametric distribution. Provided this varies smoothly over $\theta$, we can compute the Fisher information metric

$$I(\theta)_{ij} =-\mathbb{E}_{Y_\infty}(\partial_{\theta_i \theta_j} \log p(Y_\infty;\theta))$$ Let $\Theta$ be the parameter space. Now, we potentially have another Riemannian manifold, a statistical manifold $(\Theta, I)$, and we can play the same sort of game and write down a Langevin motion in the parameter space. An interesting potential is to use the negative log likelihood associated to $p_\infty$, i.e. $$\tilde{V}(\theta) := -\log L(y_1,\dotsc, y_n;\theta),$$ where $L(y_1,\dotsc, y_n;\theta) = \prod_{i=1}^n p_\infty(y_i;\theta)=e^{-\sum_{i=1}^n V(y_i)} \prod_{i=1}^n \sqrt{\det g(y_i)},$ where $Y_i=y_i$ are the observed variates of IID $Y_i\sim Y_\infty$. Let $I_n(\theta)=nI(\theta)$. Then the Langevin motion in the statistical space given by $$d\theta_t = [\nabla_{I_n} I_n^{-1}(\theta_t)-\nabla_{I_n} \tilde{V}(\theta_t)]dt+\sqrt{2I_n^{-1}(\theta_t)}dW_t$$ has steady-state $$f_\infty(\theta) \propto e^{-\tilde{V}(\theta)} \sqrt{\det I_n(\theta)}$$ $$=L(x_1,\dotsc, x_n;\theta) J(\theta)$$ where $J(\theta) = \sqrt{\det I_n(\theta)}$ is Jefferey's prior. But we recognize this from Bayesian statistics! This is just saying that $f_\infty(\theta)=f_\infty(\theta|x_1,\dotsc, x_n)$ is the posterior distribution of $\theta$ given $X_1,\dotsc, X_n$ using Jefferey's prior.

Question: I am interested in what is known about the pair $Z_t=(\theta_t, Y_t)$, i.e. solutions to the system

$$d\theta_t = [\nabla_{I_n} I_n^{-1}(\theta_t)-\nabla_{I_n} \tilde{V}(\theta_t)]dt+\sqrt{2I_n^{-1}(\theta_t)}dW_t$$

$$dY_t = [\nabla_g \cdot g^{-1}(Y_t;\theta) -\nabla_g V(Y_t;\theta)]dt+\sqrt{2g^{-1}(Y_t;\theta)}dB_t,$$ where $B$ and $W$ are independent Brownian motions. In particular, what is the steady state distribution of $Z_t$? Is there anything interesting we can say about the relationship between the "physical" motion $Y_t$ on $(M,g)$ and the "statistical" motion of $\theta_t$ on $(\Theta, I_n)$? Note if $\theta$ is a parameter of $g$ and not just the potential, then the above SDE system has a motion $Y$ on a random manifold!

(I apologize if this is too open ended and if anyone has suggestions to narrow it, I will be open to them.)

Case when only the potential is parametric: If only the potential $V$ depends on a parameter, then we can easily derive two facts. Let $Z(\theta)$ be the normalization "constant" for $p_\infty$, or as they say in statistical mechanics, the partition function: $Z(\theta) = \int_U e^{-V(y;\theta)} \sqrt{\det g(y)} dy$. Then,

  1. $-\frac{\partial \log Z}{\partial \theta_i} = \mathbb{E}\left(\frac{\partial V}{\partial \theta_i}(Y_\infty;\theta)\right)$
  2. $I(\theta)_{ij} = \operatorname{Cov}\left(\frac{\partial V}{\partial \theta_i}(Y_\infty;\theta), \frac{\partial V}{\partial \theta_j}(Y_\infty;\theta)\right)$ which matches our intuition that the Fisher information should only depend on the potential and not the metric tensor $g$, in this case.

It is hard to resist hoping that this special case might be useful or have another interesting things to say about it...

A concrete example: To give some intuition here is a simple example. Let $Y$ be a Langevin motoion with harmonic potential on Euclidean space with a diffusion scale $1/\beta$: $$dY_t = -k Y_t dt + \sqrt{2/\beta_t}dB_t$$ and $\beta_t$ solves the SDE $$d\beta_t = [2n^{-1}\beta_t+\beta_t(1-\beta_t k \overline{y^2})]dt+2n^{-1/2}\beta_t dW_t$$ where $\overline{y^2}:=\frac{1}{n}\sum_{i=1}^n y_i^2$ where $Y_i=y_i$ and $Y_i\sim Y_\infty$ IID. Note that given $\beta_t=\beta$ $Y_\infty \sim \mathcal{N}(0, (k\beta)^{-1})$ and computing the Fisher information, log-likelihood of this and all of the above necessary calculus will yield the SDE for $\beta_t$. Here is a plot the pair $(y_t, \beta_t)$ of the system when $\tilde{V}=0$ (first row) and when $\tilde{V}$ is the negative-log likelihood (second row).

The plots show the steady-state is still approached even for the Brownian motion on the statistical space ($\tilde{V}=0$).

Summary Question: Is there any known literature studying pairs of "physical" SDEs and "statistical" SDEs like this?

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