Scientists take a set of data points, say in ${\mathbb R}^2$, and, assuming that this data should fit a polynomial of degree $d$ (or an exponential, etc.), they estimate parameters.

I would think that a Bayesian update for probability distributions on a moduli space $M$ (say of degree $d$ polynomials in ${\mathbb R}^2$) would be the way to model this. Each time we are given a point $(x,y)$, we accordingly update our distribution on $M$. For example, if $M$ is the space of linear functions $y=mx+b$ then data points $(1,4), (2,5), (3,6)$ would increase the density of a distribution around the point classifying $y=x+3$.

**Question 1:** Does this perspective make any sense whatsoever? If so, is there a reference that might help mature my understanding?

I have questions about what happens when we change the model. Imagine a situation that we initially model as a degree $3$ curve in ${\mathbb R}^{\{x,y\}}$ and then realize, no, our situation is better understood as a degree $4$ curve in ${\mathbb R}^{\{x,y,z\}}$, divisible by $z$. The reason we had thought it was a degree $3$ curve in ${\mathbb R}^{\{x,y\}}$ is because we had been implicitly assuming $z=4.5$.

Suppose I want to change the model. Can I use maps of moduli spaces to make any guarantees about how our Bayesian reasoning in the new model aligns with our reasoning in the old model?

**Question 2** Can you prove or find a counterexample to the following: Given a map of moduli spaces $f:M_1\to M_2$, a distribution on $M_1$, and data in $M_1$, performing Bayesian update in $M_1$ and then pushing forward the resulting distribution to $M_2$ is the same as (commutes with) pushing the distribution and the data forward to $M_2$, and doing the Bayesian update there?