My question seems a bit large, but is actually pointing to something specific. I am talking about a connection between Differential/Algebraic topology and bayesian statistic. I come from statistics, and am working on estimating the parameters for a differential equation model. I am trying to understand whether the parameter space for the model is actually a stratified space--in the topological sense of a stratified space composed from different component manifolds. The intent of this question is to understand ways to sample from this stratified space more efficiently.
The parameters of my differential equation model are usually presented in a vector of $\mathbb{R}^n$, where each element represents a candidate value for that parameter. However, this parameter vector has a more complex structure. Some of the parameters represent transition probabilities between different groups. That means that the sum of the probabilities of starting from one group and moving to any of the other groups must be 1.0. Other elements in the vector refer to parameters that must be positive because they represent distances. Other elements might need to be integer values. So the parameter vector glues together draws from the prior distribution, just as in topology a stratified space glues together different manifolds.
The challenge in bayesian statistics is to estimate the probability of different vectors of parameters on this stratified space. The most common way to do this over the past 70 years was something like MCMC sampling. So we might break up the different sections of the parameter vector, draw random samples, and compute the likelihood of the entire vector. This method works--albeit slowly--and allows the users to sample from different chunks of the parameter space independently--based upon the definition of the prior distributions. A more modern way to solve this problem is variational inference where we create a mixture of usually guassian distributions and then adjust the parameters of the mixture to best approximate the joint distribution of the data.
I was wondering if this perspective of the parameter vector as a stratified space allows me some better options for estimating the probabilities of parameter vectors on that space? The usual idea in numerical methods is that the more structure we incorporate into our algorithms, the faster that we can compute something. So I was wondering if there is any benefit to say defining "flows" on the component manifolds in the stratified space to potentially estimate some of those probabilities faster or reduce the rejection rate for candidate parameter values, etc. There could be other approaches, though flows come to mind as one approach. Alternatively, I imagine not sampling in regions where there are "holes" in the manifold would reduce the rejection rate of sampling based methods.
I have not found any research on this question, but my lack of topology knowledge may mean I am looking in the wrong places. Hence I thought I would see if there is any merit in this idea or if anyone knows if people have looked at these types of topological ideas in statistics?