Does there exist a topology of the parameter space that is scanned during the fitting process?

In physics, an object in a spatial potential moves according to the topology of the potential, and by visually inspecting that topology and finding its global minimum, we can infer the steady-state position of the object.

When Stan searches the parameter space to find the best parameter value, is there an equivalent topology we can use to describe the parameter space? For example, say Stan is asked to fit for the parameter \sigma given data set \mathcal{D}, and it converges on \sigma=3. Is there a topology/manifold/function that we can look at that has a minimum at \sigma=3 and from that justify Stan converging on the value \sigma=3?

I’m not I quite understand what you mean by “the topology of the potential” but take a look at A Conceptual Introduction to Hamiltonian Monte Carlo, section 3 in particular.

What I mean by landscape of the parameter space is can I get something like this for it?


I guess what I’m wondering is if there’s a function of the parameter, f(\sigma), that I can extract from Stan, such that the global minimum of f occurs at \sigma_0 which best fits the data.

I’m asking this because I’m having an issue with convergence, wherein 3 out of my 4 fit chains converge to the totally wrong value. In my case, \sigma=8 is expected, and one chain gets it right, while the rest seem to be sampling a totally wrong region and not getting away from it.

I was wondering if there is a way to extract the landscape of the parameter space so I could visually inspect it to see why some chains are getting stuck. I’m assuming that when a chain gets stuck, it is because it believes that value fits the data well (note I have no priors specified, other than lower=0). To draw an analogy to the physical problem of the potential, I’m wondering if there are multiple minima in the parameter space that Stan gets stuck in. For example, given infinite time, will the three chains move toward the expected value of \sigma=8?

That would be lp__, the log posterior probability density. You can extract it from the fit object just like any parameter or generated quantity. Multiple minima can indeed be a problem.

I was thinking lp__ could be it, thank you!